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 ESTMatrix normalised stochastic compactness for a Lévy process at zerohttps://projecteuclid.org/euclid.ejp/1532570597<strong>Ross A. Maller</strong>, <strong>David M. Mason</strong>. <p><strong>Source: </strong>Electronic Journal of Probability, Volume 23, 37 pp..</p><p><strong>Abstract:</strong><br/>
We give necessary and sufficient conditions for a $d$–dimensional Lévy process $({\bf X}_t)_{t\ge 0}$ to be in the matrix normalised Feller (stochastic compactness) classes $FC$ and $FC_0$ as $t\downarrow 0$. This extends earlier results of the authors concerning convergence of a Lévy process in $\Bbb{R} ^d$ to normality, as the time parameter tends to 0. It also generalises and transfers to the Lévy case classical results of Feller and Griffin concerning real- and vector-valued random walks. The process $({\bf X}_t)$ and its quadratic variation matrix together constitute a matrix-valued Lévy process, and, in a further extension, we show that the condition derived for the process itself also guarantees the stochastic compactness of the combined matrix-valued process. This opens the way to further investigations regarding self-normalised processes.
</p>projecteuclid.org/euclid.ejp/1532570597_20181221220705Fri, 21 Dec 2018 22:07 ESTThe phase transition in the ultrametric ensemble and local stability of Dyson Brownian motionhttps://projecteuclid.org/euclid.ejp/1532570598<strong>Per von Soosten</strong>, <strong>Simone Warzel</strong>. <p><strong>Source: </strong>Electronic Journal of Probability, Volume 23, 24 pp..</p><p><strong>Abstract:</strong><br/>
We study the ultrametric random matrix ensemble, whose independent entries have variances decaying exponentially in the metric induced by the tree topology on $\mathbb{N} $, and map out the entire localization regime in terms of eigenfunction localization and Poisson statistics. Our results complement existing works on complete delocalization and random matrix universality, thereby proving the existence of a phase transition in this model. In the simpler case of the Rosenzweig-Porter model, the analysis yields a complete characterization of the transition in the local statistics. The proofs are based on the flow of the resolvents of matrices with a random diagonal component under Dyson Brownian motion, for which we establish submicroscopic stability results for short times. These results go beyond norm-based continuity arguments for Dyson Brownian motion and complement the existing analysis after the local equilibration time.
</p>projecteuclid.org/euclid.ejp/1532570598_20181221220705Fri, 21 Dec 2018 22:07 ESTSuperBrownian motion and the spatial Lambda-Fleming-Viot processhttps://projecteuclid.org/euclid.ejp/1532570599<strong>Jonathan A. Chetwynd-Diggle</strong>, <strong>Alison M. Etheridge</strong>. <p><strong>Source: </strong>Electronic Journal of Probability, Volume 23, 36 pp..</p><p><strong>Abstract:</strong><br/>
It is well known that the dynamics of a subpopulation of individuals of a rare type in a Wright-Fisher diffusion can be approximated by a Feller branching process. Here we establish an analogue of that result for a spatially distributed population whose dynamics are described by a spatial Lambda-Fleming-Viot process (SLFV). The subpopulation of rare individuals is then approximated by a superBrownian motion. This result mirrors [10], where it is shown that when suitably rescaled, sparse voter models converge to superBrownian motion. We also prove the somewhat more surprising result, that by choosing the dynamics of the SLFV appropriately we can recover superBrownian motion with stable branching in an analogous way. This is a spatial analogue of (a special case of) results of [6], who show that the generalised Fleming-Viot process that is dual to the beta-coalescent, when suitably rescaled, converges to a continuous state branching process with stable branching mechanism.
</p>projecteuclid.org/euclid.ejp/1532570599_20181221220705Fri, 21 Dec 2018 22:07 ESTThe polymorphic evolution sequence for populations with phenotypic plasticityhttps://projecteuclid.org/euclid.ejp/1532678635<strong>Martina Baar</strong>, <strong>Anton Bovier</strong>. <p><strong>Source: </strong>Electronic Journal of Probability, Volume 23, 27 pp..</p><p><strong>Abstract:</strong><br/>
In this paper we study a class of stochastic individual-based models that describe the evolution of haploid populations where each individual is characterised by a phenotype and a genotype. The phenotype of an individual determines its natural birth- and death rates as well as the competition kernel, $c(x,y)$ which describes the induced death rate that an individual of type $x$ experiences due to the presence of an individual or type $y$. When a new individual is born, with a small probability a mutation occurs, i.e. the offspring has different genotype as the parent. The novel aspect of the models we study is that an individual with a given genotype may expresses a certain set of different phenotypes, and during its lifetime it may switch between different phenotypes, with rates that are much larger then the mutation rates and that, moreover, may depend on the state of the entire population. The evolution of the population is described by a continuous-time, measure-valued Markov process. In [4], such a model was proposed to describe tumor evolution under immunotherapy. In the present paper we consider a large class of models which comprises the example studied in [4] and analyse their scaling limits as the population size tends to infinity and the mutation rate tends to zero. Under suitable assumptions, we prove convergence to a Markov jump process that is a generalisation of the polymorphic evolution sequence (PES) as analysed in [9, 11].
