We study the a.s. convergence of a sequence of random embeddings of a fixed manifold into Euclidean spaces of increasing dimensions. We show that the limit is deterministic. As a consequence, we show that many intrinsic functionals of the embedded manifolds also converge to deterministic limits. Particularly interesting examples of these functionals are given by the Lipschitz-Killing curvatures, for which we also prove unbiasedness, using the Gaussian kinematic formula.

We give a heat flow derivation for the Godbillon Vey class. In particular we prove that if $(M,g)$ is a compact Riemannian manifold with a codimension 1 foliation $\mathcal{F} $, defined by an integrable 1-form $\omega $ such that $||\omega ||=1$, then the Godbillon-Vey class can be written as $[-\mathcal{A} \omega \wedge d\omega ]_{dR}$ for an operator $\mathcal{A} :\Omega ^*(M)\rightarrow \Omega ^*(M)$ induced by the heat flow.

We show how the theory of stochastic flows allows to recover in an elementary way a well known result of Warren on the sticky Brownian motion equation.

We consider $d$-dimensional diffusion processes in multi-parameter random environments which are given by values at different $d$ points of one-dimensional $\alpha $-stable or $(r, \alpha )$-semi-stable Lévy processes. From the model, we derive some conditions of random environments that imply the dichotomy of recurrence and transience for the $d$-dimensional diffusion processes. The limiting behavior is quite different from that of a $d$-dimensional standard Brownian motion. We also consider the direct product of a one-dimensional diffusion process in a reflected non-positive Brownian environment and a one-dimensional standard Brownian motion. For the two-dimensional diffusion process, we show the transience property for almost all reflected Brownian environments.

In this note we show that a bounded degree planar triangulation is recurrent if and only if the set of accumulation points of some/any circle packing of it is polar (that is, planar Brownian motion avoids it with probability $1$). This generalizes a theorem of He and Schramm [6] who proved it when the set of accumulation points is either empty or a Jordan curve, in which case the graph has one end. We also show that this statement holds for any straight-line embedding with angles uniformly bounded away from $0$.

We prove the universality of the large deviations principle for the empirical measures of zeros of random polynomials whose coefficients are i.i.d. random variables possessing a density with respect to the Lebesgue measure on $\mathbb{C} $, $\mathbb{R} $ or $\mathbb{R} ^+$, under the assumption that the density does not vanish too fast at zero and decays at least as $\exp -|x|^{\rho }$, $\rho >0$, at infinity.

We obtain complementary recurrence/transience criteria for processes $X=(X_n)_{n \ge 0}$ with values in $\mathbb R^d_+$ fulfilling a non-linear equation $X_{n+1}=MX_n+g(X_n)+ \xi _{n+1}$. Here $M$ denotes a primitive matrix having Perron-Frobenius eigenvalue 1, and $g$ denotes some function. The conditional expectation and variance of the noise $(\xi _{n+1})_{n \ge 0}$ are such that $X$ obeys a weak form of the Markov property. The results generalize criteria for the 1-dimensional case in [5].

Consider a finite system of rank-based competing Brownian particles, where the drift and diffusion of each particle depend only on its current rank relative to other particles. We present a simple sufficient condition for absence of multiple collisions of a given order, continuing the earlier work by Bruggeman and Sarantsev (2015). Unlike in that paper, this new condition works even for infinite systems.

Let $(S_n)_{n\in \mathbb{N} }$ be a $\mathbb{Z} $-valued random walk with increments from the domain of attraction of some $\alpha $-stable law and let $(\xi (i))_{i\in \mathbb{Z} }$ be a sequence of iid random variables. We want to investigate $U$-statistics indexed by the random walk $S_n$, that is $U_n:=\sum _{1\leq i<j\leq n}h(\xi (S_i),\xi (S_j))$ for some symmetric bivariate function $h$. We will prove the weak convergence without assumption of finite variance. Additionally, under the assumption of finite moments of order greater than two, we will establish a law of the iterated logarithm for the $U$-statistic $U_n$.

