Electronic Journal of Probability Articles (Project Euclid)
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The latest articles from Electronic Journal of Probability on Project Euclid, a site for mathematics and statistics resources.en-usCopyright 2016 Cornell University LibraryEuclid-L@cornell.edu (Project Euclid Team)Mon, 25 Jan 2016 16:14 ESTMon, 25 Jan 2016 16:14 ESThttp://projecteuclid.org/collection/euclid/images/logo_linking_100.gifProject Euclid
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Lévy Classes and Self-Normalization
http://projecteuclid.org/euclid.ejp/1453756464
<strong>Davar Khoshnevisan</strong>. <p><strong>Source: </strong>Electronic Journal of Probability, Volume 1, 18 pp..</p><p><strong>Abstract:</strong><br/>
We prove a Chung's law of the iterated logarithm for recurrent linear Markov processes. In order to attain this level of generality, our normalization is random. In particular, when the Markov process in question is a diffusion, we obtain the integral test corresponding to a law of the iterated logarithm due to Knight.
</p>projecteuclid.org/euclid.ejp/1453756464_20160125161428Mon, 25 Jan 2016 16:14 ESTMonotonous subsequences and the descent process of invariant random permutationshttps://projecteuclid.org/euclid.ejp/1543287754<strong>Mohamed Slim Kammoun</strong>. <p><strong>Source: </strong>Electronic Journal of Probability, Volume 23, 31 pp..</p><p><strong>Abstract:</strong><br/>
It is known from the work of Baik, Deift and Johansson [3] that we have Tracy-Widom fluctuations for the longest increasing subsequence of uniform permutations. In this paper, we prove that this result holds also in the case of the Ewens distribution and more generally for a class of random permutations with distribution invariant under conjugation. Moreover, we obtain the convergence of the first components of the associated Young tableaux to the Airy Ensemble as well as the global convergence to the Vershik-Kerov-Logan-Shepp shape. Using similar techniques, we also prove that the limiting descent process of a large class of random permutations is stationary, one-dependent and determinantal.
</p>projecteuclid.org/euclid.ejp/1543287754_20181221220705Fri, 21 Dec 2018 22:07 ESTExistence and uniqueness results for BSDE with jumps: the whole nine yardshttps://projecteuclid.org/euclid.ejp/1545102139<strong>Antonis Papapantoleon</strong>, <strong>Dylan Possamaï</strong>, <strong>Alexandros Saplaouras</strong>. <p><strong>Source: </strong>Electronic Journal of Probability, Volume 23, 68 pp..</p><p><strong>Abstract:</strong><br/>
This paper is devoted to obtaining a wellposedness result for multidimensional BSDEs with possibly unbounded random time horizon and driven by a general martingale in a ﬁltration only assumed to satisfy the usual hypotheses, i.e. the ﬁltration may be stochastically discontinuous. We show that for stochastic Lipschitz generators and unbounded, possibly inﬁnite, time horizon, these equations admit a unique solution in appropriately weighted spaces. Our result allows in particular to obtain a wellposedness result for BSDEs driven by discrete–time approximations of general martingales.
</p>projecteuclid.org/euclid.ejp/1545102139_20181221220705Fri, 21 Dec 2018 22:07 ESTDoubly Reflected BSDEs and $\mathcal{E} ^{{f}}$-Dynkin games: beyond the right-continuous casehttps://projecteuclid.org/euclid.ejp/1545102140<strong>Miryana Grigorova</strong>, <strong>Peter Imkeller</strong>, <strong>Youssef Ouknine</strong>, <strong>Marie-Claire Quenez</strong>. <p><strong>Source: </strong>Electronic Journal of Probability, Volume 23, 38 pp..</p><p><strong>Abstract:</strong><br/>
We formulate a notion of doubly reflected BSDE in the case where the barriers $\xi $ and $\zeta $ do not satisfy any regularity assumption and with general filtration. Under a technical assumption (a Mokobodzki-type condition), we show existence and uniqueness of the solution. In the case where $\xi $ is right upper-semicontinuous and $\zeta $ is right lower-semicontinuous, the solution is characterized in terms of the value of a corresponding $\boldsymbol{\mathcal {E}} ^f$-Dynkin game, i.e. a game problem over stopping times with (non-linear) $f$-expectation, where $f$ is the driver of the doubly reflected BSDE. In the general case where the barriers do not satisfy any regularity assumptions, the solution of the doubly reflected BSDE is related to the value of “an extension” of the previous non-linear game problem over a larger set of “stopping strategies” than the set of stopping times. This characterization is then used to establish a comparison result and a priori estimates with universal constants.
</p>projecteuclid.org/euclid.ejp/1545102140_20181221220705Fri, 21 Dec 2018 22:07 ESTA central limit theorem for the gossip processhttps://projecteuclid.org/euclid.ejp/1545102141<strong>A.D. Barbour</strong>, <strong>Adrian Röllin</strong>. <p><strong>Source: </strong>Electronic Journal of Probability, Volume 23, 37 pp..</p><p><strong>Abstract:</strong><br/>
The Aldous gossip process represents the dissemination of information in geographical space as a process of locally deterministic spread, augmented by random long range transmissions. Starting from a single initially informed individual, the proportion of individuals informed follows an almost deterministic path, but for a random time shift, caused by the stochastic behaviour in the very early stages of development. In this paper, it is shown that, even with the extra information available after a substantial development time, this broad description remains accurate to first order. However, the precision of the prediction is now much greater, and the random time shift is shown to have an approximately normal distribution, with mean and variance that can be computed from the current state of the process.
</p>projecteuclid.org/euclid.ejp/1545102141_20181221220705Fri, 21 Dec 2018 22:07 ESTThe law of a point process of Brownian excursions in a domain is determined by the law of its tracehttps://projecteuclid.org/euclid.ejp/1545210235<strong>Wei Qian</strong>, <strong>Wendelin Werner</strong>. <p><strong>Source: </strong>Electronic Journal of Probability, Volume 23, 23 pp..</p><p><strong>Abstract:</strong><br/>
We show the result that is stated in the title of the paper, which has consequences about decomposition of Brownian loop-soup clusters in two dimensions.
</p>projecteuclid.org/euclid.ejp/1545210235_20181221220705Fri, 21 Dec 2018 22:07 ESTThe convex hull of a planar random walk: perimeter, diameter, and shapehttps://projecteuclid.org/euclid.ejp/1545447916<strong>James McRedmond</strong>, <strong>Andrew R. Wade</strong>. <p><strong>Source: </strong>Electronic Journal of Probability, Volume 23, 24 pp..</p><p><strong>Abstract:</strong><br/>
We study the convex hull of the first $n$ steps of a planar random walk, and present large-$n$ asymptotic results on its perimeter length $L_n$, diameter $D_n$, and shape. In the case where the walk has a non-zero mean drift, we show that $L_n / D_n \to 2$ a.s., and give distributional limit theorems and variance asymptotics for $D_n$, and in the zero-drift case we show that the convex hull is infinitely often arbitrarily well-approximated in shape by any unit-diameter compact convex set containing the origin, and then $\liminf _{n \to \infty } L_n/D_n =2$ and $\limsup _{n \to \infty } L_n /D_n = \pi $, a.s. Among the tools that we use is a zero-one law for convex hulls of random walks.
