SOME LIMIT PROPERTIES OF LOCAL TIME FOR RANDOM WALK

Let X,X1,X2 be i. i. d. random variables with EX^2＋δ〈∞ （for some δ〉0）. Consider a one dimensional random walk S={Sn}n≥0, starting from S0 =0. Let ζ＊ （n）=supx∈zζ（x,n）,ζ（x,n） =#{0≤k≤n：[Sk]=x}. A strong approximation of ζ（n） by the local time for Wiener process is presented and the limsup type and liminf-type laws of iterated logarithm of the maximum local time ζ＊（n） are obtained. Furthermore,the precise asymptoties in the law of iterated logarithm of ζ＊（n） is proved.     

Almost-sure central limit theorem for a Markov model of random walk in dynamical random environment

Summary We consider a model of random walk on ℤ^ν, ν≥2, in a dynamical random environment described by a field ξ={ξ _t (x): (t,x)∈ℤ^ν+1}. The random walk transition probabilities are taken as P(X _{t +1}= y|X _t = x,ξ _t =η) =P ₀( y−x)+ c(y−x;η(x)). We assume that the variables {ξ _t (x):(t,x) ∈ℤ^ν+1} are i.i.d., that both P ₀(u) and c(u;s) are finite range in u, and that the random term c(u;·) is small and with zero average. We prove that the C.L.T. holds almost-surely, with the same parameters as for P ₀, for all ν≥2. For ν≥3 there is a finite random (i.e., dependent on ξ) correction to the average of X _t , and there is a corresponding random correction of order to the C.L.T.. For ν≥5 there is a finite random correction to the covariance matrix of X _t and a corresponding correction of order to the C.L.T.. Proofs are based on some new L ^p estimates for a class of functionals of the field. Received: 4 January 1996/In revised form: 26 May 1997     

Central limit theorems for a branching random walk with a random environment in time

We consider a branching random walk with a random environment in time, in which the offspring distribution of a particle of generation n and the distribution of the displacements of its children depend on an environment indexed by the time n. The environment is supposed to be independent and identically distributed. For A ?, let Z_n(A) be the number of particles of generation n located in A. We show central limit theorems for the counting measure Z_n(·) with appropriate normalization.     

Local asymptotics for the time of first return to the origin of transient random walk

We consider a transient random walk on Z^d which is asymptotically stable, without centering, in a sense which allows different norming for each component. The paper is devoted to the asymptotics of the probability of the first return to the origin of such a random walk at time n.     

Local limit theorem for the maximum of asymptotically stable random walks

Let {S _n ; n?≥?0} be an asymptotically stable random walk and let M _n denote it’s maximum in the first n steps. We show that the asymptotic behaviour of local probabilities for M _n can be approximated by the density of the maximum of the corresponding stable process if and only if the renewal mass-function based on ascending ladder heights is regularly varying at infinity. We also give some conditions on the random walk, which guarantee the desired regularity of the renewal mass-function. Finally, we give an example of a random walk, for which the local limit theorem for M _n does not hold.     

Limit theorem for a random walk in a reversible random environment

Institute of Mathematics and Cybernetics, Academy of Sciences of the Lithuanian SSR. Translated from Litovskii Matematicheskii Sbornik (Lietuvos Matematikos Rinkinys), Vol. 29, No. 4, pp. 627–644, October–December, 1989.     

A tauberian theorem for random walk

LetX ₁,X ₂,... be independent random variables, all with the same distribution symmetric about 0; $$S_n = \sum\limits_{i = 1}^n {X_i } $$ It is shown that if for some fixed intervalI, constant 1<a≦2 and slowly varying functionM one has $$\sum\limits_{k = 1}^n {P\{ S_k \in I\} \sim \frac{{n^{1 - 1/\alpha } }}{{M(n)}}} (n \to \infty )$$ then theX _i belong to the domain of attraction of a symmetric stable law.     

On the limit law of a random walk conditioned to reach a high level

We consider a random walk with a negative drift and with a jump distribution which under Cramér’s change of measure belongs to the domain of attraction of a spectrally positive stable law. If conditioned to reach a high level and suitably scaled, this random walk converges in law to a nondecreasing Markov process which can be interpreted as a spectrally positive Lévy process conditioned not to overshoot level 1.     

On the limit behavior of the distribution of the crossing number of a strip by sample paths of a random walk

We find asymptotic representations for the distribution of the crossing number of an expanding strip by sample paths of a random walk in the case when the crossing number is finite with probability 1. The results are obtained under various restrictions on the rate of decrease at infinity for the distribution tails.     

On the first passage time of a simple random walk on a tree

We consider a simple random walk on a tree. Exact expressions are obtained for the expectation and the variance of the first passage time, thereby recovering the known result that these are integers. A relationship of the mean first passage matrix with the distance matrix is established and used to derive a formula for the inverse of the mean first passage matrix.     

Large deviations of the first passage time for a random walk with semiexponentially distributed jumps

Suppose that ξ, ξ(1), ξ(2), ... are independent identically distributed random variables such that ?ξ is semiexponential; i.e., $P( - \xi \geqslant t) = e^{ - t^\beta L(t)} $ is a slowly varying function as t → ∞ possessing some smoothness properties. Let E ξ = 0, D ξ = 1, and S(k) = ξ(1) + ? + ξ(k). Given d > 0, define the first upcrossing time η ₊(u) = inf{k ≥ 1: S(k) + kd > u} at nonnegative level u ≥ 0 of the walk S(k) + kd with positive drift d > 0. We prove that, under general conditions, the following relation is valid for $u = (n) \in \left[ {0, dn - N_n \sqrt n } \right]$ : 0.1 $P(\eta + (u) > n) \sim \frac{{E\eta + (u)}}{n}P(S(n) \leqslant x) as n \to \infty $ , where x = u ? nd < 0 and an arbitrary fixed sequence N _n not exceeding $d\sqrt n $ tends to ∞. The conditions under which we prove (0.1) coincide exactly with the conditions under which the asymptotic behavior of the probability P(S(n) ≤ x) for $x \leqslant - \sqrt n $ was found in [1] (for $x \in \left[ { - \sqrt n ,0} \right]$ it follows from the central limit theorem).     

Approximation of the expectation of the first exit time from an interval for a random walk

We obtain asymptotic expansions for the expectation of the first exit time from an expanding strip for a random walk trajectory. We suppose that the distribution of random walk jumps satisfies the Cramér condition on the existence of an exponential moment.     

Asymptotic expansions for the distribution of the crossing number of a strip by a Markov-modulated random walk

We obtain complete asymptotic expansions for the distribution of the crossing number of a strip in n steps by sample paths of a random walk defined on a finite Markov chain. We assume that the Cramér condition holds for the distribution of jumps and the width of the strip grows with n. The method consists in finding factorization representations of the moment generating functions of the distributions under study, isolating the main terms in the asymptotics of the representations, and inverting those main terms by the modified saddle-point method.     

Hitting time of a half-line by two-dimensional random walk

We consider the probability that a two-dimensional random walk starting from the origin never returns to the half-line {(x₁,x₂)|x₁0,x₂=0} before time n. It is proved that for aperiodic random walk with mean zero and finite 2+(>2)-th absolute moment, this probability times n^1/4 converges to some positive constant c^* as . We show that c^* is expressed by using the characteristic function of the increment of the random walk. For the simple random walk, this expression gives Mathematics Subject Classification (2000):60G50, 60E10