We consider the decreasing and the increasing $r$-excessive functions $\varphi _r$ and $\psi _r$ that are associated with a one-dimensional conservative regular continuous strong Markov process $X$ with values in an interval with endpoints $\alpha < \beta $. We prove that the $r$-excessive local martingale $\bigl ( e^{-r (t \wedge T_\alpha )} \varphi _r (X_{t \wedge T_\alpha }) \bigr )$ $\bigl ($resp., $\bigl ( e^{-r (t \wedge T_\beta )} \psi _r (X_{t \wedge T_\beta }) \bigr ) \bigr )$ is a strict local martingale if the boundary point $\alpha $ (resp., $\beta $) is inaccessible and entrance, and a martingale otherwise.

In this paper, we consider the product space for two processes with independent increments under nonlinear expectations. By introducing a discretization method, we construct a nonlinear expectation under which the given two processes can be seen as a new process with independent increments.

The connective constant of a transitive graph is the exponential growth rate of its number of self-avoiding walks. We prove that the set of connective constants of the so-called Cayley graphs contains a Cantor set. In particular, this set has the cardinality of the continuum.

A conjecture of Benjamini & Schramm from 1996 states that any finitely generated group that is not a finite extension of $\mathbb{Z} $ has a non-trivial percolation phase. Our main results prove this conjecture for certain groups, and in particular prove that any group with a non-trivial homomorphism into the additive group of real numbers satisfies the conjecture. We use this to reduce the conjecture to the case of hereditary just-infinite groups.

The novelty here is mainly in the methods used, combining the methods of EIT and evolving sets, and using the algebraic properties of the group to apply these methods.

Given an infinite connected graph, a way to randomly perturb its metric is to assign random i.i.d. lengths to the edges. An open question attributed to Furstenberg ([14]) is whether there exists a bi-infinite geodesic in first passage percolation on the euclidean lattice of dimension at least 2. Although the answer is generally conjectured to be negative, we give a positive answer for graphs satisfying some negative curvature assumption. Assuming only strict positivity and finite expectation of the random lengths, we prove that if a graph $X$ has bounded degree and contains a Morse geodesic (e.g. is non-elementary Gromov hyperbolic), then almost surely, there exists a bi-infinite geodesic in first passage percolation on $X$.

Erratum for Optimal linear drift for the speed of convergence of an hypoelliptic diffusion, A. Guillin, and P. Monmarché, Electron. Commun. Probab. 21 (2016), paper no. 74, 14 pp. doi:10.1214/16-ECP25.

We consider self-averaging sequences in which each term is a weighted average over previous terms. For several sequences of this kind it is known that they do not converge to a limit. These sequences share the property that $n$th term is mainly based on terms around a fixed fraction of $n$. We give a probabilistic interpretation to such sequences and give weak conditions under which it is natural to expect non-convergence. Our methods are illustrated by application to the group Russian roulette problem.

We present a new proof of a fundamental result concerning cycles of random permutations which gives some intuition for the connection between Touchard polynomials and the Poisson distribution. We also introduce a rather novel permutation statistic and study its distribution. This quantity, indexed by $m$, is the number of sets of size $m$ fixed by the permutation. This leads to a new and simpler derivation of the exponential generating function for the number of covers of certain multisets.

Biggins [Uniform convergence of martingales in the branching random walk. Ann. Probab., 20(1):137–151, 1992] proved local uniform convergence of additive martingales in $d$-dimensional supercritical branching random walks at complex parameters $\lambda $ from an open set $\Lambda \subseteq \mathbb{C} ^d$. We investigate the martingales corresponding to parameters from the boundary $\partial \Lambda $ of $\Lambda $. The boundary can be decomposed into several parts. We demonstrate by means of an example that there may be a part of the boundary, on which the martingales do not exist. Where the martingales exist, they may diverge, vanish in the limit or converge to a non-degenerate limit. We provide mild sufficient conditions for each of these three types of limiting behaviors to occur. The arguments that give convergence to a non-degenerate limit also apply in $\Lambda $ and require weaker moment assumptions than the ones used by Biggins.

In this paper, we will give some remarks on links between the spectral gap of the Ornstein-Uhlenbeck operator on the Riemannian path space with lower and upper bounds of the Ricci curvature on the base manifold; this work was motivated by a recent work of A. Naber on the characterization of the bound of the Ricci curvature by analysis of path spaces.