</p>projecteuclid.org/euclid.ejp/1545447916_20181221220705Fri, 21 Dec 2018 22:07 ESTStrong solutions of mean-field stochastic differential equations with irregular drifthttps://projecteuclid.org/euclid.ejp/1545447917<strong>Martin Bauer</strong>, <strong>Thilo Meyer-Brandis</strong>, <strong>Frank Proske</strong>. <p><strong>Source: </strong>Electronic Journal of Probability, Volume 23, 35 pp..</p><p><strong>Abstract:</strong><br/>
We investigate existence and uniqueness of strong solutions of mean-field stochastic differential equations with irregular drift coefficients. Our direct construction of strong solutions is mainly based on a compactness criterion employing Malliavin Calculus together with some local time calculus. Furthermore, we establish regularity properties of the solutions such as Malliavin differentiablility as well as Sobolev differentiability and Hölder continuity in the initial condition. Using this properties we formulate an extension of the Bismut-Elworthy-Li formula to mean-field stochastic differential equations to get a probabilistic representation of the first order derivative of an expectation functional with respect to the initial condition.
</p>projecteuclid.org/euclid.ejp/1545447917_20181221220705Fri, 21 Dec 2018 22:07 ESTFront evolution of the Fredrickson-Andersen one spin facilitated modelhttps://projecteuclid.org/euclid.ejp/1546571126<strong>Oriane Blondel</strong>, <strong>Aurelia Deshayes</strong>, <strong>Cristina Toninelli</strong>. <p><strong>Source: </strong>Electronic Journal of Probability, Volume 24, 32 pp..</p><p><strong>Abstract:</strong><br/>
The Fredrickson-Andersen one spin facilitated model (FA-1f) on $\mathbb Z$ belongs to the class of kinetically constrained spin models (KCM). Each site refreshes with rate one its occupation variable to empty (respectively occupied) with probability $q$ (respectively $p=1-q$), provided at least one nearest neighbor is empty. Here, we study the non equilibrium dynamics of FA-1f started from a conﬁguration entirely occupied on the left half-line and focus on the evolution of the front, namely the position of the leftmost zero. We prove, for $q$ larger than a threshold $\bar q<1$, a law of large numbers and a central limit theorem for the front, as well as the convergence to an invariant measure of the law of the process seen from the front.
</p>projecteuclid.org/euclid.ejp/1546571126_20190103220527Thu, 03 Jan 2019 22:05 ESTHeavy subtrees of Galton-Watson trees with an application to Apollonian networkshttps://projecteuclid.org/euclid.ejp/1549357219<strong>Luc Devroye</strong>, <strong>Cecilia Holmgren</strong>, <strong>Henning Sulzbach</strong>. <p><strong>Source: </strong>Electronic Journal of Probability, Volume 24, 44 pp..</p><p><strong>Abstract:</strong><br/>
We study heavy subtrees of conditional Galton-Watson trees. In a standard Galton-Watson tree conditional on its size being $n$, we order all children by their subtree sizes, from large (heavy) to small. A node is marked if it is among the $k$ heaviest nodes among its siblings. Unmarked nodes and their subtrees are removed, leaving only a tree of marked nodes, which we call the $k$-heavy tree. We study various properties of these trees, including their size and the maximal distance from any original node to the $k$-heavy tree. In particular, under some moment condition, the $2$-heavy tree is with high probability larger than $cn$ for some constant $c > 0$, and the maximal distance from the $k$-heavy tree is $O(n^{1/(k+1)})$ in probability. As a consequence, for uniformly random Apollonian networks of size $n$, the expected size of the longest simple path is $\Omega (n)$. We also show that the length of the heavy path (that is, $k=1$) converges (after rescaling) to the corresponding object in Aldous’ Brownian continuum random tree.
</p>projecteuclid.org/euclid.ejp/1549357219_20190205040031Tue, 05 Feb 2019 04:00 ESTDifferentiability of SDEs with drifts of super-linear growthhttps://projecteuclid.org/euclid.ejp/1549616424<strong>Peter Imkeller</strong>, <strong>Gonçalo dos Reis</strong>, <strong>William Salkeld</strong>. <p><strong>Source: </strong>Electronic Journal of Probability, Volume 24, 43 pp..</p><p><strong>Abstract:</strong><br/>
We close an unexpected gap in the literature of Stochastic Differential Equations (SDEs) with drifts of super linear growth and with random coefficients, namely, we prove Malliavin and Parametric Differentiability of such SDEs. The former is shown by proving Stochastic Gâteaux Differentiability and Ray Absolute Continuity. This method enables one to take limits in probability rather than mean square or almost surely bypassing the potentially non-integrable error terms from the unbounded drift. This issue is strongly linked with the difficulties of the standard methodology of [13, Lemma 1.2.3] for this setting. Several examples illustrating the range and scope of our results are presented.
We close with parametric differentiability and recover representations linking both derivatives as well as a Bismut-Elworthy-Li formula.
</p>projecteuclid.org/euclid.ejp/1549616424_20190208040027Fri, 08 Feb 2019 04:00 ESTA stability approach for solving multidimensional quadratic BSDEshttps://projecteuclid.org/euclid.ejp/1549616425<strong>Jonathan Harter</strong>, <strong>Adrien Richou</strong>. <p><strong>Source: </strong>Electronic Journal of Probability, Volume 24, 51 pp..</p><p><strong>Abstract:</strong><br/>
We establish an existence and uniqueness result for a class of multidimensional quadratic backward stochastic differential equations (BSDE). This class is characterized by constraints on some uniform a priori estimate on solutions of a sequence of approximated BSDEs. We also present effective examples of applications. Our approach relies on the strategy developed by Briand and Elie in [Stochastic Process. Appl. 123 2921–2939] concerning scalar quadratic BSDEs.
</p>projecteuclid.org/euclid.ejp/1549616425_20190208040027Fri, 08 Feb 2019 04:00 ESTQuantitative CLTs for symmetric $U$-statistics using contractionshttps://projecteuclid.org/euclid.ejp/1549681361<strong>Christian Döbler</strong>, <strong>Giovanni Peccati</strong>. <p><strong>Source: </strong>Electronic Journal of Probability, Volume 24, 43 pp..</p><p><strong>Abstract:</strong><br/>
We consider sequences of symmetric $U$-statistics, not necessarily Hoeffding-degenerate, both in a one- and multi-dimensional setting, and prove quantitative central limit theorems (CLTs) based on the use of contraction operators . Our results represent an explicit counterpart to analogous criteria that are available for sequences of random variables living on the Gaussian, Poisson or Rademacher chaoses, and are perfectly tailored for geometric applications. As a demonstration of this fact, we develop explicit bounds for subgraph counting in generalised random graphs on Euclidean spaces; special attention is devoted to the so-called ‘dense parameter regime’ for uniformly distributed points, for which we deduce CLTs that are new even in their qualitative statement, and that substantially extend classical findings by Jammalamadaka and Janson (1986) and Bhattacharaya and Ghosh (1992).
</p>projecteuclid.org/euclid.ejp/1549681361_20190208220246Fri, 08 Feb 2019 22:02 ESTBehavior of the empirical Wasserstein distance in ${\mathbb R}^d$ under moment conditionshttps://projecteuclid.org/euclid.ejp/1550113245<strong>Jérôme Dedecker</strong>, <strong>Florence Merlevède</strong>. <p><strong>Source: </strong>Electronic Journal of Probability, Volume 24, 32 pp..</p><p><strong>Abstract:</strong><br/>
We establish some deviation inequalities, moment bounds and almost sure results for the Wasserstein distance of order $p\in [1, \infty )$ between the empirical measure of independent and identically distributed ${\mathbb R}^d$-valued random variables and the common distribution of the variables. We only assume the existence of a (strong or weak) moment of order $rp$ for some $r>1$, and we discuss the optimality of the bounds.