</p>projecteuclid.org/euclid.ejp/1532678635_20181221220705Fri, 21 Dec 2018 22:07 ESTCutoff for lamplighter chains on fractalshttps://projecteuclid.org/euclid.ejp/1532678636<strong>Amir Dembo</strong>, <strong>Takashi Kumagai</strong>, <strong>Chikara Nakamura</strong>. <p><strong>Source: </strong>Electronic Journal of Probability, Volume 23, 21 pp..</p><p><strong>Abstract:</strong><br/>
We show that the total-variation mixing time of the lamplighter random walk on fractal graphs exhibit sharp cutoff when the underlying graph is transient (namely of spectral dimension greater than two). In contrast, we show that such cutoff can not occur for strongly recurrent underlying graphs (i.e. of spectral dimension less than two).
</p>projecteuclid.org/euclid.ejp/1532678636_20181221220705Fri, 21 Dec 2018 22:07 ESTNumerical scheme for Dynkin games under model uncertaintyhttps://projecteuclid.org/euclid.ejp/1532678637<strong>Yan Dolinsky</strong>, <strong>Benjamin Gottesman</strong>. <p><strong>Source: </strong>Electronic Journal of Probability, Volume 23, 20 pp..</p><p><strong>Abstract:</strong><br/>
We introduce an efficient numerical scheme for continuous time Dynkin games under model uncertainty. We use the Skorokhod embedding in order to construct recombining tree approximations. This technique allows us to determine convergence rates and to construct numerically optimal stopping strategies. We apply our method to several examples of game options.
</p>projecteuclid.org/euclid.ejp/1532678637_20181221220705Fri, 21 Dec 2018 22:07 ESTBernstein-gamma functions and exponential functionals of Lévy processeshttps://projecteuclid.org/euclid.ejp/1532678638<strong>Pierre Patie</strong>, <strong>Mladen Savov</strong>. <p><strong>Source: </strong>Electronic Journal of Probability, Volume 23, 101 pp..</p><p><strong>Abstract:</strong><br/>
In this work we analyse the solution to the recurrence equation \[ M_\Psi (z+1)=\frac{-z} {\Psi (-z)}M_\Psi (z), \quad M_\Psi (1)=1, \] defined on a subset of the imaginary line and where $-\Psi $ is any continuous negative definite function. Using the analytic Wiener-Hopf method we solve this equation as a product of functions that extend the gamma function and are in bijection with the Bernstein functions. We call these functions Bernstein-gamma functions. We establish universal Stirling type asymptotic in terms of the constituting Bernstein function. This allows the full understanding of the decay of $\vert M_\Psi (z)\vert $ along imaginary lines and an access to quantities important for many studies in probability and analysis.
This functional equation is a central object in several recent studies ranging from analysis and spectral theory to probability theory. As an application of the results above, we study from a global perspective the exponential functionals of Lévy processes whose Mellin transform satisfies the equation above. Although these variables have been intensively studied, our new approach based on a combination of probabilistic and analytical techniques enables us to derive comprehensive properties and strengthen several results on the law of these random variables for some classes of Lévy processes that could be found in the literature. These encompass smoothness for its density, regularity and analytical properties, large and small asymptotic behaviour, including asymptotic expansions and bounds. We furnish a thorough study of the weak convergence of exponential functionals on a finite time horizon when the latter expands to infinity. We deliver intertwining relation between members of the class of positive self-similar semigroups.
</p>projecteuclid.org/euclid.ejp/1532678638_20181221220705Fri, 21 Dec 2018 22:07 ESTGOE statistics for Anderson models on antitrees and thin boxes in $\mathbb{Z} ^3$ with deformed Laplacianhttps://projecteuclid.org/euclid.ejp/1533715241<strong>Christian Sadel</strong>. <p><strong>Source: </strong>Electronic Journal of Probability, Volume 23, 24 pp..</p><p><strong>Abstract:</strong><br/>
Sequences of certain finite graphs - special types of antitrees - are constructed along which the Anderson model shows GOE statistics, i.e. a re-scaled eigenvalue process converges to the ${\mathrm{Sine} }_1$ process. The Anderson model on the graph is a random matrix being the sum of the adjacency matrix and a random diagonal matrix with independent identically distributed entries along the diagonal. The strength of the randomness stays fixed, there is no re-scaling with matrix size. These considered random matrices giving GOE statistics can also be viewed as random Schrödinger operators $\mathcal{P} \Delta +\mathcal{V} $ on thin finite boxes in $\mathbb{Z} ^3$ where the Laplacian $\Delta $ is deformed by a projection $\mathcal{P} $ commuting with $\Delta $.