We are concerned with exploring the probabilities of first order statements for Galton-Watson trees with $Poisson(c)$ offspring distribution. Fixing a positive integer $k$, we exploit the $k$-move Ehrenfeucht game on rooted trees for this purpose. Let $\Sigma $, indexed by $1 \leq j \leq m$, denote the finite set of equivalence classes arising out of this game, and $D$ the set of all probability distributions over $\Sigma $. Let $x_{j}(c)$ denote the true probability of the class $j \in \Sigma $ under $Poisson(c)$ regime, and $\vec{x} (c)$ the true probability vector over all the equivalence classes. Then we are able to define a natural recursion function $\Gamma $, and a map $\Psi = \Psi _{c}: D \rightarrow D$ such that $\vec{x} (c)$ is a fixed point of $\Psi _{c}$, and starting with any distribution $\vec{x} \in D$, we converge to this fixed point via $\Psi $ because it is a contraction. We show this both for $c \leq 1$ and $c > 1$, though the techniques for these two ranges are quite different.

In this paper we study the existence phase transition of the random fractal ball model and the random fractal box model. We show that both of these are in the empty phase at the critical point of this phase transition.

We prove an upper bound of $1.5321 n \log n$ for the mixing time of the random-to-random insertion shuffle, improving on the best known upper bound of $2 n \log n$. Our proof is based on the analysis of a non-Markovian coupling.

In [2] foundations for diffusion approximation via Stein’s method are laid. This paper has been cited more than 130 times and is a cornerstone in the area of Stein’s method (see, for example, its use in [1] or [7]). A semigroup argument is used in [2] to solve a Stein equation for Gaussian diffusion approximation. We prove that, contrary to the claim in [2], the semigroup considered therein is not strongly continuous on the Banach space of continuous, real-valued functions on $D[0,1]$ growing slower than a cubic, equipped with an appropriate norm. We also provide a proof of the exact formulation of the solution to the Stein equation of interest, which does not require the aforementioned strong continuity. This shows that the main results of [2] hold true.

In this article, we provide a Stein type characterization for $G$-normal distributions: Let $\mathcal{N} [\varphi ]=\sup _{\mu \in \Theta }\mu [\varphi ],\ \varphi \in C_{b,Lip}(\mathbb{R} ),$ be a sublinear expectation. $\mathcal{N} $ is $G$-normal if and only if for any $\varphi \in C_b^2(\mathbb{R} )$, we have \[ \int _\mathbb{R} [\frac{x} {2}\varphi '(x)-G(\varphi ''(x))]\mu ^\varphi (dx)=0, \] where $\mu ^\varphi $ is a realization of $\varphi $ associated with $\mathcal{N} $, i.e., $\mu ^\varphi \in \Theta $ and $\mu ^\varphi [\varphi ]=\mathcal{N} [\varphi ]$.

We prove a weak law of large numbers for the capacity of the range of simple random walks on $\mathbb{Z} ^{4}$. On $\mathbb{Z} ^{3}$, we show that the capacity, properly scaled, converges in distribution towards the corresponding quantity for three dimensional Brownian motion. The paper answers two of the three open questions raised by Asselah, Schapira and Sousi in [2, Section 6].

In this paper we consider Bernoulli percolation on an infinite connected bounded degrees graph $G$. Assuming the uniqueness of the infinite open cluster and a quasi-multiplicativity of crossing probabilities, we prove the existence of Kesten’s incipient infinite cluster. We show that our assumptions are satisfied if $G$ is a slab $\mathbb{Z} ^2\times \{0,\ldots ,k\}^{d-2}$ ($d\geq 2$, $k\geq 0$). We also argue that the quasi-multiplicativity assumption should hold for $G=\mathbb{Z} ^d$ when $d<6$, but not when $d>6$.

A completely elementary and self-contained proof of convergence of Gaussian multiplicative chaos is given. The argument shows further that the limiting random measure is nontrivial in the entire subcritical phase $(\gamma < \sqrt{2d} )$ and that the limit is universal (i.e., the limiting measure is independent of the regularisation of the underlying field).

In this paper we give efficient constants in the Fuk-Nagaev inequalities. Next we derive new upper bounds on the weak norms of martingales from our Fuk-Nagaev type inequality.