</p>projecteuclid.org/euclid.ejp/1550113245_20190213220052Wed, 13 Feb 2019 22:00 ESTProfile of a self-similar growth-fragmentationhttps://projecteuclid.org/euclid.ejp/1550199785<strong>François Gaston Ged</strong>. <p><strong>Source: </strong>Electronic Journal of Probability, Volume 24, 21 pp..</p><p><strong>Abstract:</strong><br/>
A self-similar growth-fragmentation describes the evolution of particles that grow and split as time passes. Its genealogy yields a self-similar continuum tree endowed with an intrinsic measure. Extending results of Haas [13] for pure fragmentations, we relate the existence of an absolutely continuous profile to a simple condition in terms of the index of self-similarity and the so-called cumulant of the growth-fragmentation. When absolutely continuous, we approximate the profile by a function of the small fragments, and compute the Hausdorff dimension in the singular case. Applications to Boltzmann random planar maps are emphasized, exploiting recently established connections between growth-fragmentations and random maps [1, 5].
</p>projecteuclid.org/euclid.ejp/1550199785_20190214220311Thu, 14 Feb 2019 22:03 ESTSpectral conditions for equivalence of Gaussian random fields with stationary incrementshttps://projecteuclid.org/euclid.ejp/1550199786<strong>Abolfazl Safikhani</strong>, <strong>Yimin Xiao</strong>. <p><strong>Source: </strong>Electronic Journal of Probability, Volume 24, 19 pp..</p><p><strong>Abstract:</strong><br/>
This paper studies the problem of equivalence of Gaussian measures induced by Gaussian random fields (GRFs) with stationary increments and proves a sufficient condition for the equivalence in terms of the behavior of the spectral measures at infinity. The main results extend those of Stein (2004), Van Zanten (2007, 2008) and are applicable to a rich family of nonstationary space-time models with possible anisotropy behavior.
</p>projecteuclid.org/euclid.ejp/1550199786_20190214220311Thu, 14 Feb 2019 22:03 ESTUniversality of the least singular value for sparse random matriceshttps://projecteuclid.org/euclid.ejp/1550221265<strong>Ziliang Che</strong>, <strong>Patrick Lopatto</strong>. <p><strong>Source: </strong>Electronic Journal of Probability, Volume 24, 53 pp..</p><p><strong>Abstract:</strong><br/>
We study the distribution of the least singular value associated to an ensemble of sparse random matrices. Our motivating example is the ensemble of $N\times N$ matrices whose entries are chosen independently from a Bernoulli distribution with parameter $p$. These matrices represent the adjacency matrices of random Erdős–Rényi digraphs and are sparse when $p\ll 1$. We prove that in the regime $pN\gg 1$, the distribution of the least singular value is universal in the sense that it is independent of $p$ and equal to the distribution of the least singular value of a Gaussian matrix ensemble. We also prove the universality of the joint distribution of multiple small singular values. Our methods extend to matrix ensembles whose entries are chosen from arbitrary distributions that may be correlated, complex valued, and have unequal variances.
</p>projecteuclid.org/euclid.ejp/1550221265_20190215040117Fri, 15 Feb 2019 04:01 ESTConvergence of the empirical spectral distribution of Gaussian matrix-valued processeshttps://projecteuclid.org/euclid.ejp/1550286034<strong>Arturo Jaramillo</strong>, <strong>Juan Carlos Pardo</strong>, <strong>José Luis Pérez</strong>. <p><strong>Source: </strong>Electronic Journal of Probability, Volume 24, 22 pp..</p><p><strong>Abstract:</strong><br/>
For a given normalized Gaussian symmetric matrix-valued process $Y^{(n)}$, we consider the process of its eigenvalues $\{(\lambda _{1}^{(n)}(t),\dots , \lambda _{n}^{(n)}(t)); t\ge 0\}$ as well as its corresponding process of empirical spectral measures $\mu ^{(n)}=(\mu _{t}^{(n)}; t\geq 0)$. Under some mild conditions on the covariance function associated to $Y^{(n)}$, we prove that the process $\mu ^{(n)}$ converges in probability to a deterministic limit $\mu $, in the topology of uniform convergence over compact sets. We show that the process $\mu $ is characterized by its Cauchy transform, which is a rescaling of the solution of a Burgers’ equation. Our results extend those of Rogers and Shi [14] for the free Brownian motion and Pardo et al. [12] for the non-commutative fractional Brownian motion when $H>1/2$ whose arguments use strongly the non-collision of the eigenvalues. Our methodology does not require the latter property and in particular explains the remaining case of the non-commutative fractional Brownian motion for $H< 1/2$ which, up to our knowledge, was unknown.
</p>projecteuclid.org/euclid.ejp/1550286034_20190215220041Fri, 15 Feb 2019 22:00 ESTSmall-time fluctuations for the bridge in a model class of hypoelliptic diffusions of weak Hörmander typehttps://projecteuclid.org/euclid.ejp/1550480425<strong>Karen Habermann</strong>. <p><strong>Source: </strong>Electronic Journal of Probability, Volume 24, 19 pp..</p><p><strong>Abstract:</strong><br/>
We study the small-time asymptotics for hypoelliptic diffusion processes conditioned by their initial and final positions, in a model class of diffusions satisfying a weak Hörmander condition where the diffusivity is constant and the drift is linear. We show that, while the diffusion bridge can exhibit a blow-up behaviour in the small time limit, we can still make sense of suitably rescaled fluctuations which converge weakly. We explicitly describe the limit fluctuation process in terms of quantities associated to the unconditioned diffusion. In the discussion of examples, we also find an expression for the bridge from $0$ to $0$ in time $1$ of an iterated Kolmogorov diffusion.
</p>projecteuclid.org/euclid.ejp/1550480425_20190218040039Mon, 18 Feb 2019 04:00 ESTNon-asymptotic error bounds for the multilevel Monte Carlo Euler method applied to SDEs with constant diffusion coefficienthttps://projecteuclid.org/euclid.ejp/1550653271<strong>Benjamin Jourdain</strong>, <strong>Ahmed Kebaier</strong>. <p><strong>Source: </strong>Electronic Journal of Probability, Volume 24, 34 pp..</p><p><strong>Abstract:</strong><br/>
In this paper, we are interested in deriving non-asymptotic error bounds for the multilevel Monte Carlo method. As a first step, we deal with the explicit Euler discretization of stochastic differential equations with a constant diffusion coefficient. We prove that, as long as the deviation is below an explicit threshold, a Gaussian-type concentration inequality optimal in terms of the variance holds for the multilevel estimator. To do so, we use the Clark-Ocone representation formula and derive bounds for the moment generating functions of the squared difference between a crude Euler scheme and a finer one and of the squared difference of their Malliavin derivatives.
</p>projecteuclid.org/euclid.ejp/1550653271_20190220040137Wed, 20 Feb 2019 04:01 ESTNon asymptotic variance bounds and deviation inequalities by optimal transporthttps://projecteuclid.org/euclid.ejp/1550653272<strong>Kevin Tanguy</strong>. <p><strong>Source: </strong>Electronic Journal of Probability, Volume 24, 18 pp..</p><p><strong>Abstract:</strong><br/>
The purpose of this note is to show how simple Optimal Transport arguments, on the real line, can be used in Superconcentration theory. This methodology is efficient to produce sharp non-asymptotic variance bounds for various functionals (maximum, median, $l^p$ norms) of standard Gaussian random vectors in $\mathbb{R} ^n$. The flexibility of this approach can also provide exponential deviation inequalities reflecting preceding variance bounds. As a further illustration, usual laws from Extreme theory and Coulomb gases are studied.