</p>projecteuclid.org/euclid.ejp/1533715241_20181221220705Fri, 21 Dec 2018 22:07 ESTTraffic distributions of random band matriceshttps://projecteuclid.org/euclid.ejp/1536717736<strong>Benson Au</strong>. <p><strong>Source: </strong>Electronic Journal of Probability, Volume 23, 48 pp..</p><p><strong>Abstract:</strong><br/>
We study random band matrices within the framework of traffic probability. As a starting point, we revisit the familiar case of permutation invariant Wigner matrices and compare the situation to the general case in the absence of this invariance. Here, we find a departure from the usual free probabilistic universality of the joint distribution of independent Wigner matrices. We further prove general Markov-type concentration inequalities for the joint traffic distribution. We then extend our analysis to random band matrices and investigate the extent to which the joint traffic distribution of independent copies of these matrices deviates from the Wigner case.
</p>projecteuclid.org/euclid.ejp/1536717736_20181221220705Fri, 21 Dec 2018 22:07 ESTDirichlet form associated with the $\Phi _3^4$ modelhttps://projecteuclid.org/euclid.ejp/1536717737<strong>Rongchan Zhu</strong>, <strong>Xiangchan Zhu</strong>. <p><strong>Source: </strong>Electronic Journal of Probability, Volume 23, 31 pp..</p><p><strong>Abstract:</strong><br/>
We construct the Dirichlet form associated with the dynamical $\Phi ^4_3$ model obtained in [23, 7] and [37]. This Dirichlet form on cylinder functions is identiﬁed as a classical gradient bilinear form. As a consequence, this classical gradient bilinear form is closable and then by a well-known result its closure is also a quasi-regular Dirichlet form, which means that there exists another (Markov) diﬀusion process, which also admits the $\Phi ^4_3$ ﬁeld measure as an invariant (even symmetrizing) measure.
</p>projecteuclid.org/euclid.ejp/1536717737_20181221220705Fri, 21 Dec 2018 22:07 ESTA $q$-deformation of the symplectic Schur functions and the Berele insertion algorithmhttps://projecteuclid.org/euclid.ejp/1536717738<strong>Ioanna Nteka</strong>. <p><strong>Source: </strong>Electronic Journal of Probability, Volume 23, 23 pp..</p><p><strong>Abstract:</strong><br/>
A randomisation of the Berele insertion algorithm is proposed, where the insertion of a letter to a symplectic Young tableau leads to a distribution over the set of symplectic Young tableaux. Berele’s algorithm provides a bijection between words from an alphabet and a symplectic Young tableau along with a recording oscillating tableau. The randomised version of the algorithm is achieved by introducing a parameter $0< q <1$. The classic Berele algorithm corresponds to letting the parameter $q\to 0$. The new version provides a probabilistic framework that allows to prove Littlewood-type identities for a $q$-deformation of the symplectic Schur functions. These functions correspond to multilevel extensions of the continuous $q$-Hermite polynomials. Finally, we show that when both the original and the $q$-modified insertion algorithms are applied to a random word then the shape of the symplectic Young tableau evolves as a Markov chain on the set of partitions.
</p>projecteuclid.org/euclid.ejp/1536717738_20181221220705Fri, 21 Dec 2018 22:07 ESTA random walk approach to linear statistics in random tournament ensembleshttps://projecteuclid.org/euclid.ejp/1536717739<strong>Christopher H. Joyner</strong>, <strong>Uzy Smilansky</strong>. <p><strong>Source: </strong>Electronic Journal of Probability, Volume 23, 37 pp..</p><p><strong>Abstract:</strong><br/>
We investigate the linear statistics of random matrices with purely imaginary Bernoulli entries of the form $H_{pq} = \overline{H} _{qp} = \pm i$, that are either independently distributed or exhibit global correlations imposed by the condition $\sum _{q} H_{pq} = 0$. These are related to ensembles of so-called random tournaments and random regular tournaments respectively. Specifically, we construct a random walk within the space of matrices and show that the induced motion of the first $k$ traces in a Chebyshev basis converges to a suitable Ornstein-Uhlenbeck process. Coupling this with Stein’s method allows us to compute the rate of convergence to a Gaussian distribution in the limit of large matrix dimension.