We first study a model, introduced recently in [4], of a critical branching random walk in an IID random environment on the $d$-dimensional integer lattice. The walker performs critical (0-2) branching at a lattice point if and only if there is no ‘obstacle’ placed there. The obstacles appear at each site with probability $p\in [0,1)$ independently of each other. We also consider a similar model, where the offspring distribution is subcritical.

Let $S_n$ be the event of survival up to time $n$. We show that on a set of full $\mathbb P_p$-measure, as $n\to \infty $, $P^{\omega }(S_n)\sim 2/(qn)$ in the critical case, while this probability is asymptotically stretched exponential in the subcritical case.

Hence, the model exhibits ‘self-averaging’ in the critical case but not in the subcritical one. I.e., in the first case, the asymptotic tail behavior is the same as in a ‘toy model’ where space is removed, while in the second, the spatial survival probability is larger than in the corresponding toy model, suggesting spatial strategies.

A spine decomposition of the branching process along with known results on random walks are utilized.

Consider a Poisson process on $\mathbb{R} $ with intensity $f$ where $0 \leq f(x)<\infty $ for ${x}\geq 0$ and ${f(x)}=0$ for $x<0$. The “points” of the process represent sleeping frogs. In addition, there is one active frog initially located at the origin. At time ${t}=0$ this frog begins performing Brownian motion with leftward drift $\lambda $ (i.e. its motion is a random process of the form ${B}_{t}-\lambda{t} $). Any time an active frog arrives at a point where a sleeping frog is residing, the sleeping frog becomes active and begins performing Brownian motion with leftward drift $\lambda $, independently of the motion of all of the other active frogs. This paper establishes sharp conditions on the intensity function $f$ that determine whether the model is transient (meaning the probability that infinitely many frogs return to the origin is 0), or non-transient (meaning this probability is greater than 0). A discrete model with $\text{Poiss} (f(n))$ sleeping frogs at positive integer points (and where activated frogs perform biased random walks on $\mathbb{Z} $) is also examined. In this case as well, we obtain a similar sharp condition on $f$ corresponding to transience of the model.

A piecewise constant local martingale $M$ with boundedly many jumps is a uniformly integrable martingale if and only if $M_\infty ^-$ is integrable.

We prove that Kendall’s Rank correlation matrix converges to the Marčenko Pastur law, under the assumption that observations are i.i.d random vectors $X_1, \ldots , X_n$ with components that are independent and absolutely continuous with respect to the Lebesgue measure. This is the first result on the empirical spectral distribution of a multivariate $U$-statistic.

The first aim of the present note is to quantify the speed of convergence of a conditioned process toward its $Q$-process under suitable assumptions on the quasi-stationary distribution of the process. Conversely, we prove that, if a conditioned process converges uniformly to a conservative Markov process which is itself ergodic, then it admits a unique quasi-stationary distribution and converges toward it exponentially fast, uniformly in its initial distribution. As an application, we provide a conditional ergodic theorem.

We consider Malliavin calculus based on the Itô chaos decomposition of square integrable random variables on the Lévy space. We show that when a random variable satisfies a certain measurability condition, its differentiability and fractional differentiability can be determined by weighted Lebesgue spaces. The measurability condition is satisfied for all random variables if the underlying Lévy process is a compound Poisson process on a finite time interval.

In this short note, we consider a system of two rotors, one of which interacts with a Langevin heat bath. We show that the system relaxes to its invariant measure (steady state) no faster than a stretched exponential $\exp (-c t^{1/2})$. This indicates that the exponent $1/2$ obtained earlier by the present authors and J.-P. Eckmann for short chains of rotors is optimal.

We find sufficient conditions for convergence of a continuous-time Robbins-Monro type stochastic approximation procedure in infinite dimensional Hilbert spaces in terms of Lyapunova functions, the variational approach to stochastic partial differential equations being used as the main tool.

This note provides a large deviation principle for a class of biorthogonal ensembles. We extend the results of Eichelsbacher, Sommerauer and Stolz to a more general type of interactions. In particular, our result covers the case of the singular values of lower triangular random matrices with independent entries introduced by Cheliotis and implies a variational formulation for the Dykema–Haagerup distribution.