</p>projecteuclid.org/euclid.ejp/1550653272_20190220040137Wed, 20 Feb 2019 04:01 ESTCan the stochastic wave equation with strong drift hit zero?https://projecteuclid.org/euclid.ejp/1550653273<strong>Kevin Lin</strong>, <strong>Carl Mueller</strong>. <p><strong>Source: </strong>Electronic Journal of Probability, Volume 24, 26 pp..</p><p><strong>Abstract:</strong><br/>
We study the stochastic wave equation with multiplicative noise and singular drift:
\[\partial _{t}u(t,x)=\Delta u(t,x)+u^{-\alpha }(t,x)+g(u(t,x))\dot{W} (t,x)\]
where $x$ lies in the circle $\mathbf{R} /J\mathbf{Z} $ and $u(0,x)>0$. We show that
(i) If $0<\alpha <1$ then with positive probability, $u(t,x)=0$ for some $(t,x)$.
(ii) If $\alpha >3$ then with probability one, $u(t,x)\ne 0$ for all $(t,x)$.
</p>projecteuclid.org/euclid.ejp/1550653273_20190220040137Wed, 20 Feb 2019 04:01 ESTAsymptotic properties of expansive Galton-Watson treeshttps://projecteuclid.org/euclid.ejp/1550826098<strong>Romain Abraham</strong>, <strong>Jean-François Delmas</strong>. <p><strong>Source: </strong>Electronic Journal of Probability, Volume 24, 51 pp..</p><p><strong>Abstract:</strong><br/>
We consider a super-critical Galton-Watson tree $\tau $ whose non-degenerate offspring distribution has finite mean. We consider the random trees $\tau _n$ distributed as $\tau $ conditioned on the $n$-th generation, $Z_n$, to be of size $a_n\in{\mathbb N} $. We identify the possible local limits of $\tau _n$ as $n$ goes to infinity according to the growth rate of $a_n$. In the low regime, the local limit $\tau ^0$ is the Kesten tree, in the moderate regime the family of local limits, $\tau ^\theta $ for $\theta \in (0, +\infty )$, is distributed as $\tau $ conditionally on $\{W=\theta \}$, where $W$ is the (non-trivial) limit of the renormalization of $Z_n$. In the high regime, we prove the local convergence towards $\tau ^\infty $ in the Harris case (finite support of the offspring distribution) and we give a conjecture for the possible limit when the offspring distribution has some exponential moments. When the offspring distribution has a fat tail, the problem is open. The proof relies on the strong ratio theorem for Galton-Watson processes. Those latter results are new in the low regime and high regime, and they can be used to complete the description of the (space-time) Martin boundary of Galton-Watson processes. Eventually, we consider the continuity in distribution of the local limits $(\tau ^\theta , \theta \in [0, \infty ])$.
</p>projecteuclid.org/euclid.ejp/1550826098_20190222040150Fri, 22 Feb 2019 04:01 ESTExceedingly large deviations of the totally asymmetric exclusion processhttps://projecteuclid.org/euclid.ejp/1550826099<strong>Stefano Olla</strong>, <strong>Li-Cheng Tsai</strong>. <p><strong>Source: </strong>Electronic Journal of Probability, Volume 24, 71 pp..</p><p><strong>Abstract:</strong><br/>
Consider the Totally Asymmetric Simple Exclusion Process (TASEP) on the integer lattice $ \mathbb{Z} $. We study the functional Large Deviations of the integrated current $ \mathsf{h} (t,x) $ under the hyperbolic scaling of space and time by $ N $, i.e., $ \mathsf{h} _{N}(t,\xi ) := \frac{1} {N}\mathsf{h} (Nt,N\xi ) $. As hinted by the asymmetry in the upper- and lower-tail large deviations of the exponential Last Passage Percolation, the TASEP exhibits two types of deviations. One type of deviations occur with probability $ \exp (-O(N)) $, referred to as speed-$ N $; while the other with probability $ \exp (-O(N^{2})) $, referred to as speed-$ N^2 $. In this work we study the speed-$ N^2 $ functional Large Deviation Principle (LDP) of the TASEP, and establish (non-matching) large deviation upper and lower bounds.
</p>projecteuclid.org/euclid.ejp/1550826099_20190222040150Fri, 22 Feb 2019 04:01 ESTCramér’s estimate for stable processes with power drifthttps://projecteuclid.org/euclid.ejp/1551150461<strong>Christophe Profeta</strong>, <strong>Thomas Simon</strong>. <p><strong>Source: </strong>Electronic Journal of Probability, Volume 24, 21 pp..</p><p><strong>Abstract:</strong><br/>
We investigate the upper tail probabilities of the all-time maximum of a stable Lévy process with a power negative drift. The asymptotic behaviour is shown to be exponential in the spectrally negative case and polynomial otherwise, with explicit exponents and constants. Analogous results are obtained, at a less precise level, for the fractionally integrated stable Lévy process. We also study the lower tail probabilities of the integrated stable Lévy process in the presence of a power positive drift.
</p>projecteuclid.org/euclid.ejp/1551150461_20190225220749Mon, 25 Feb 2019 22:07 ESTFinding the seed of uniform attachment treeshttps://projecteuclid.org/euclid.ejp/1551323285<strong>Gábor Lugosi</strong>, <strong>Alan S. Pereira</strong>. <p><strong>Source: </strong>Electronic Journal of Probability, Volume 24, 15 pp..</p><p><strong>Abstract:</strong><br/>
A uniform attachment tree is a random tree that is generated dynamically. Starting from a fixed “seed” tree, vertices are added sequentially by attaching each vertex to an existing vertex chosen uniformly at random. Upon observing a large (unlabeled) tree, one wishes to find the initial seed. We investigate to what extent seed trees can be recovered, at least partially. We consider three types of seeds: a path, a star, and a random uniform attachment tree. We propose and analyze seed-finding algorithms for all three types of seed trees.
</p>projecteuclid.org/euclid.ejp/1551323285_20190227220813Wed, 27 Feb 2019 22:08 ESTScaling limits of population and evolution processes in random environmenthttps://projecteuclid.org/euclid.ejp/1552013626<strong>Vincent Bansaye</strong>, <strong>Maria-Emilia Caballero</strong>, <strong>Sylvie Méléard</strong>. <p><strong>Source: </strong>Electronic Journal of Probability, Volume 24, 38 pp..</p><p><strong>Abstract:</strong><br/>
We propose a general method for investigating scaling limits of finite dimensional Markov chains to diffusions with jumps. The results of tightness, identification and convergence in law are based on the convergence of suitable characteristics of the chain transition. We apply these results to population processes recursively defined as sums of independent random variables. Two main applications are developed. First, we extend the Wright-Fisher model to independent and identically distributed random environments and show its convergence, under a large population assumption, to a Wright-Fisher diffusion in random environment. Second, we obtain the convergence in law of generalized Galton-Watson processes with interaction in random environment to solutions of stochastic differential equations with jumps.
