Limit Theorems For Random Walks Research Articles

Предельные теоремы для случайного блуждания в полуплоскости с перескоком границы

Предельные теоремы для случайного блуждания в полуплоскости с перескоком границы

Dispersion and Limit Theorems for Random Walks Associated with Hypergeometric Functions of Type BC

The spherical functions of the non-compact Grassmann manifolds $$G_{p,q}({\mathbb {F}})=G/K$$ over the (skew-)fields $${\mathbb {F}}={\mathbb {R}}, {\mathbb {C}}, {\mathbb {H}}$$ with rank $$q\ge 1$$ and dimension parameter $$p>q$$ can be described as Heckman–Opdam hypergeometric functions of type BC, where the double coset space G / / K is identified with the Weyl chamber $$ C_q^B\subset {\mathbb {R}}^q$$ of type B. The corresponding product formulas and Harish-Chandra integral representations were recently written down by M. Rosler and the author in an explicit way such that both formulas can be extended analytically to all real parameters $$p\in [2q-1,\infty [$$ , and that associated commutative convolution structures $$*_p$$ on $$C_q^B$$ exist. In this paper, we study the associated moment functions and the dispersion of probability measures on $$C_q^B$$ with the aid of this generalized integral representation. This leads to strong laws of large numbers and central limit theorems for associated time-homogeneous random walks on $$(C_q^B, *_p)$$ where the moment functions and the dispersion appear in order to determine drift vectors and covariance matrices of these limit laws explicitly. For integers p, all results have interpretations for G-invariant random walks on the Grassmannians G / K. Besides the BC-cases, we also study the spaces $$GL(q,{\mathbb {F}})/U(q,{\mathbb {F}})$$ , which are related to Weyl chambers of type A, and for which corresponding results hold. For the rank-one-case $$q=1$$ , the results of this paper are well known in the context of Jacobi-type hypergroups on $$[0,\infty [$$ .

Central limit theorem on hyperbolic groups

We prove a central limit theorem for random walks with finite variance on Gromov hyperbolic groups.

A quenched functional central limit theorem for random walks in random environments under [formula omitted

We prove a quenched central limit theorem for random walks in i.i.d. weakly elliptic random environments in the ballistic regime. Such theorems have been proved recently by Rassoul-Agha and Seppäläinen in Rassoul-Agha and Seppäläinen (2009) and Berger and Zeitouni in Berger and Zeitouni (2008) under the assumption of large finite moments for the regeneration time. In this paper, with the extra (T)γ condition of Sznitman we reduce the moment condition to E(τ2(lnτ)1+m)<+∞ for m>1+1/γ, which allows the inclusion of new non-uniformly elliptic examples such as Dirichlet random environments.

A Central Limit Theorem for Random Walks on the Dual of a Compact Grassmannian

We consider compact Grassmann manifolds $G/K$ over the real, complex or quaternionic numbers whose spherical functions are Heckman-Opdam polynomials of type $BC$. From an explicit integral representation of these polynomials we deduce a sharp Mehler-Heine formula, that is an approximation of the Heckman-Opdam polynomials in terms of Bessel functions, with a precise estimate on the error term. This result is used to derive a central limit theorem for random walks on the semi-lattice parametrizing the dual of $G/K$, which are constructed by successive decompositions of tensor powers of spherical representations of $G$. The limit is the distribution of a Laguerre ensemble in random matrix theory. Most results of this paper are established for a larger continuous set of multiplicity parameters beyond the group cases.

Regular sequences and random walks in affine buildings

We dene and characterise regular sequences in ane buildings, thereby giving the padic analogue of the fundamental work of Kaimanovich on regular sequences in symmetric spaces. As applications we prove limit theorems for random walks on ane buildings and their automorphism groups.

Limit theorems for random walks that avoid bounded sets, with applications to the largest gap problem

Consider a centred random walk in dimension one with a positive finite variance σ2, and let τB be the hitting time for a bounded Borel set B with a non-empty interior. We prove the asymptotic Px(τB>n)∼2/πσ−1VB(x)n−1/2 and provide an explicit formula for the limit VB as a function of the initial position x of the walk. We also give a functional limit theorem for the walk conditioned to avoid B by the time n. As a main application, we consider the case that B is an interval and study the size of the largest gap Gn (maximal spacing) within the range of the walk by the time n. We prove a limit theorem for Gn, which is shown to be of the constant order, and describe its limit distribution. In addition, we prove an analogous result for the number of non-visited sites within the range of an integer-valued random walk.