We prove an integration by parts formula for the probability measure on the pinned path space induced by the Semi-classical Riemmanian Brownian Bridge, over a manifold with a pole, followed by a discussion on its equivalence with the Brownian Bridge measure.

We consider the limit behavior of an excited random walk (ERW), i.e., a random walk whose transition probabilities depend on the number of times the walk has visited to the current state. We prove that an ERW being naturally scaled converges in distribution to an excited Brownian motion that satisfies an SDE, where the drift of the unknown process depends on its local time. Similar result was obtained by Raimond and Schapira, their proof was based on the Ray-Knight type theorems. We propose a new method based on a study of the Radon-Nikodym density of the ERW distribution with respect to the distribution of a symmetric random walk.

The aim of this short note is to obtain the existence, uniqueness and moment upper bounds of the solution to a stochastic heat equation with measure initial data, without using the iteration method in [1, 2, 3].

Let $X_{m} = G_{1}\ldots G_{m}$ denote the product of $m$ independent random matrices of size $N \times N$, with each matrix in the product consisting of independent standard Gaussian variables. Denoting by $N_{\mathbb{R} }(m)$ the total number of real eigenvalues of $X_{m}$, we show that for $m$ fixed \[ \mathbb{E} (N_{\mathbb{R} }(m)) = \sqrt{\frac {2Nm}{\pi }} +O(\log (N)), \qquad N \to \infty . \] This generalizes a well-known result of Edelman et al. [10] to all $m>1$. Furthermore, we show that the normalized global density of real eigenvalues converges weakly in expectation to the density of the random variable $|U|^{m}B$ where $U$ is uniform on $[-1,1]$ and $B$ is Bernoulli on $\{-1,1\}$. This proves a conjecture of Forrester and Ipsen [13]. The results are obtained by the asymptotic analysis of a certain Meijer G-function.

The aim of this article is to provide a simple sampling procedure to reconstruct any monotone path from its signature. For every $N$, we sample a lattice path of $N$ steps with weights given by the coefficient of the corresponding word in the signature. We show that these weights on lattice paths satisfy the large deviations principle. In particular, this implies that the probability of picking up a “wrong” path is exponentially small in $N$. The argument relies on a probabilistic interpretation of the signature for monotone paths.

For any Borel probability measure on $\mathbb{R} ^n$, we may define a family of eccentricity tensors. This new notion, together with a tensorization trick, allows us to prove an energy minimization property for rotationally invariant probability measures. We use this theory to give a new proof of the Welch bounds, and to improve upon them for collections of real vectors. In addition, we are able to give elementary proofs for two theorems characterizing probability measures optimizing one-parameter families of energy integrals on the sphere. We are also able to explain why a phase transition occurs for optimizers of these two families.

This paper studies a new type of filtering problem, where the diffusion coefficient of the observation noise is strictly positive only in the interior of the bounded interval where observation takes its values. We derive a Zakai and a Kushner–Stratonovich equation, and prove uniqueness of the measure–valued solution of the Zakai equation.

Recently Lubetzky and Peres showed that simple random walks on a sequence of $d$-regular Ramanujan graphs $G_n=(V_n,E_n)$ of increasing sizes exhibit cutoff in total variation around the diameter lower bound $\frac{d} {d-2}\log _{d-1}|V_n| $. We provide a different argument under the assumption that for some $r(n) \gg 1$ the maximal number of simple cycles in a ball of radius $r(n)$ in $G_n$ is uniformly bounded in $n$.

In this paper, we use partial differential equation (PDE) techniques and probabilistic approaches to study the lower capacity of the ball for the Itô process driven by $G$-Brownian motion ($G$-Itô process). In particular, the lower bound on the lower capacity of certain balls is obtained. As an application, we prove a strict comparison theorem in $G$-expectation framework.

We prove a version of the moving particle lemma for the exclusion process on any finite weighted graph, based on the octopus inequality of Caputo, Liggett, and Richthammer. In light of their proof of Aldous’ spectral gap conjecture, we conjecture that our moving particle lemma is optimal in general. Our result can be applied to graphs which lack translational invariance, including, but not limited to, fractal graphs. An application of our result is the proof of local ergodicity for the exclusion process on a class of weighted graphs, the details of which are reported in a follow-up paper [arXiv:1705.10290].