</p>projecteuclid.org/euclid.ejp/1552013626_20190307215413Thu, 07 Mar 2019 21:54 ESTDirected, cylindric and radial Brownian webshttps://projecteuclid.org/euclid.ejp/1553133829<strong>David Coupier</strong>, <strong>Jean-François Marckert</strong>, <strong>Viet Chi Tran</strong>. <p><strong>Source: </strong>Electronic Journal of Probability, Volume 24, 48 pp..</p><p><strong>Abstract:</strong><br/>
The Brownian web (BW) is a collection of coalescing Brownian paths $(W_{(x,t)},(x,t) \in \mathbb{R} ^2)$ indexed by the plane. It appears in particular as continuous limit of various discrete models of directed forests of coalescing random walks and navigation schemes. Radial counterparts have been considered but global invariance principles are hard to establish. In this paper, we consider cylindrical forests which in some sense interpolate between the directed and radial forests: we keep the topology of the plane while still taking into account the angular component. We define in this way the cylindric Brownian web (CBW), which is locally similar to the planar BW but has several important macroscopic differences. For example, in the CBW, the coalescence time between two paths admits exponential moments and the CBW as its dual contain each a.s. a unique bi-infinite path. This pair of bi-infinite paths is distributed as a pair of reflected Brownian motions on the cylinder. Projecting the CBW on the radial plane, we obtain a radial Brownian web (RBW), i.e. a family of coalescing paths where under a natural parametrization, the angular coordinate of a trajectory is a Brownian motion. Recasting some of the discrete radial forests of the literature on the cylinder, we propose rescalings of these forests that converge to the CBW, and deduce the global convergence of the corresponding rescaled radial forests to the RBW. In particular, a modification of the radial model proposed in Coletti and Valencia is shown to converge to the CBW.
</p>projecteuclid.org/euclid.ejp/1553133829_20190320220357Wed, 20 Mar 2019 22:03 EDTGlobal fluctuations for 1D log-gas dynamics. Covariance kernel and supporthttps://projecteuclid.org/euclid.ejp/1553155301<strong>Jeremie Unterberger</strong>. <p><strong>Source: </strong>Electronic Journal of Probability, Volume 24, 28 pp..</p><p><strong>Abstract:</strong><br/>
We consider the hydrodynamic limit in the macroscopic regime of the coupled system of stochastic differential equations,
\[ d\lambda _t^i=\frac{1} {\sqrt{N} } dW_t^i - V'(\lambda _t^i) dt+ \frac{\beta } {2N} \sum _{j\not =i} \frac{dt} {\lambda ^i_t-\lambda ^j_t}, \qquad i=1,\ldots ,N, \qquad \mbox{(0.1)} \]
with $\beta >1$, sometimes called generalized Dyson’s Brownian motion , describing the dissipative dynamics of a log-gas of $N$ equal charges with equilibrium measure corresponding to a $\beta $-ensemble, with sufficiently regular convex potential $V$. The limit $N\to \infty $ is known to satisfy a mean-field Mc Kean-Vlasov equation. Fluctuations around this limit have been shown [39] to define a Gaussian process solving some explicit martingale problem written in terms of a generalized transport equation.
We prove a series of results concerning either the Mc Kean-Vlasov equation for the density $\rho _t$, notably regularity results and time-evolution of the support, or the associated hydrodynamic fluctuation process, whose space-time covariance kernel we compute explicitly.
</p>projecteuclid.org/euclid.ejp/1553155301_20190321040150Thu, 21 Mar 2019 04:01 EDTMixing times for the simple exclusion process in ballistic random environmenthttps://projecteuclid.org/euclid.ejp/1553155302<strong>Dominik Schmid</strong>. <p><strong>Source: </strong>Electronic Journal of Probability, Volume 24, 25 pp..</p><p><strong>Abstract:</strong><br/>
We consider the exclusion process on segments of the integers in a site-dependent random environment. We assume to be in the ballistic regime in which a single particle has positive linear speed. Our goal is to study the mixing time of the exclusion process when the number of particles is linear in the size of the segment. We investigate the order of the mixing time depending on the support of the environment distribution. In particular, we prove for nestling environments that the order of the mixing time is different than in the case of a single particle.
</p>projecteuclid.org/euclid.ejp/1553155302_20190321040150Thu, 21 Mar 2019 04:01 EDTThe speed of critically biased random walk in a one-dimensional percolation modelhttps://projecteuclid.org/euclid.ejp/1553306439<strong>Jan-Erik Lübbers</strong>, <strong>Matthias Meiners</strong>. <p><strong>Source: </strong>Electronic Journal of Probability, Volume 24, 29 pp..</p><p><strong>Abstract:</strong><br/>
We consider biased random walks in a one-dimensional percolation model. This model goes back to Axelson-Fisk and Häggström and exhibits the same phase transition as biased random walk on the infinite cluster of supercritical Bernoulli bond percolation on $\mathbb{Z} ^d$, namely, for some critical value $\lambda _{\mathrm{c} }>0$ of the bias, it holds that the asymptotic linear speed $\overline{\mathrm {v}} $ of the walk is strictly positive if the bias $\lambda $ is strictly smaller than $\lambda _{\mathrm{c} }$, whereas $\overline{\mathrm {v}} =0$ if $\lambda \geq \lambda _{\mathrm{c} }$.
We show that at the critical bias $\lambda = \lambda _{\mathrm{c} }$, the displacement of the random walk from the origin is of order $n/\log n$. This is in accordance with simulation results by Dhar and Stauffer for biased random walk on the infinite cluster of supercritical bond percolation on $\mathbb{Z} ^d$.
Our result is based on fine estimates for the tails of suitable regeneration times. As a by-product of these estimates we also obtain the order of fluctuations of the walk in the sub-ballistic and in the ballistic, nondiffusive phase.
</p>projecteuclid.org/euclid.ejp/1553306439_20190322220110Fri, 22 Mar 2019 22:01 EDTSplitting tessellations in spherical spaceshttps://projecteuclid.org/euclid.ejp/1553565775<strong>Daniel Hug</strong>, <strong>Christoph Thäle</strong>. <p><strong>Source: </strong>Electronic Journal of Probability, Volume 24, 60 pp..</p><p><strong>Abstract:</strong><br/>
The concept of splitting tessellations and splitting tessellation processes in spherical spaces of dimension $d\geq 2$ is introduced. Expectations, variances and covariances of spherical curvature measures induced by a splitting tessellation are studied using tools from spherical integral geometry. Also the spherical pair-correlation function of the $(d-1)$-dimensional Hausdorff measure is computed explicitly and compared to its analogue for Poisson great hypersphere tessellations. Finally, the typical cell distribution and the distribution of the typical spherical maximal face of any dimension $k\in \{1,\ldots ,d-1\}$ are expressed as mixtures of the related distributions of Poisson great hypersphere tessellations. This in turn is used to determine the expected length and the precise birth time distribution of the typical maximal spherical segment of a splitting tessellation.
</p>projecteuclid.org/euclid.ejp/1553565775_20190325220308Mon, 25 Mar 2019 22:03 EDTSecond Errata to “Processes on Unimodular Random Networks”https://projecteuclid.org/euclid.ejp/1553565776<strong>David Aldous</strong>, <strong>Russell Lyons</strong>. <p><strong>Source: </strong>Electronic Journal of Probability, Volume 24, 2 pp..</p><p><strong>Abstract:</strong><br/>
We correct a few more minor errors in our paper, Electron. J. Probab. 12 , Paper 54 (2007), 1454–1508.