</p>projecteuclid.org/euclid.ejp/1536717739_20181221220705Fri, 21 Dec 2018 22:07 ESTSpin systems from loop soupshttps://projecteuclid.org/euclid.ejp/1536717740<strong>Tim van de Brug</strong>, <strong>Federico Camia</strong>, <strong>Marcin Lis</strong>. <p><strong>Source: </strong>Electronic Journal of Probability, Volume 23, 17 pp..</p><p><strong>Abstract:</strong><br/>
We study spin systems defined by the winding of a random walk loop soup. For a particular choice of loop soup intensity, we show that the corresponding spin system is reflection-positive and is dual, in the Kramers-Wannier sense, to the spin system $\mathrm{sgn} (\varphi )$ where $\varphi $ is a discrete Gaussian free field.
In general, we show that the spin correlation functions have conformally covariant scaling limits corresponding to the one-parameter family of functions studied by Camia, Gandolfi and Kleban ( Nuclear Physics B 902 , 2016) and defined in terms of the winding of the Brownian loop soup. These functions have properties consistent with the behavior of correlation functions of conformal primaries in a conformal field theory. Here, we prove that they do correspond to correlation functions of continuum fields (random generalized functions) for values of the intensity of the Brownian loop soup that are not too large.
</p>projecteuclid.org/euclid.ejp/1536717740_20181221220705Fri, 21 Dec 2018 22:07 ESTUniqueness for the 3-state antiferromagnetic Potts model on the treehttps://projecteuclid.org/euclid.ejp/1536717741<strong>Andreas Galanis</strong>, <strong>Leslie Ann Goldberg</strong>, <strong>Kuan Yang</strong>. <p><strong>Source: </strong>Electronic Journal of Probability, Volume 23, 43 pp..</p><p><strong>Abstract:</strong><br/>
The antiferromagnetic $q$-state Potts model is perhaps the most canonical model for which the uniqueness threshold on the tree is not yet understood, largely because of the absence of monotonicities. Jonasson established the uniqueness threshold in the zero-temperature case, which corresponds to the $q$-colourings model. In the permissive case (where the temperature is positive), the Potts model has an extra parameter $\beta \in (0,1)$, which makes the task of analysing the uniqueness threshold even harder and much less is known.
In this paper, we focus on the case $q=3$ and give a detailed analysis of the Potts model on the tree by refining Jonasson’s approach. In particular, we establish the uniqueness threshold on the $d$-ary tree for all values of $d\geq 2$. When $d\geq 3$, we show that the 3-state antiferromagnetic Potts model has uniqueness for all $\beta \geq 1-3/(d+1)$. The case $d=2$ is critical since it relates to the 3-colourings model on the binary tree ($\beta =0$), which has non-uniqueness. Nevertheless, we show that the Potts model has uniqueness for all $\beta \in (0,1)$ on the binary tree. Both of these results are tight since it is known that uniqueness does not hold in the complementary regime.
Our proof technique gives for general $q>3$ an analytical condition for proving uniqueness based on the two-step recursion on the tree, which we conjecture to be sufficient to establish the uniqueness threshold for all non-critical cases ($q\neq d+1$).
</p>projecteuclid.org/euclid.ejp/1536717741_20181221220705Fri, 21 Dec 2018 22:07 ESTThe dimension of the range of a transient random walkhttps://projecteuclid.org/euclid.ejp/1536717742<strong>Nicos Georgiou</strong>, <strong>Davar Khoshnevisan</strong>, <strong>Kunwoo Kim</strong>, <strong>Alex D. Ramos</strong>. <p><strong>Source: </strong>Electronic Journal of Probability, Volume 23, 31 pp..</p><p><strong>Abstract:</strong><br/>
We find formulas for the macroscopic Minkowski and Hausdorff dimensions of the range of an arbitrary transient walk in $\mathbb{Z} ^d$. This endeavor solves a problem of Barlow and Taylor (1991).
</p>projecteuclid.org/euclid.ejp/1536717742_20181221220705Fri, 21 Dec 2018 22:07 ESTExistence and uniqueness of reflecting diffusions in cuspshttps://projecteuclid.org/euclid.ejp/1536717743<strong>Cristina Costantini</strong>, <strong>Thomas G. Kurtz</strong>. <p><strong>Source: </strong>Electronic Journal of Probability, Volume 23, 21 pp..</p><p><strong>Abstract:</strong><br/>
We consider stochastic differential equations with (oblique) reflection in a 2-dimensional domain that has a cusp at the origin, i.e. in a neighborhood of the origin has the form $\{(x_1,x_2):0<x_1\leq \delta _0,\psi _1(x_1)<x_2<\psi _ 2(x_1)\}$, with $\psi _1(0)=\psi _2(0)=0$, $\psi _1'(0)=\psi _2'(0)=0$.
