Some properties of random walk paths

For random walk on the d-dimensional integer lattice we consider again the problem of deciding when a set is recurrent, that is visited infinitely often with probability one by the random walk in question. Some special cases are considered, among them the following: for d = 2, what sequences (n_j) have the property that with probability one the random walk visits the origin for infinitely many n_j. A related problem, which is however not a special case of the recurrence problem, is to decide for what sequences (n_j) the states visited by the random walk at times n_j are all distinct, with only a finite number of exceptions. This problem is dealt with in the final part of the paper.     

A random walk in higher arithmetic

A random walk technique is applied to linear differential equations.     

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.     

Some properties of random fields related to stochastic integrals with respect to strong martingales

We investigate the properties of random fields related to stochastic intergals with respect to a two-parameter strong square-integrable martingale in the case where the martingale and the integrands depend on the limits of integration. We prove inequalities for the moments of the uniform norm and the modulus of continuity of trajectories of modifications of such fields. Bibliography: 8 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 320, 2004, pp. 80–96.     

Some arithmetic properties of recurrence sequences

Translated from Matematicheskie Zametki, Vol. 47, No. 6, pp. 124–131, June, 1990.     

Some properties of set-valued stochastic integrals

The present paper is devoted to properties of set-valued stochastic integrals defined as some special type of set-valued random variables. In particular, it is shown that if the probability base is separable or probability measure is nonatomic then defined set-valued stochastic integrals can be represented by a sequence of Itô?s integrals of nonanticipative selectors of integrated set-valued processes. Immediately from Michael?s continuous selection theorem it follows that the indefinite set-valued stochastic integrals possess some continuous selections. The problem of integrably boundedness of set-valued stochastic integrals is considered. Some remarks dealing with stochastic differential inclusions are also given.     

Reinforced random walk

Summary Leta _i,i1, be a sequence of nonnegative numbers. Difine a nearest neighbor random motion =X ₀,X ₁, ... on the integers as follows. Initially the weight of each interval (i, i+1), i an integer, equals 1. If at timen an interval (i, i+1) has been crossed exactlyk times by the motion, its weight is . Given (X ₀,X ₁, ...,X _n)=(i₀, i₁, ..., i_n), the probability thatX _n+1 isi _n–1 ori _n+1 is proportional to the weights at timen of the intervals (i _n–1,i _n) and (i _n,ii_n+1). We prove that either visits all integers infinitely often a.s. or visits a finite number of integers, eventually oscillating between two adjacent integers, a.s., and that X _n /n=0 a.s. For much more general reinforcement schemes we proveP ( visits all integers infinitely often)+P ( has finite range)=1.Supported by a National Science Foundation Grant     

Vertex-reinforced random walk

Summary This paper considers a class of non-Markovian discrete-time random processes on a finite state space {1,...,d}. The transition probabilities at each time are influenced by the number of times each state has been visited and by a fixed a priori likelihood matrix,R, which is real, symmetric and nonnegative. LetS _i (n) keep track of the number of visits to statei up to timen, and form the fractional occupation vector,V(n), where . It is shown thatV(n) converges to to a set of critical points for the quadratic formH with matrixR, and that under nondegeneracy conditions onR, there is a finite set of points such that with probability one,V(n)p for somep in the set. There may be more than onep in this set for whichP(V(n)p)>0. On the other handP(V(n)p)=0 wheneverp fails in a strong enough sense to be maximum forH.This research was supported by an NSF graduate fellowship and by an NSF postdoctoral fellowship     

Spectral properties of evolutionary operators in branching random walk models

We introduce a model of continuous-time branching random walk on a finite-dimensional integer lattice with finitely many branching sources of three types and study the spectral properties of the operator describing the evolution of the mean numbers of particles both at an arbitrary source and on the entire lattice. For the leading positive eigenvalue of the operator, we obtain existence conditions determining exponential growth in the number of particles in this model.     

Some properties of random evolutions

We study asymptotic properties of matrix-valued random evolutions and consider an example of evolutions of this type.Deceased.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 47, No. 10, pp. 1333–1337, October, 1995.     

集值映射Pettis-Aumann积分的一些性质

对集值映射的Pettis-Aumann型积分的性质作了讨论,给出了Pettis-Aumann积分型积分的Fatou引理形式以及其它一些收敛定理,探讨了带参数集值映射的积分性质。     

Some properties of complex matrix-variate generalized Dirichlet integrals

Dirichlet integrals and the associated Dirichlet statistical densities are widely used in various areas. Generalizations of Dirichlet integrals and Dirichlet models to matrix-variate cases, when the matrices are real symmetric positive definite or hermitian positive definite, are available [4]. Real scalar variables case of the Dirichlet models are generalized in various directions. One such generalization of the type-2 or inverted Dirichlet is looked into in this article. Matrix-variate analogue, when the matrices are hermitian positive definite, are worked out along with some properties which are mathematically and statistically interesting.