</p>projecteuclid.org/euclid.ejp/1553565776_20190325220308Mon, 25 Mar 2019 22:03 EDTQuantitative contraction rates for Markov chains on general state spaceshttps://projecteuclid.org/euclid.ejp/1553565777<strong>Andreas Eberle</strong>, <strong>Mateusz B. Majka</strong>. <p><strong>Source: </strong>Electronic Journal of Probability, Volume 24, 36 pp..</p><p><strong>Abstract:</strong><br/>
We investigate the problem of quantifying contraction coefficients of Markov transition kernels in Kantorovich ($L^1$ Wasserstein) distances. For diffusion processes, relatively precise quantitative bounds on contraction rates have recently been derived by combining appropriate couplings with carefully designed Kantorovich distances. In this paper, we partially carry over this approach from diffusions to Markov chains. We derive quantitative lower bounds on contraction rates for Markov chains on general state spaces that are powerful if the dynamics is dominated by small local moves. For Markov chains on $\mathbb R^d$ with isotropic transition kernels, the general bounds can be used efficiently together with a coupling that combines maximal and reflection coupling. The results are applied to Euler discretizations of stochastic differential equations with non-globally contractive drifts, and to the Metropolis adjusted Langevin algorithm for sampling from a class of probability measures on high dimensional state spaces that are not globally log-concave.
</p>projecteuclid.org/euclid.ejp/1553565777_20190325220308Mon, 25 Mar 2019 22:03 EDTMultivariate approximation in total variation using local dependencehttps://projecteuclid.org/euclid.ejp/1553565778<strong>A.D. Barbour</strong>, <strong>A. Xia</strong>. <p><strong>Source: </strong>Electronic Journal of Probability, Volume 24, 35 pp..</p><p><strong>Abstract:</strong><br/>
We establish two theorems for assessing the accuracy in total variation of multivariate discrete normal approximation to the distribution of an integer valued random vector $W$. The first is for sums of random vectors whose dependence structure is local. The second applies to random vectors $W$ resulting from integrating the ${\mathbb Z}^d$-valued marks of a marked point process with respect to its ground process. The error bounds are of magnitude comparable to those given in [Rinott & Rotar (1996)], but now with respect to the stronger total variation distance. Instead of requiring the summands to be bounded, we make third moment assumptions. We demonstrate the use of the theorems in four applications: monochrome edges in vertex coloured graphs, induced triangles and $2$-stars in random geometric graphs, the times spent in different states by an irreducible and aperiodic finite Markov chain, and the maximal points in different regions of a homogeneous Poisson point process.
</p>projecteuclid.org/euclid.ejp/1553565778_20190325220308Mon, 25 Mar 2019 22:03 EDTA random walk with catastropheshttps://projecteuclid.org/euclid.ejp/1553565779<strong>Iddo Ben-Ari</strong>, <strong>Alexander Roitershtein</strong>, <strong>Rinaldo B. Schinazi</strong>. <p><strong>Source: </strong>Electronic Journal of Probability, Volume 24, 21 pp..</p><p><strong>Abstract:</strong><br/>
Random population dynamics with catastrophes (events pertaining to possible elimination of a large portion of the population) has a long history in the mathematical literature. In this paper we study an ergodic model for random population dynamics with linear growth and binomial catastrophes: in a catastrophe, each individual survives with some fixed probability, independently of the rest. Through a coupling construction, we obtain sharp two-sided bounds for the rate of convergence to stationarity which are applied to show that the model exhibits a cutoff phenomenon.
</p>projecteuclid.org/euclid.ejp/1553565779_20190325220308Mon, 25 Mar 2019 22:03 EDTOn Stein’s method for multivariate self-decomposable laws with finite first momenthttps://projecteuclid.org/euclid.ejp/1553565780<strong>Benjamin Arras</strong>, <strong>Christian Houdré</strong>. <p><strong>Source: </strong>Electronic Journal of Probability, Volume 24, 33 pp..</p><p><strong>Abstract:</strong><br/>
We develop a multidimensional Stein methodology for non-degenerate self-decomposable random vectors in $\mathbb{R} ^d$ having ﬁnite ﬁrst moment. Building on previous univariate ﬁndings, we solve an integro-partial differential Stein equation by a mixture of semigroup and Fourier analytic methods. Then, under a second moment assumption, we introduce a notion of Stein kernel and an associated Stein discrepancy speciﬁcally designed for inﬁnitely divisible distributions. Combining these new tools, we obtain quantitative bounds on smooth-Wasserstein distances between a probability measure in $\mathbb{R} ^d$ and a non-degenerate self-decomposable target law with ﬁnite second moment. Finally, under an appropriate Poincaré-type inequality assumption, we investigate, via variational methods, the existence of Stein kernels. In particular, this leads to quantitative versions of classical results on characterizations of probability distributions by variational functionals.
</p>projecteuclid.org/euclid.ejp/1553565780_20190325220308Mon, 25 Mar 2019 22:03 EDTProbability measure-valued polynomial diffusionshttps://projecteuclid.org/euclid.ejp/1553565781<strong>Christa Cuchiero</strong>, <strong>Martin Larsson</strong>, <strong>Sara Svaluto-Ferro</strong>. <p><strong>Source: </strong>Electronic Journal of Probability, Volume 24, 32 pp..</p><p><strong>Abstract:</strong><br/>
We introduce a class of probability measure-valued diffusions, coined polynomial , of which the well-known Fleming–Viot process is a particular example. The defining property of finite dimensional polynomial processes considered in [8, 21] is transferred to this infinite dimensional setting. This leads to a representation of conditional marginal moments via a finite dimensional linear PDE, whose spatial dimension corresponds to the degree of the moment. As a result, the tractability of finite dimensional polynomial processes are preserved in this setting. We also obtain a representation of the corresponding extended generators, and prove well-posedness of the associated martingale problems. In particular, uniqueness is obtained from the duality relationship with the PDEs mentioned above.
</p>projecteuclid.org/euclid.ejp/1553565781_20190325220308Mon, 25 Mar 2019 22:03 EDTInvasion percolation on Galton-Watson treeshttps://projecteuclid.org/euclid.ejp/1554256913<strong>Marcus Michelen</strong>, <strong>Robin Pemantle</strong>, <strong>Josh Rosenberg</strong>. <p><strong>Source: </strong>Electronic Journal of Probability, Volume 24, 35 pp..</p><p><strong>Abstract:</strong><br/>
We consider invasion percolation on Galton-Watson trees. On almost every Galton-Watson tree, the invasion cluster almost surely contains only one infinite path. This means that for almost every Galton-Watson tree, invasion percolation induces a probability measure on infinite paths from the root. We show that under certain conditions of the progeny distribution, this measure is absolutely continuous with respect to the limit uniform measure. This confirms that invasion percolation, an efficient self-tuning algorithm, may be used to sample approximately from the limit uniform distribution. Additionally, we analyze the forward maximal weights along the backbone of the invasion cluster and prove a limit law for the process.
</p>projecteuclid.org/euclid.ejp/1554256913_20190402220219Tue, 02 Apr 2019 22:02 EDTRapid social connectivityhttps://projecteuclid.org/euclid.ejp/1554775411<strong>Itai Benjamini</strong>, <strong>Jonathan Hermon</strong>. <p><strong>Source: </strong>Electronic Journal of Probability, Volume 24, 33 pp..</p><p><strong>Abstract:</strong><br/>
Given a graph $G=(V,E)$, consider Poisson($|V|$) walkers performing independent lazy simple random walks on $G$ simultaneously, where the initial position of each walker is chosen independently with probability proportional to the degrees. When two walkers visit the same vertex at the same time they are declared to be acquainted. The social connectivity time $\mathrm{SC} (G)$ is defined as the first time in which there is a path of acquaintances between every pair of walkers. It is shown that when the average degree of $G$ is $d$, with high probability \[ c\log |V| \le \mathrm{SC} (G) \le C d^{1+5 \cdot 1_{G \text{ is not regular} } } \log ^3 |V|. \] When $G$ is regular the lower bound is improved to $\mathrm{SC} (G) \ge \log |V| -6 \log \log |V| $, with high probability. We determine $\mathrm{SC} (G)$ up to a constant factor in the cases that $G$ is an expander and when it is the $n$-cycle.