Given a vector field $g$ of directions of reflection at the boundary points other than the origin, defining directions of reflection at the origin $g^i(0):=\lim _{x_1\rightarrow 0^{+}}g (x_1,\psi _i(x_1))$, $ i=1,2$, and assuming there exists a vector $e^{*}$ such that $\langle e^{*},g^i(0)\rangle >0$, $i=1,2$, and $e^{*}_1>0$, we prove weak existence and uniqueness of the solution starting at the origin and strong existence and uniqueness starting away from the origin.
Our proof uses a new scaling result and a coupling argument.
</p>projecteuclid.org/euclid.ejp/1536717743_20181221220705Fri, 21 Dec 2018 22:07 ESTThe random matrix hard edge: rare events and a transitionhttps://projecteuclid.org/euclid.ejp/1536717744<strong>Diane Holcomb</strong>. <p><strong>Source: </strong>Electronic Journal of Probability, Volume 23, 20 pp..</p><p><strong>Abstract:</strong><br/>
We study properties of the point process that appears as the local limit at the random matrix hard edge. We show a transition from the hard edge to bulk behavior and give a central limit theorem and large deviation result for the number of points in a growing interval $[0,\lambda ]$ as $\lambda \to \infty $. We study these results for the square root of the hard edge process. In this setting many of these behaviors mimic those of the $\mathrm{Sine} _\beta $ process.
</p>projecteuclid.org/euclid.ejp/1536717744_20181221220705Fri, 21 Dec 2018 22:07 ESTOn the speed of once-reinforced biased random walk on treeshttps://projecteuclid.org/euclid.ejp/1536717745<strong>Andrea Collevecchio</strong>, <strong>Mark Holmes</strong>, <strong>Daniel Kious</strong>. <p><strong>Source: </strong>Electronic Journal of Probability, Volume 23, 32 pp..</p><p><strong>Abstract:</strong><br/>
We study the asymptotic behaviour of once-reinforced biased random walk (ORbRW) on Galton-Watson trees. Here the underlying (unreinforced) random walk has a bias towards or away from the root. We prove that in the setting of multiplicative once-reinforcement the ORbRW can be recurrent even when the underlying biased random walk is ballistic. We also prove that, on Galton-Watson trees without leaves, the speed is positive in the transient regime. Finally, we prove that, on regular trees, the speed of the ORbRW is monotone decreasing in the reinforcement parameter when the underlying random walk has high speed, and the reinforcement parameter is small.
</p>projecteuclid.org/euclid.ejp/1536717745_20181221220705Fri, 21 Dec 2018 22:07 ESTCost functionals for large (uniform and simply generated) random treeshttps://projecteuclid.org/euclid.ejp/1536717746<strong>Jean-François Delmas</strong>, <strong>Jean-Stéphane Dhersin</strong>, <strong>Marion Sciauveau</strong>. <p><strong>Source: </strong>Electronic Journal of Probability, Volume 23, 36 pp..</p><p><strong>Abstract:</strong><br/>
Additive tree functionals allow to represent the cost of many divide-and-conquer algorithms. We give an invariance principle for such tree functionals for the Catalan model (random tree uniformly distributed among the full binary ordered trees with given number of nodes) and for simply generated trees (including random tree uniformly distributed among the ordered trees with given number of nodes). In the Catalan model, this relies on the natural embedding of binary trees into the Brownian excursion and then on elementary $ L^2$ computations. We recover results first given by Fill and Kapur (2004) and then by Fill and Janson (2009). In the simply generated case, we use convergence of conditioned Galton-Watson trees towards stable Lévy trees, which provides less precise results but leads us to conjecture a different phase transition value between “global” and “local” regimes. We also recover results first given by Janson (2003 and 2016) in the Brownian case and give a generalization to the stable case.
</p>projecteuclid.org/euclid.ejp/1536717746_20181221220705Fri, 21 Dec 2018 22:07 ESTBerry–Esseen bounds for typical weighted sumshttps://projecteuclid.org/euclid.ejp/1536976980<strong>S.G. Bobkov</strong>, <strong>G.P. Chistyakov</strong>, <strong>F. Götze</strong>. <p><strong>Source: </strong>Electronic Journal of Probability, Volume 23, 22 pp..</p><p><strong>Abstract:</strong><br/>
Under correlation-type conditions, we derive upper bounds of order $\frac{1} {\sqrt{n} }$ for the Kolmogorov distance between the distributions of weighted sums of dependent summands and the normal law.