In this paper we look at the asymptotic number of $r$-caterpillars for $\Lambda $-coalescents which come down from infinity, under a regularly varying assumption. An $r$-caterpillar is a functional of the coalescent process started from $n$ individuals which, roughly speaking, is a block of the coalescent at some time, formed by one line of descend to which $r-1$ singletons have merged one by one. We show that the number of $r$-caterpillars, suitably scaled, converge to an explicit constant as the sample size $n$ goes to $\infty $.

Consider the stochastic heat equation $\dot{u} =\frac 12 u''+\sigma (u)\xi $ on $(0\,,\infty )\times \mathbb{R} $ subject to $u(0)\equiv 1$, where $\sigma :\mathbb{R} \to \mathbb{R} $ is a Lipschitz (local) function that does not vanish at $1$, and $\xi $ denotes space-time white noise. It is well known that $u$ has continuous sample functions [22]; as a result, $\lim _{t\downarrow 0}u(t\,,x)= 1$ almost surely for every $x\in \mathbb{R} $.

The corresponding fluctuations are also known [14, 16, 20]: For every fixed $x\in \mathbb{R} $, $t\mapsto u(t\,,x)$ looks locally like a fixed multiple of fractional Brownian motion (fBm) with index $1/4$. In particular, an application of Fubini’s theorem implies that, on an $x$-set of full Lebesgue measure, the short-time behavior of the peaks of the random function $t\mapsto u(t\,,x)$ are governed by the law of the iterated logarithm for fBm, up to possibly a suitable normalization constant. By contrast, the main result of this paper claims that, on an $x$-set of full Hausdorff dimension, the short-time peaks of $t\mapsto u(t\,,x)$ follow a non-iterated logarithm law, and that those peaks contain a rich multifractal structure a.s.

Large-time variations of these results were predicted in the physics literature a number of years ago and proved very recently in [10, 11]. To the best of our knowledge, the short-time results of the present paper are observed here for the first time.

In the $R$-spread out, $d$-dimensional voter model, each site $x$ of $\mathbb{Z} ^d$ has state (or ‘opinion’) 0 or 1 and, with rate 1, updates its opinion by copying that of some site $y$ chosen uniformly at random among all sites within distance $R$ from $x$. If $d \geq 3$, the set of (extremal) stationary measures of this model is given by a family $\mu _{\alpha , R}$, where $\alpha \in [0,1]$. Configurations sampled from this measure are polynomially correlated fields of 0’s and 1’s in which the density of 1’s is $\alpha $ and the correlation weakens as $R$ becomes larger. We study these configurations from the point of view of nearest neighbor site percolation on $\mathbb{Z} ^d$, focusing on asymptotics as $R \to \infty $. In [RV15], we have shown that, if $R$ is large, there is a critical value $\alpha _c(R)$ such that there is percolation if $\alpha > \alpha _c(R)$ and no percolation if $\alpha < \alpha _c(R)$. Here we prove that, as $R \to \infty $, $\alpha _c(R)$ converges to the critical probability for Bernoulli site percolation on $\mathbb{Z} ^d$. Our proof relies on a new upper bound on the joint occurrence of events under $\mu _{\alpha ,R}$ which is of independent interest.

This note presents families of inequalities for the Gaussian measure of convex sets which extend the recently proven Gaussian correlation inequality in various directions.

We consider the problem of finding the Perron–Frobenius eigenvector of a primitive matrix. Dividing each of the rows of the matrix by the sum of the elements in the row, the resulting new matrix is stochastic. We give a formula for the normalized Perron–Frobenius eigenvector of the original matrix, in terms of a realization of the Markov chain defined by the associated stochastic matrix. This formula is a generalization of the classical formula for the invariant probability measure of a Markov chain.

We provide necessary and sufficient first order geometric conditions for the stochastic invariance of a closed subset of $\mathbb{R} ^d$ with respect to a jump-diffusion under weak regularity assumptions on the coefficients. Our main result extends the recent characterization proved in Abi Jaber, Bouchard and Illand (2016) to jump-diffusions. We also derive an equivalent formulation in the semimartingale framework.