</p>projecteuclid.org/euclid.ejp/1554775411_20190408220351Mon, 08 Apr 2019 22:03 EDTContinuous-state branching processes with competition: duality and reflection at infinityhttps://projecteuclid.org/euclid.ejp/1554775412<strong>Clément Foucart</strong>. <p><strong>Source: </strong>Electronic Journal of Probability, Volume 24, 38 pp..</p><p><strong>Abstract:</strong><br/>
The boundary behavior of continuous-state branching processes with quadratic competition is studied in whole generality. We first observe that despite competition, explosion can occur for certain branching mechanisms. We obtain a necessary and sufficient condition for $\infty $ to be accessible in terms of the branching mechanism and the competition parameter $c>0$. We show that when $\infty $ is inaccessible, it is always an entrance boundary. In the case where $\infty $ is accessible, explosion can occur either by a single jump to $\infty $ (the process at $z$ jumps to $\infty $ at rate $\lambda z$ for some $\lambda >0$) or by accumulation of large jumps over finite intervals. We construct a natural extension of the minimal process and show that when $\infty $ is accessible and $0\leq \frac{2\lambda } {c}<1$, the extended process is reflected at $\infty $. In the case $\frac{2\lambda } {c}\geq 1$, $\infty $ is an exit of the extended process. When the branching mechanism is not the Laplace exponent of a subordinator, we show that the process with reflection at $\infty $ gets extinct almost surely. Moreover absorption at $0$ is almost sure if and only if Grey’s condition is satisfied. When the branching mechanism is the Laplace exponent of a subordinator, necessary and sufficient conditions are given for a stationary distribution to exist. The Laplace transform of the latter is provided. The study is based on classical time-change arguments and on a new duality method relating logistic CSBPs with certain generalized Feller diffusions.
</p>projecteuclid.org/euclid.ejp/1554775412_20190408220351Mon, 08 Apr 2019 22:03 EDTDistances between zeroes and critical points for random polynomials with i.i.d. zeroeshttps://projecteuclid.org/euclid.ejp/1554775413<strong>Zakhar Kabluchko</strong>, <strong>Hauke Seidel</strong>. <p><strong>Source: </strong>Electronic Journal of Probability, Volume 24, 25 pp..</p><p><strong>Abstract:</strong><br/>
Consider a random polynomial $Q_n$ of degree $n+1$ whose zeroes are i.i.d. random variables $\xi _0,\xi _1,\ldots ,\xi _n$ in the complex plane. We study the pairing between the zeroes of $Q_n$ and its critical points, i.e. the zeroes of its derivative $Q_n'$. In the asymptotic regime when $n\to \infty $, with high probability there is a critical point of $Q_n$ which is very close to $\xi _0$. We localize the position of this critical point by proving that the difference between $\xi _0$ and the critical point has approximately complex Gaussian distribution with mean $1/(nf(\xi _0))$ and variance of order $\log n \cdot n^{-3}$. Here, $f(z)= \mathbb E [\frac 1 {z-\xi _k}]$ is the Cauchy–Stieltjes transform of the $\xi _k$’s. We also state some conjectures on critical points of polynomials with dependent zeroes, for example the Weyl polynomials and characteristic polynomials of random matrices.
</p>projecteuclid.org/euclid.ejp/1554775413_20190408220351Mon, 08 Apr 2019 22:03 EDTWasserstein-2 bounds in normal approximation under local dependencehttps://projecteuclid.org/euclid.ejp/1554775414<strong>Xiao Fang</strong>. <p><strong>Source: </strong>Electronic Journal of Probability, Volume 24, 14 pp..</p><p><strong>Abstract:</strong><br/>
We obtain a general bound for the Wasserstein-2 distance in normal approximation for sums of locally dependent random variables. The proof is based on an asymptotic expansion for expectations of second-order differentiable functions of the sum. We apply the main result to obtain Wasserstein-2 bounds in normal approximation for sums of $m$-dependent random variables, U-statistics and subgraph counts in the Erdős-Rényi random graph. We state a conjecture on Wasserstein-$p$ bounds for any positive integer $p$ and provide supporting arguments for the conjecture.
</p>projecteuclid.org/euclid.ejp/1554775414_20190408220351Mon, 08 Apr 2019 22:03 EDTRescaled Whittaker driven stochastic differential equations converge to the additive stochastic heat equationhttps://projecteuclid.org/euclid.ejp/1554775415<strong>Yu-Ting Chen</strong>. <p><strong>Source: </strong>Electronic Journal of Probability, Volume 24, 33 pp..</p><p><strong>Abstract:</strong><br/>
We study some linear SDEs arising from the two-dimensional $q$-Whittaker driven particle system on the torus as $q\to 1$. The main result proves that the SDEs along certain characteristics converge to the additive stochastic heat equation. Extensions for the SDEs with generalized coefficients and in other spatial dimensions are also obtained. Our proof views the limiting process after recentering as a process of the convolution of a space-time white noise and the Fourier transform of the heat kernel. Accordingly we turn to similar space-time stochastic integrals defined by the SDEs, but now the convolution and the Fourier transform are broken. To obtain tightness of these induced integrals, we bound the oscillations of complex exponentials arising from divergence of the characteristics, with two methods of different nature.
</p>projecteuclid.org/euclid.ejp/1554775415_20190408220351Mon, 08 Apr 2019 22:03 EDTConfinement of Brownian polymers under geometric area tiltshttps://projecteuclid.org/euclid.ejp/1554775416<strong>Pietro Caputo</strong>, <strong>Dmitry Ioffe</strong>, <strong>Vitali Wachtel</strong>. <p><strong>Source: </strong>Electronic Journal of Probability, Volume 24, 21 pp..</p><p><strong>Abstract:</strong><br/>
We consider confinement properties of families of non-colliding Brownian bridges above a hard wall, which are subject to geometrically growing self-potentials of tilted area type. The model is introduced in order to mimic level lines of $2+1$ discrete Solid-On-Solid random interfaces above a hard wall.
</p>projecteuclid.org/euclid.ejp/1554775416_20190408220351Mon, 08 Apr 2019 22:03 EDTRandom walk in cooling random environment: ergodic limits and concentration inequalitieshttps://projecteuclid.org/euclid.ejp/1554775418<strong>Luca Avena</strong>, <strong>Yuki Chino</strong>, <strong>Conrado da Costa</strong>, <strong>Frank den Hollander</strong>. <p><strong>Source: </strong>Electronic Journal of Probability, Volume 24, 35 pp..</p><p><strong>Abstract:</strong><br/>
In previous work by Avena and den Hollander [3], a model of a random walk in a dynamic random environment was proposed where the random environment is resampled from a given law along a given sequence of times. In the regime where the increments of the resampling times diverge, which is referred to as the cooling regime , a weak law of large numbers and certain fluctuation properties were derived under the annealed measure, in dimension one. In the present paper we show that a strong law of large numbers and a quenched large deviation principle hold as well. In the cooling regime, the random walk can be represented as a sum of independent variables, distributed as the increments of a random walk in a static random environment over diverging periods of time. Our proofs require suitable multi-layer decompositions of sums of random variables controlled by moment bounds and concentration estimates. Along the way we derive two results of independent interest, namely, concentration inequalities for the random walk in the static random environment and an ergodic theorem that deals with limits of sums of triangular arrays representing the structure of the cooling regime. We close by discussing our present understanding of homogenisation effects as a function of the cooling scheme, and by hinting at what can be done in higher dimensions. We argue that, while the cooling scheme does not affect the speed in the strong law of large numbers nor the rate function in the large deviation principle, it does affect the fluctuation properties.