Local limit theorem for symmetric random walks in Gromov-hyperbolic groups

Completing a strategy of Gouezel and Lalley, we prove a local limit theorem for the random walk generated by any symmetric finitely supported probability measure on a non-elementary Gromov-hyperbolic group: denoting by $R$ the inverse of the spectral radius of the random walk, the probability to return to the identity at time $n$ behaves like $C R^{-n}n^{-3/2}$. An important step in the proof is to extend Ancona's results on the Martin boundary up to the spectral radius: we show that the Martin boundary for $R$-harmonic functions coincides with the geometric boundary of the group. In an appendix, we explain how the symmetry assumption of the measure can be dispensed with for surface groups.

Limit theorems for random walks under irregular conductance

For a general one-dimensional random walk with state-dependent transition probabilities, we present weak limits of the empirical moments of conductance along the path of the random walk. In particular we obtain remarkably simple quenched convergences under random conductance model.

Local Limit Theorems for random walks in a 1D random environment

We consider random walks in a one-dimensional i.i.d. random environment with jumps to the nearest neighbours. For almost all environments, we prove a quenched Local Limit Theorem (LLT) for the position of the walk if the diffusivity condition is satisfied. As a corollary, we obtain the annealed version of the LLT and a new proof of the theorem of Lalley which states that the distribution of the environment viewed from the particle has a limit for a. e. environment.

Limit theorems for random walks on a strip in subdiffusive regimes

We study the asymptotic behaviour of occupation times of a transient random walk (RW) in a quenched random environment (RE) on a strip in a subdiffusive regime. The asymptotic behaviour of hitting times, which is a more traditional object of study, is exactly the same. As a particular case, we solve a long standing problem of describing the asymptotic behaviour of a RW with bounded jumps on a one-dimensional lattice. Our technique results from the development of ideas from our previous work (Dolgopyat and Goldsheid 2012 Commun. Math. Phys. 315 241–77) on the simple RWs in RE and those used in Bolthausen and Goldsheid (2000 Commun. Math. Phys. 214 429–47; 2008 Commun. Math. Phys. 278 253–88) and Goldsheid (2008 Probab. Theory Relat. Fields 141 471–511) for the study of random walks on a strip.

A local limit theorem for random walks in balanced environments

Central limit theorems for random walks in quenched random environments have attracted plenty of attention in the past years. More recently still, finer local limit theorems - yielding a Gaussian density multiplied by a highly oscillatory modulating factor - for such models have been obtained. In the one-dimensional nearest-neighbor case with i.i.d. transition probabilities, local limits of uniformly elliptic ballistic walks are now well understood. We complete the picture by proving a similar result for the only recurrent case, namely the balanced one, in which such a walk is diffusive. The method of proof is, out of necessity, entirely different from the ballistic case.

Quenched central limit theorems for random walks in random scenery

Random walks in random scenery are processes defined by Zn:=∑k=1nωSk where S:=(Sk,k≥0) is a random walk evolving in Zd and ω:=(ωx,x∈Zd) is a sequence of i.i.d. real random variables. Under suitable assumptions on the random walk S and the random scenery ω, almost surely with respect to ω, the correctly renormalized sequence (Zn)n≥1 is proved to converge in distribution to a centered Gaussian law with explicit variance.

Conditioned random walks and Lévy processes

Let X1, X2, … be independent, identically distributed, zero mean random variables with (−α)-regularly varying tails, α>1. For S n = ∑ i = 1 n X i , it is known that under these distributional assumptions, ℙ(Sn>x)∼ nℙ(X1>x) as x→∞, uniformly for x⩾cn for any constant c>0. Here, we show that the process Mn=max {Si−iμ:i⩽n}, for any constant μ⩾0, behaves in a similar manner. This allows us to generalize Durrett's results [‘Conditioned limit theorems for random walks with negative drift’, Z. Wahrscheinlichkeitstheorie verw. Gebiete 52 (1980) 277–287], by showing that, without any further assumptions, both (n−1S[nt], 0⩽t⩽1|Sn>na) and (n−1 S[nt], 0⩽t⩽1|Mn>na) for any constant a>0 converge weakly to a simple process consisting of a single ‘large jump’. We show that similar results hold for general Lévy processes, extending the work of Konstantopoulos and Richardson [‘Conditional limit theorems for spectrally positive Lévy processes’, Adv. in Appl. Probab. 34 (2002) 158–178] who dealt with the special case of spectrally positive processes.