We obtain the optimal proxy variance for the sub-Gaussianity of Beta distribution, thus proving upper bounds recently conjectured by Elder (2016). We provide different proof techniques for the symmetrical (around its mean) case and the non-symmetrical case. The technique in the latter case relies on studying the ordinary differential equation satisfied by the Beta moment-generating function known as the confluent hypergeometric function. As a consequence, we derive the optimal proxy variance for the Dirichlet distribution, which is apparently a novel result. We also provide a new proof of the optimal proxy variance for the Bernoulli distribution, and discuss in this context the proxy variance relation to log-Sobolev inequalities and transport inequalities.

We prove a Donsker-type theorem for vector processes of functionals of correlated Wiener integrals. This includes the case of correlated geometric fractional Brownian motions of arbitrary Hurst parameters in $(0,1)$ driven by the same Brownian motion. Starting from a Donsker-type approximation of Wiener integrals of Volterra type by disturbed binary random walks, the continuous and discrete Wiener chaos representation in terms of Wick calculus is effective. The main result is the compatibility of these continuous and discrete stochastic calculi via these multivariate limit theorems.

In this paper, under the Dobrushin’s uniqueness condition, we obtain explicit estimates of the geometrical convergence rate for the random scan Gibbs sampler in the Wasserstein metric.

We demonstrate a relationship between the heat kernel on a finite weighted Abelian Cayley graph and Gaussian functions on lattices. This can be used to prove a new inequality for the heat kernel on such a graph: when $t \leq t'$, \[ \frac{H_t(u, v)} {H_t(u,u)} \leq \frac{H_{t'}(u, v)} {H_{t'}(u,u)}. \] This was an open problem posed by Regev and Shinkar.

We consider multidimensional quadratic BSDEs with bounded and unbounded terminal conditions. We provide sufficient conditions which guarantee existence and uniqueness of solutions. In particular, these conditions are satisfied if the terminal condition or the dependence in the system are small enough.

We consider a system of annihilating particles where particles start from the points of a Poisson process on either the full-line or positive half-line and move at constant i.i.d. speeds until collision. When two particles collide, they annihilate. We assume the law of speeds to be symmetric. We prove almost sure annihilation of positive-speed particles started from the positive half-line, and existence of a regime of survival of zero-speed particles on the full-line in the case when speeds can only take 3 values. We also state open questions.

Gaussian Volterra processes are processes of the form $(X_{t}:=\int _{\mathbf{T} }k(t,s)\mathrm{d} W_{s})_{t\in \mathbf{T} }$ where $(W_{t})_{t\in \mathbf{T} }$ is Brownian motion, and $k$ is a deterministic Volterra kernel. On integrating the kernel $k$ an information loss may occur, in the sense that the filtration of the Volterra process needs to be enlarged in order to recover the filtration of the driving Brownian motion. In this note we describe such enlargement of filtrations in terms of the Volterra kernel. For kernels of the form $k(t,s)=k(t-s)$ we provide a simple criterion to ensure that the aforementioned filtrations coincide.

The Beta coalescents are stochastic processes modeling the genealogy of a population. They appear as the rescaled limits of the genealogical trees of numerous stochastic population models. In this article, we take interest in the number of blocs at small times in the Beta coalescent. Berestycki, Berestycki and Schweinsberg [2] proved a law of large numbers for this quantity. Recently, Limic and Talarczyk [9] proved that a functional central limit theorem holds as well. We give here a simple proof for an unidimensional version of this result, using a coupling between Beta coalescents and continuous-time branching processes.

Limit theorems for the magnetization of Curie-Weiss model have been studied extensively by Ellis and Newman. To refine these results, Chen, Fang and Shao prove Cramér type moderate deviation theorems for non-critical cases by using Stein method. In this paper, we consider the same question for the remaining case - the critical Curie-Weiss model. By direct and simple arguments based on Laplace method, we provide an explicit formula of the error and deduce a Cramér type result.

For large random matrices $X$ with independent, centered entries but not necessarily identical variances, the eigenvalue density of $XX^*$ is well-approximated by a deterministic measure on $\mathbb{R} $. We show that the density of this measure has only square and cubic-root singularities away from zero. We also extend the bulk local law in [5] to the vicinity of these singularities.