</p>projecteuclid.org/euclid.ejp/1536976980_20181221220705Fri, 21 Dec 2018 22:07 ESTWeighted dependency graphshttps://projecteuclid.org/euclid.ejp/1537257885<strong>Valentin Féray</strong>. <p><strong>Source: </strong>Electronic Journal of Probability, Volume 23, 65 pp..</p><p><strong>Abstract:</strong><br/>
The theory of dependency graphs is a powerful toolbox to prove asymptotic normality of sums of random variables. In this article, we introduce a more general notion of weighted dependency graphs and give normality criteria in this context. We also provide generic tools to prove that some weighted graph is a weighted dependency graph for a given family of random variables.
To illustrate the power of the theory, we give applications to the following objects: uniform random pair partitions, the random graph model $G(n,M)$, uniform random permutations, the symmetric simple exclusion process and multilinear statistics on Markov chains. The application to random permutations gives a bivariate extension of a functional central limit theorem of Janson and Barbour. On Markov chains, we answer positively an open question of Bourdon and Vallée on the asymptotic normality of subword counts in random texts generated by a Markovian source.
</p>projecteuclid.org/euclid.ejp/1537257885_20181221220705Fri, 21 Dec 2018 22:07 ESTTrees within trees: simple nested coalescentshttps://projecteuclid.org/euclid.ejp/1537257886<strong>Airam Blancas</strong>, <strong>Jean-Jil Duchamps</strong>, <strong>Amaury Lambert</strong>, <strong>Arno Siri-Jégousse</strong>. <p><strong>Source: </strong>Electronic Journal of Probability, Volume 23, 27 pp..</p><p><strong>Abstract:</strong><br/>
We consider the compact space of pairs of nested partitions of $\mathbb{N} $, where by analogy with models used in molecular evolution, we call “gene partition” the finer partition and “species partition” the coarser one. We introduce the class of nondecreasing processes valued in nested partitions, assumed Markovian and with exchangeable semigroup. These processes are said simple when each partition only undergoes one coalescence event at a time (but possibly the same time). Simple nested exchangeable coalescent (SNEC) processes can be seen as the extension of $\Lambda $-coalescents to nested partitions. We characterize the law of SNEC processes as follows. In the absence of gene coalescences, species blocks undergo $\Lambda $-coalescent type events and in the absence of species coalescences, gene blocks lying in the same species block undergo i.i.d. $\Lambda $-coalescents. Simultaneous coalescence of the gene and species partitions are governed by an intensity measure $\nu _s$ on $(0,1]\times{\mathcal M} _1 ([0,1])$ providing the frequency of species merging and the law in which are drawn (independently) the frequencies of genes merging in each coalescing species block. As an application, we also study the conditions under which a SNEC process comes down from infinity.
</p>projecteuclid.org/euclid.ejp/1537257886_20181221220705Fri, 21 Dec 2018 22:07 ESTMetastability of hard-core dynamics on bipartite graphshttps://projecteuclid.org/euclid.ejp/1537495434<strong>Frank den Hollander</strong>, <strong>Francesca R. Nardi</strong>, <strong>Siamak Taati</strong>. <p><strong>Source: </strong>Electronic Journal of Probability, Volume 23, 65 pp..</p><p><strong>Abstract:</strong><br/>
We study the metastable behaviour of a stochastic system of particles with hard-core interactions in a high-density regime. Particles sit on the vertices of a bipartite graph. New particles appear subject to a neighbourhood exclusion constraint, while existing particles disappear, all according to independent Poisson clocks. We consider the regime in which the appearance rates are much larger than the disappearance rates, and there is a slight imbalance between the appearance rates on the two parts of the graph. Starting from the configuration in which the weak part is covered with particles, the system takes a long time before it reaches the configuration in which the strong part is covered with particles. We obtain a sharp asymptotic estimate for the expected transition time, show that the transition time is asymptotically exponentially distributed, and identify the size and shape of the critical droplet representing the bottleneck for the crossover. For various types of bipartite graphs the computations are made explicit. Proofs rely on potential theory for reversible Markov chains, and on isoperimetric results.
</p>projecteuclid.org/euclid.ejp/1537495434_20181221220705Fri, 21 Dec 2018 22:07 ESTExistence and continuity of the flow constant in first passage percolationhttps://projecteuclid.org/euclid.ejp/1537927580<strong>Raphaël Rossignol</strong>, <strong>Marie Théret</strong>. <p><strong>Source: </strong>Electronic Journal of Probability, Volume 23, 42 pp..</p><p><strong>Abstract:</strong><br/>
We consider the model of i.i.d. first passage percolation on $\mathbb{Z} ^d$, where we associate with the edges of the graph a family of i.i.d. random variables with common distribution $G$ on $ [0,+\infty ]$ (including $+\infty $). Whereas the time constant is associated to the study of $1$-dimensional paths with minimal weight, namely geodesics, the flow constant is associated to the study of $(d-1)$-dimensional surfaces with minimal weight. In this article, we investigate the existence of the flow constant under the only hypothesis that $G(\{+\infty \} ) < p_c(d)$ (in particular without any moment assumption), the convergence of some natural maximal flows towards this constant, and the continuity of this constant with regard to the distribution $G$.
