Asymptotics for Random Walks with Dependent Heavy-Tailed Increments

We consider a random walk {S _n} with dependent heavy-tailed increments and negative drift. We study the asymptotics for the tail probability P{sup _n S _n >x} as x. If the increments of {S _n} are independent then the exact asymptotic behavior of P{sup _n S _n >x} is well known. We investigate the case in which the increments are given as a one-sided asymptotically stationary linear process. The tail behavior of sup _n S _n turns out to depend heavily on the coefficients of this linear process.     

On Large Deviations of Multivariate Heavy-Tailed Random Walks

Let {S _n ;n=1,2,…} be a random walk in R ^d and E(S ₁)=(μ ₁,…,μ _d ). Let a _j >μ _j for j=1,…,d and A=(a ₁,∞)×⋅⋅⋅×(a _d ,∞). We are interested in the probability P(S _n /n∈A) for large n in the case where the components of S ₁ are heavy tailed. An objective is to associate an exact power with the aforementioned probability. We also derive sharper asymptotic bounds for the probability and show that in essence, the occurrence of the event {S _n /n∈A} is caused by large single increments of the components in a specific way.      

Estimates for the Tail Probability of the Supremum of a Random Walk with Independent Increments

The authors investigate the tail probability of the supremum of a random walk with independent increments and obtain some equivalent assertions in the case that the increments are independent and identically distributed random variables with O-subexponential integrated distributions. A uniform upper bound is derived for the distribution of the supremum of a random walk with independent but non-identically distributed increments, whose tail distributions are dominated by a common tail distribution with an O-subexponential integrated distribution.     

Boundaries and Harmonic Functions for Random Walks with Random Transition Probabilities

The usual random walk on a group (homogeneous both in time and in space) is determined by a probability measure on the group. In a random walk with random transition probabilities this single measure is replaced with a stationary sequence of measures, so that the resulting (random) Markov chains are still space homogeneous, but no longer time homogeneous. We study various notions of measure theoretical boundaries associated with this model and establish an analogue of the Poisson formula for (random) bounded harmonic functions. Under natural conditions on transition probabilities we identify these boundaries for several classes of groups with hyperbolic properties and prove the boundary triviality (i.e., the absence of non-constant random bounded harmonic functions) for groups of subexponential growth, in particular, for nilpotent groups.     

Asymptotics for the Tail Probability of Random Sums with a Heavy-Tailed Random Number and Extended Negatively Dependent Summands

Let {X,X_k:k≥1} be a sequence of extended negatively dependent random variables with a common distribution F satisfying EX 0.Let r be a nonnegative integer-valued random variable,independent of {X,X_k:k≥1}.In this paper,the authors obtain the necessary and sufficient conditions for the random sums S_r =(?)X_n to have a consistently varying tail when the random number t has a heavier tail than the summands,i.e.,(P(Xx))/(P(r x))→0as x→∞     

On subexponential tails for the maxima of negatively driven compound renewal and Lévy processes

We study subexponential tail asymptotics for the distribution of the maximum $M_t?{sup}_{u\in [0,t]}{X}_u$ of a process $X_t$ with negative drift for the entire range of $t>0$. We consider compound renewal processes with linear drift and Lévy processes. For both processes we also formulate and prove the principle of a single big jump for their maxima. The class of compound renewal processes with drift particularly includes the Cramér–Lundberg renewal risk process.     

随机环境中有界跳幅随机游动常返性暂留性的另一证明

假定环境是平稳遍历的,对具有有限跳幅的随机环境中的随机游动,该文给出了其常返性暂留性的另一证明.Bremont(2002)的文章中,通过计算逃逸概率的方法给出了证明,而该文的证明采用了鞅收敛定理的方法.     

Random walks in a strongly sparse random environment

The integer points (sites) of the real line are marked by the positions of a standard random walk with positive integer jumps. We say that the set of marked sites is weakly, moderately or strongly sparse depending on whether the jumps of the random walk are supported by a bounded set, have finite or infinite mean, respectively. Focussing on the case of strong sparsity and assuming additionally that the distribution tail of the jumps is regularly varying at infinity we consider a nearest neighbor random walk on the set of integers having jumps $\pm 1$ with probability $1∕2$ at every nonmarked site, whereas a random drift is imposed at every marked site. We prove new distributional limit theorems for the so defined random walk in a strongly sparse random environment, thereby complementing results obtained recently in Buraczewski et al. (2019) for the case of moderate sparsity and in Matzavinos et al. (2016) for the case of weak sparsity. While the random walk in a strongly sparse random environment exhibits either the diffusive scaling inherent to a simple symmetric random walk or a wide range of subdiffusive scalings, the corresponding limit distributions are non-stable.     

Discrete and Continuous Time Extremes of Gaussian Processes

Joint distribution of maximums of a Gaussian stationary process in continuous time and in uniform grid on the real axis is studied. When the grid is sufficiently sparse, maxima are asymptotically independent. When the grid is sufficiently tight, the maximums asymptotically coincide. In the boundary case which we call Pickands grid, the limit distribution is non-degenerate. It calculated in terms of a Pickands type constant.AMS 2000 Subject Classification. Primary—60G70, Secondary—60G15*Partially supported by the Scientific foundation of the Netherlands, RFFI grant 0401-00700 and grant DFG 436 RUS 113/722.     

On Some Expectation and Derivative Operators Related to Integral Representations of Random Variables with Respect to a PII Process

Given a process with independent increments X (not necessarily a martingale) and a large class of square integrable r.v. H = f(X _T ), f being the Fourier transform of a finite measure μ, we provide a direct expression for Kunita-Watanabe and Föllmer-Schweizer decompositions of H. The representation is expressed by means of two significant maps: the expectation and derivative operators related to the characteristics of X. We also evaluate the expression for the variance optimal error when hedging the claim H with underlying process X. Those questions are motivated by finding the solution of the celebrated problem of global and local quadratic risk minimization in mathematical finance.     

Nonparametric estimation and testing time-homogeneity for processes with independent increments

We consider a nonparametric estimation problem for the Lévy measure of time-inhomogeneous process with independent increments. We derive the functional asymptotic normality and efficiency, in an ℓ^∞-space, of generalized Nelson–Aalen estimators. Also we propose some asymptotically distribution free tests for time-homogeneity of the Lévy measure. Our result is a fruit of the empirical process theory and the martingale theory.