We consider shifts $\Pi _{n,m}$ of a partially exchangeable random partition $\Pi _\infty $ of $\mathbb{N} $ obtained by restricting $\Pi _\infty $ to $\{n+1,n+2,\dots , n+m\}$ and then subtracting $n$ from each element to get a partition of $[m]:= \{1, \ldots , m \}$. We show that for each fixed $m$ the distribution of $\Pi _{n,m}$ converges to the distribution of the restriction to $[m]$ of the exchangeable random partition of $\mathbb{N} $ with the same ranked frequencies as $\Pi _\infty $. As a consequence, the partially exchangeable random partition $\Pi _\infty $ is exchangeable if and only if $\Pi _\infty $ is stationary in the sense that for each fixed $m$ the distribution of $\Pi _{n,m}$ on partitions of $[m]$ is the same for all $n$. We also describe the evolution of the frequencies of a partially exchangeable random partition under the shift transformation. For an exchangeable random partition with proper frequencies, the time reversal of this evolution is the heaps process studied by Donnelly and others.

We study frozen percolation on the (planar) triangular lattice, where connected components stop growing (“freeze”) as soon as their “size” becomes at least $N$, for some parameter $N \geq 1$. The size of a connected component can be measured in several natural ways, and we consider the two particular cases of diameter and volume (i.e. number of sites).

Diameter-frozen and volume-frozen percolation have been studied in previous works ([5, 15] and [6, 4], resp.), and they display radically different behaviors. These works adopt the rule that the boundary of a frozen cluster stays vacant forever, and we investigate the influence of these “boundary rules” in the present paper. We prove the (somewhat surprising) result that they strongly matter in the diameter case, and we discuss briefly the volume case.

We use techniques of Malliavin calculus to study the convergence in law of a family of generalized Hermite processes $Z_\gamma $ with kernels defined by parameters $\gamma $ taking values in a tetrahedral region $\Delta $ of $\mathbb{R} ^q$. We prove that, as $\gamma $ converges to a face of $\Delta $, the process $Z_\gamma $ converges to a compound Gaussian distribution with random variance given by the square of a Hermite process of one lower rank. The convergence in law is shown to be stable. This work generalizes a previous result of Bai and Taqqu, who proved the result in the case $q=2$ and without stability.

One major obstacle in applications of Stein’s method for compound Poisson approximation is the availability of so-called magic factors (bounds on the solution of the Stein equation) with favourable dependence on the parameters of the approximating compound Poisson random variable. In general, the best such bounds have an exponential dependence on these parameters, though in certain situations better bounds are available. In this paper, we extend the region for which well-behaved magic factors are available for compound Poisson approximation in the Kolmogorov metric, allowing useful compound Poisson approximation theorems to be established in some regimes where they were previously unavailable. To illustrate the advantages offered by these new bounds, we consider applications to runs, reliability systems, Poisson mixtures and sums of independent random variables.

The trie-based radix sort algorithm stores pairwise different infinite binary strings in the leaves of a binary tree in a way that the Ulam-Harris coding of each leaf equals a prefix (that is, an initial segment) of the corresponding string, with the prefixes being of minimal length so that they are pairwise different. We investigate the radix sort tree chains – the tree-valued Markov chains that arise when successively storing the finite collections of random infinite binary strings $Z_1,\ldots , Z_n$, $n=1,2,\ldots $ according to the trie-based radix sort algorithm, where the source strings $Z_1, Z_2,\ldots $ are independent and identically distributed. We establish a bijective correspondence between the full Doob–Martin boundary of the radix sort tree chain with a symmetric Bernoulli source (that is, each $Z_k$ is a fair coin-tossing sequence) and the family of radix sort tree chains for which the common distribution of the $Z_k$ is a diffuse probability measure on $\{0,1\}^\infty $. In essence, our result characterizes all the ways that it is possible to condition such a chain of radix sort trees consistently on its behavior “in the large”.

We consider the branching random walk $\{\mathcal R^N_z: z\in V_N\}$ with Gaussian increments indexed over a two-dimensional box $V_N$ of side length $N$, and we study the first passage percolation where each vertex is assigned weight $e^{\gamma \mathcal R^N_z}$ for $\gamma >0$. We show that for $\gamma >0$ sufficiently small but fixed, the expected FPP distance between the left and right boundaries is at most $O(N^{1 - \gamma ^2/10})$.