</p>projecteuclid.org/euclid.ejp/1537927580_20181221220705Fri, 21 Dec 2018 22:07 ESTLimit theorems for free Lévy processeshttps://projecteuclid.org/euclid.ejp/1538618571<strong>Octavio Arizmendi</strong>, <strong>Takahiro Hasebe</strong>. <p><strong>Source: </strong>Electronic Journal of Probability, Volume 23, 36 pp..</p><p><strong>Abstract:</strong><br/>
We consider different limit theorems for additive and multiplicative free Lévy processes. The main results are concerned with positive and unitary multiplicative free Lévy processes at small times, showing convergence to log free stable laws for many examples. The additive case is much easier, and we establish the convergence at small or large times to free stable laws. During the investigation we found out that a log free stable law with index $1$ coincides with the Dykema-Haagerup distribution. We also consider limit theorems for positive multiplicative Boolean Lévy processes at small times, obtaining log Boolean stable laws in the limit.
</p>projecteuclid.org/euclid.ejp/1538618571_20181221220705Fri, 21 Dec 2018 22:07 ESTNatural parametrization of SLE: the Gaussian free field point of viewhttps://projecteuclid.org/euclid.ejp/1539828067<strong>Stéphane Benoist</strong>. <p><strong>Source: </strong>Electronic Journal of Probability, Volume 23, 16 pp..</p><p><strong>Abstract:</strong><br/>
We provide another construction of the natural parametrization of SLE$_\kappa $ [9, 8] for $\kappa < 4$. We construct it as the expectation of the quantum time [14], which is a random measure carried by SLE in an ambient Gaussian free field. This quantum time was built as the push forward on the SLE curve of the Liouville boundary measure, which is a natural field-dependent measure supported on the boundary of the domain. We moreover show that the quantum time can be reconstructed as a chaos on any measure on the trace of SLE with the right Markovian covariance property. This provides another proof that the natural parametrization is characterized by its Markovian covariance property.
</p>projecteuclid.org/euclid.ejp/1539828067_20181221220705Fri, 21 Dec 2018 22:07 ESTDisconnection by level sets of the discrete Gaussian free field and entropic repulsionhttps://projecteuclid.org/euclid.ejp/1540260051<strong>Maximilian Nitzschner</strong>. <p><strong>Source: </strong>Electronic Journal of Probability, Volume 23, 21 pp..</p><p><strong>Abstract:</strong><br/>
We derive asymptotic upper and lower bounds on the large deviation probability that the level set of the Gaussian free field on $\mathbb{Z} ^d$, $d \geq 3$, below a level $\alpha $ disconnects the discrete blow-up of a compact set $A$ from the boundary of the discrete blow-up of a box that contains $A$, when the level set of the Gaussian free field above $\alpha $ is in a strongly percolative regime. These bounds substantially strengthen the results of [21], where $A$ was a box and the convexity of $A$ played an important role in the proof. We also derive an asymptotic upper bound on the probability that the average of the Gaussian free field well inside the discrete blow-up of $A$ is above a certain level when disconnection occurs. The derivation of the upper bounds uses the solidification estimates for porous interfaces that were derived in the work [15] of A.-S. Sznitman and the author to treat a similar disconnection problem for the vacant set of random interlacements. If certain critical levels for the Gaussian free field coincide, an open question at the moment, the asymptotic upper and lower bounds that we obtain for the disconnection probability match in principal order, and conditioning on disconnection lowers the average of the Gaussian free field well inside the discrete blow-up of $A$, which can be understood as entropic repulsion.