</p>projecteuclid.org/euclid.ejp/1554775418_20190408220351Mon, 08 Apr 2019 22:03 EDTAnnealed scaling relations for Voronoi percolationhttps://projecteuclid.org/euclid.ejp/1554861841<strong>Hugo Vanneuville</strong>. <p><strong>Source: </strong>Electronic Journal of Probability, Volume 24, 71 pp..</p><p><strong>Abstract:</strong><br/>
We prove annealed scaling relations for planar Voronoi percolation. To our knowledge, this is the first result of this kind for a continuum percolation model. We are mostly inspired by the proof of scaling relations for Bernoulli percolation by Kesten [22]. Along the way, we show an annealed quasi-multiplicativity property by relying on the quenched box-crossing property proved by Ahlberg, Griffiths, Morris and Tassion [3]. Intermediate results also include the study of quenched and annealed notions of pivotal events and the extension of the quenched box-crossing property of [3] to the near-critical regime.
</p>projecteuclid.org/euclid.ejp/1554861841_20190409220426Tue, 09 Apr 2019 22:04 EDTFluctuation theory for Lévy processes with completely monotone jumpshttps://projecteuclid.org/euclid.ejp/1555034439<strong>Mateusz Kwaśnicki</strong>. <p><strong>Source: </strong>Electronic Journal of Probability, Volume 24, 40 pp..</p><p><strong>Abstract:</strong><br/>
We study the Wiener–Hopf factorization for Lévy processes $X_{t}$ with completely monotone jumps. Extending previous results of L.C.G. Rogers, we prove that the space-time Wiener–Hopf factors are complete Bernstein functions of both the spatial and the temporal variable. As a corollary, we prove complete monotonicity of: (a) the tail of the distribution function of the supremum of $X_{t}$ up to an independent exponential time; (b) the Laplace transform of the supremum of $X_{t}$ up to a fixed time $T$, as a function of $T$. The proof involves a detailed analysis of the holomorphic extension of the characteristic exponent $f(\xi )$ of $X_{t}$, including a peculiar structure of the curve along which $f(\xi )$ takes real values.
</p>projecteuclid.org/euclid.ejp/1555034439_20190411220117Thu, 11 Apr 2019 22:01 EDTErgodicity of some classes of cellular automata subject to noisehttps://projecteuclid.org/euclid.ejp/1555034440<strong>Irène Marcovici</strong>, <strong>Mathieu Sablik</strong>, <strong>Siamak Taati</strong>. <p><strong>Source: </strong>Electronic Journal of Probability, Volume 24, 44 pp..</p><p><strong>Abstract:</strong><br/>
Cellular automata (CA) are dynamical systems on symbolic configurations on the lattice. They are also used as models of massively parallel computers. As dynamical systems, one would like to understand the effect of small random perturbations on the dynamics of CA. As models of computation, they can be used to study the reliability of computation against noise.
We consider various families of CA (nilpotent, permutive, gliders, CA with a spreading symbol, surjective, algebraic) and prove that they are highly unstable against noise, meaning that they forget their initial conditions under slightest positive noise. This is manifested as the ergodicity of the resulting probabilistic CA. The proofs involve a collection of different techniques (couplings, entropy, Fourier analysis), depending on the dynamical properties of the underlying deterministic CA and the type of noise.
</p>projecteuclid.org/euclid.ejp/1555034440_20190411220117Thu, 11 Apr 2019 22:01 EDTA note on concentration for polynomials in the Ising modelhttps://projecteuclid.org/euclid.ejp/1555466612<strong>Radosław Adamczak</strong>, <strong>Michał Kotowski</strong>, <strong>Bartłomiej Polaczyk</strong>, <strong>Michał Strzelecki</strong>. <p><strong>Source: </strong>Electronic Journal of Probability, Volume 24, 22 pp..</p><p><strong>Abstract:</strong><br/>
We present precise multilevel exponential concentration inequalities for polynomials in Ising models satisfying the Dobrushin condition. The estimates have the same form as two-sided tail estimates for polynomials in Gaussian variables due to Latała. In particular, for quadratic forms we obtain a Hanson–Wright type inequality.
We also prove concentration results for convex functions and estimates for nonnegative definite quadratic forms, analogous as for quadratic forms in i.i.d. Rademacher variables, for more general random vectors satisfying the approximate tensorization property for entropy.
</p>projecteuclid.org/euclid.ejp/1555466612_20190416220404Tue, 16 Apr 2019 22:04 EDTAsymptotic representation theory and the spectrum of a random geometric graph on a compact Lie grouphttps://projecteuclid.org/euclid.ejp/1555466613<strong>Pierre-Loïc Méliot</strong>. <p><strong>Source: </strong>Electronic Journal of Probability, Volume 24, 85 pp..</p><p><strong>Abstract:</strong><br/>
Let $G$ be a compact Lie group, $N\geq 1$ and $L>0$. The random geometric graph on $G$ is the random graph $\Gamma _{\mathrm{geom} }(N,L)$ whose vertices are $N$ random points $g_1,\ldots ,g_N$ chosen under the Haar measure of $G$, and whose edges are the pairs $\{g_i,g_j\}$ with $d(g_i,g_j)\leq L$, $d$ being the distance associated to the standard Riemannian structure on $G$. In this paper, we describe the asymptotic behavior of the spectrum of the adjacency matrix of $\Gamma _{\mathrm{geom} }(N,L)$, when $N$ goes to infinity.
1. If $L$ is fixed and $N \to + \infty $ (Gaussian regime), then the largest eigenvalues of $\Gamma _{\mathrm{geom} }(N,L)$ converge after an appropriate renormalisation towards certain explicit linear combinations of values of Bessel functions.
2. If $L = O(N^{-\frac{1} {\dim G}})$ and $N \to +\infty $ (Poissonian regime), then the geometric graph $\Gamma _{\mathrm{geom} }(N,L)$ converges in the local Benjamini–Schramm sense, which implies the weak convergence in probability of the spectral measure of $\Gamma _{\mathrm{geom} }(N,L)$.
In both situations, the representation theory of the group $G$ provides us with informations on the limit of the spectrum, and conversely, the computation of this limiting spectrum involves many classical tools from representation theory: Weyl’s character formula and the weight lattice in the Gaussian regime, and a degeneration of these objects in the Poissonian regime. The representation theoretic approach allows one to understand precisely how the degeneration from the Gaussian to the Poissonian regime occurs, and the article is written so as to highlight this degeneration phenomenon. In the Poissonian regime, this approach leads us to an algebraic conjecture on certain functionals of the irreducible representations of $G$.
</p>projecteuclid.org/euclid.ejp/1555466613_20190416220404Tue, 16 Apr 2019 22:04 EDT