Limit Theorems for Random Walk under the Assumption of Maxima Large Deviation

We obtain an asymptotic as $n,m\rightarrow\infty$ of large $(\ge \theta n)$ deviations for the Shepp statistic, $\rho_{m,n}:=\max_{k\le m}\max_{i\le n} (S_{k+i}-S_k)$, in explicit form, concluding with a constant, for the random walk $S_n$ with steps obeying the Cramér condition. We introduce limit theorems for $\rho_{m,n}$ and $\tau_n(\theta):=\min\{m:\max_{k\le n}(S_{k+m}-S_{m})\ge \theta n\}$, as well as functional limit theorems under the assumption of large deviation of the Shepp statistic.

A Central Limit Theorem for Random Walk in a Random Environment on a Marked Galton-Watson Tree.

Models of random walks in a random environment were introduced at first by Chernoff in 1967 in order to study biological mechanisms. The original model has been intensively studied since then and is now well understood. In parallel, similar models of random processes in a random environment have been studied. In this article we focus on a model of random walk on random marked trees, following a model introduced by R. Lyons and R. Pemantle (1992). Our point of view is a bit different yet, as we consider a very general way of constructing random trees with random transition probabilities on them. We prove an analogue of R. Lyons and R. Pemantle's recurrence criterion in this setting, and we study precisely the asymptotic behavior, under restrictive assumptions. Our last result is a generalization of a result of Y. Peres and O. Zeitouni (2006) concerning biased random walks on Galton-Watson trees.

A local limit theorem for random walks in random scenery and on randomly oriented lattices

Random walks in random scenery are processes defined by $Z_n:=\sum_{k=1}^n\xi_{X_1+...+X_k}$, where $(X_k,k\ge 1)$ and $(\xi_y,y\in\mathbb Z)$ are two independent sequences of i.i.d. random variables. We assume here that their distributions belong to the normal domain of attraction of stable laws with index $\alpha\in (0,2]$ and $\beta\in (0,2]$ respectively. These processes were first studied by H. Kesten and F. Spitzer, who proved the convergence in distribution when $\alpha\neq 1$ and as $n\to \infty$, of $n^{-\delta}Z_n$, for some suitable $\delta>0$ depending on $\alpha$ and $\beta$. Here we are interested in the convergence, as $n\to \infty$, of $n^\delta{\mathbb P}(Z_n=\lfloor n^{\delta} x\rfloor)$, when $x\in \RR$ is fixed. We also consider the case of random walks on randomly oriented lattices for which we obtain similar results.

Almost sure absolute continuity of Bernoulli convolutions

La continuite absolue, presque surement, est demontree dans une classe de convolutions de Bernoulli symetrique, etendant un resultat de Peres et Solomyak.

Central limit theorem and recurrence for random walks in bistochastic random environments

We prove the annealed central limit theorem for random walks in bistochastic random environments on Zd with zero-local drift. The proof is based on a “dynamicist’s interpretation” of the system and requires a much weaker condition than the customary uniform ellipticity. Moreover, recurrence is derived for d≤2.

Bessel Convolutions on Matrix Cones: Algebraic Properties and Random Walks

Bessel-type convolution algebras of measures on the matrix cones of positive semidefinite q×q-matrices over ℝ,ℂ,ℍ were introduced recently by Rösler. These convolutions depend on a continuous parameter, generate commutative hypergroups, and have Bessel functions of matrix argument as characters. In this paper, we study the algebraic structure of these hypergroups. In particular, the subhypergroups, quotients, and automorphisms are classified. The algebraic properties are partially related to the properties of random walks on these matrix Bessel hypergroups. In particular, known properties of Wishart distributions, which form Gaussian convolution semigroups on these hypergroups, are put into a new light. Moreover, limit theorems for random walks are presented. In particular, we obtain strong laws of large numbers and a central limit theorem with Wishart distributions as limits.