</p>projecteuclid.org/euclid.ejp/1540260051_20181221220705Fri, 21 Dec 2018 22:07 ESTRandom surface growth and Karlin-McGregor polynomialshttps://projecteuclid.org/euclid.ejp/1540260052<strong>Theodoros Assiotis</strong>. <p><strong>Source: </strong>Electronic Journal of Probability, Volume 23, 81 pp..</p><p><strong>Abstract:</strong><br/>
We consider consistent dynamics for non-intersecting birth and death chains, originating from dualities of stochastic coalescing flows and one dimensional orthogonal polynomials. As corollaries, we obtain unified and simple probabilistic proofs of certain key intertwining relations between multivariate Markov chains on the levels of some branching graphs. Special cases include the dynamics on the Gelfand-Tsetlin graph considered in the seminal work of Borodin and Olshanski in [10] and the ones on the BC-type graph recently studied by Cuenca in [17]. Moreover, we introduce a general inhomogeneous random growth process with a wall that includes as special cases the ones considered by Borodin and Kuan [8] and Cerenzia [15], that are related to the representation theory of classical groups and also the Jacobi growth process more recently studied by Cerenzia and Kuan [16]. Its most important feature is that, this process retains the determinantal structure of the ones studied previously and for the fully packed initial condition we are able to calculate its correlation kernel explicitly in terms of a contour integral involving orthogonal polynomials. At a certain scaling limit, at a finite distance from the wall, one obtains for a single level discrete determinantal ensembles associated to continuous orthogonal polynomials, that were recently introduced by Borodin and Olshanski in [11], and that depend on the inhomogeneities.
</p>projecteuclid.org/euclid.ejp/1540260052_20181221220705Fri, 21 Dec 2018 22:07 ESTA family of random sup-measures with long-range dependencehttps://projecteuclid.org/euclid.ejp/1540260053<strong>Olivier Durieu</strong>, <strong>Yizao Wang</strong>. <p><strong>Source: </strong>Electronic Journal of Probability, Volume 23, 24 pp..</p><p><strong>Abstract:</strong><br/>
A family of self-similar and translation-invariant random sup-measures with long-range dependence are investigated. They are shown to arise as the limit of the empirical random sup-measure of a stationary heavy-tailed process, inspired by an infinite urn scheme, where same values are repeated at several random locations. The random sup-measure reflects the long-range dependence nature of the original process, and in particular characterizes how locations of extremes appear as long-range clusters represented by random closed sets. A limit theorem for the corresponding point-process convergence is established.
</p>projecteuclid.org/euclid.ejp/1540260053_20181221220705Fri, 21 Dec 2018 22:07 ESTLarge deviations for small noise diffusions in a fast markovian environmenthttps://projecteuclid.org/euclid.ejp/1540951492<strong>Amarjit Budhiraja</strong>, <strong>Paul Dupuis</strong>, <strong>Arnab Ganguly</strong>. <p><strong>Source: </strong>Electronic Journal of Probability, Volume 23, 33 pp..</p><p><strong>Abstract:</strong><br/>
A large deviation principle is established for a two-scale stochastic system in which the slow component is a continuous process given by a small noise finite dimensional Itô stochastic differential equation, and the fast component is a finite state pure jump process. Previous works have considered settings where the coupling between the components is weak in a certain sense. In the current work we study a fully coupled system in which the drift and diffusion coefficient of the slow component and the jump intensity function and jump distribution of the fast process depend on the states of both components. In addition, the diffusion can be degenerate. Our proofs use certain stochastic control representations for expectations of exponential functionals of finite dimensional Brownian motions and Poisson random measures together with weak convergence arguments. A key challenge is in the proof of the large deviation lower bound where, due to the interplay between the degeneracy of the diffusion and the full dependence of the coefficients on the two components, the associated local rate function has poor regularity properties.
</p>projecteuclid.org/euclid.ejp/1540951492_20181221220705Fri, 21 Dec 2018 22:07 ESTA simple method for the existence of a density for stochastic evolutions with rough coefficientshttps://projecteuclid.org/euclid.ejp/1542942364<strong>Marco Romito</strong>. <p><strong>Source: </strong>Electronic Journal of Probability, Volume 23, 43 pp..</p><p><strong>Abstract:</strong><br/>
We extend the validity of a simple method for the existence of a density for stochastic differential equations, first introduced in [15], by proving local estimates for the density, existence for the density with summable drift, and by improving the regularity of the density.
</p>projecteuclid.org/euclid.ejp/1542942364_20181221220705Fri, 21 Dec 2018 22:07 ESTPrecise large deviations for random walk in random environmenthttps://projecteuclid.org/euclid.ejp/1542942365<strong>Dariusz Buraczewski</strong>, <strong>Piotr Dyszewski</strong>. <p><strong>Source: </strong>Electronic Journal of Probability, Volume 23, 26 pp..</p><p><strong>Abstract:</strong><br/>
We study one-dimensional nearest neighbour random walk in site-dependent random environment. We establish precise (sharp) large deviations in the so-called ballistic regime, when the random walk drifts to the right with linear speed. In the sub-ballistic regime, when the speed is sublinear, we describe the precise probability of slowdown.
</p>projecteuclid.org/euclid.ejp/1542942365_20181221220705Fri, 21 Dec 2018 22:07 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 EST