Tail expansions for the distribution of the maximum of a random walk with negative drift and regularly varying increments

Markov chains with heavy-tailed increments and asymptotically zero drift

We study the recurrence/transience phase transition for Markov chains on $\mathbb{R}_+$, $\mathbb{R}$, and $\mathbb{R}^2$ whose increments have heavy tails with exponent in $(1,2)$ and asymptotically zero mean. This is the infinite-variance analogue of the classical Lamperti problem. On $\mathbb{R}_+$, for example, we show that if the tail of the positive increments is about $c y^{-α}$ for an exponent $α\in (1,2)$ and if the drift at $x$ is about $b x^{-γ}$, then the critical regime has $γ= α-1$ and recurrence/transience is determined by the sign of $b + cπ\textrm{cosec} (πα)$. On $\mathbb{R}$…

Efficient Rare-Event Simulation for Multiple Jump Events in Regularly Varying Random Walks and Compound Poisson Processes

We propose a class of strongly efficient rare event simulation estimators for random walks and compound Poisson processes with a regularly varying increment/jump-size distribution in a general large deviations regime. Our estimator is based on an importance sampling strategy that hinges on the heavy-tailed sample path large deviations result recently established in Rhee, Blanchet, and Zwart (2016). The new estimators are straightforward to implement and can be used to systematically evaluate the probability of a wide range of rare events with bounded relative error. They are "universal" in the…

Tail asymptotics for the area under the excursion of a random walk with heavy-tailed increments

We study tail behaviour of the distribution of the area under the positive excursion of a random walk which has negative drift and heavy-tailed increments. We determine the asymptotics for tail probabilities for the area.

The cluster index of regularly varying sequences with applications to limit theory for functions of multivariate Markov chains

We introduce the cluster index of a multivariate regularly varying stationary sequence and characterize the index in terms of the spectral tail process. This index plays a major role in limit theory for partial sums of regularly varying sequences. We illustrate the use of the cluster index by characterizing infinite variance stable limit distributions and precise large deviation results for sums of multivariate functions acting on a stationary Markov chain under a drift condition.

Moment conditions for a sequence with negative drift to be uniformly bounded in L^r

Suppose a sequence of random variables {X_n} has negative drift when above a certain threshold and has increments bounded in L^p. When p>2 this implies that EX_n is bounded above by a constant independent of n and the particular sequence {X_n}. When p=<2 there are counterexamples showing this does not hold. In general, increments bounded in L^p lead to a uniform L^r bound on X_n^+ for any r<p-1, but not for r>=p-1. These results are motivated by questions about stability of queueing networks.

Precise large deviations for dependent regularly varying sequences

We study a precise large deviation principle for a stationary regularly varying sequence of random variables. This principle extends the classical results of A.V. Nagaev (1969) and S.V. Nagaev (1979) for iid regularly varying sequences. The proof uses an idea of Jakubowski (1993,1997) in the context of centra limit theorems with infinite variance stable limits. We illustrate the principle for \sv\ models, functions of a Markov chain satisfying a polynomial drift condition and solutions of linear and non-linear stochastic recurrence equations.

Anomalous recurrence properties of many-dimensional zero-drift random walks

Famously, a $d$-dimensional, spatially homogeneous random walk whose increments are non-degenerate, have finite second moments, and have zero mean is recurrent if $d \in \{1,2\}$ but transient if $d \geq 3$. Once spatial homogeneity is relaxed, this is no longer true. We study a family of zero-drift spatially non-homogeneous random walks (Markov processes) whose increment covariance matrix is asymptotically constant along rays from the origin, and which, in any ambient dimension $d \geq 2$, can be adjusted so that the walk is either transient or recurrent. Natural examples are provided by rand…

Veraverbeke's theorem at large - On the maximum of some processes with negative drift and heavy tail innovations

Veraverbeke's (1977) theorem relates the tail of the distribution of the supremum of a random walk with negative drift to the tail of the distribution of its increments, or equivalently, the probability that a centered random walk with heavy-tail increments hits a moving linear boundary. We study similar problems for more general processes. In particular, we derive an analogue of Veraverbeke's theorem for fractional integrated ARMA models without prehistoric influence, when the innovations have regularly varying tails. Furthermore, we prove some limit theorems for the trajectory of the process…

A central limit theorem for two-dimensional random walks in a cone

We prove that a planar random walk with bounded increments and mean zero which is conditioned to stay in a cone converges weakly to the corresponding Brownian meander if and only if the tail distribution of the exit time from the cone is regularly varying. This condition is satisfied in many natural examples.

Upper bounds for the maximum of a random walk with negative drift

Consider a random walk $S_n=\sum_{i=0}^n X_i$ with negative drift. This paper deals with upper bounds for the maximum $M=\max_{n\ge 1}S_n$ of this random walk in different settings of power moment existences. As it is usual for deriving upper bounds, we truncate summands. Therefore we use an approach of splitting the time axis by stopping times into intervals of random but finite length and then choose a level of truncation on each interval. Hereby we can reduce the problem of finding upper bounds for $M$ to the problem of finding upper bounds for $M_τ=\max_{n\le τ}S_n$. In addition we test ou…

Bounds on the running maximum of a random walk with small drift

We derive a lower bound for the probability that a random walk with i.i.d.\ increments and small negative drift $μ$ exceeds the value $x>0$ by time $N$. When the moment generating functions are bounded in an interval around the origin, this probability can be bounded below by $1-O(x|μ| \log N)$. The approach is elementary and does not use strong approximation theorems.

Convex hulls of planar random walks

For the perimeter length $L_n$ and the area $A_n$ of the convex hull of the first $n$ steps of a planar random walk, this thesis study $n \to \infty$ mean and variance asymptotics and establish distributional limits. The results apply to random walks both with drift (the mean of random walk increments) and with no drift under mild moments assumptions on the increments. Assuming increments of the random walk have finite second moment and non-zero mean, Snyder and Steele showed that $n^{-1} L_n$ converges almost surely to a deterministic limit, and proved an upper bound on the variance Var$[ L_n…

具有对数正态增量的 Lévy 过程与随机游走的样本路径大偏差

Sample-Path Large Deviations for Lévy Processes and Random Walks with Lognormal Increments

本文研究对数正态增量 Lévy 过程与随机游走的样本路径大偏差，建立一维及独立坐标多维过程的 extended LDP，并构造反例说明其在 J_1 与 M_1' 拓扑下不能加强为标准 LDP。

Asymptotic expansions for functions of the increments of certain Gaussian processes

Let $G=\{G(x),x\ge 0\}$ be a mean zero Gaussian process with stationary increments and set $σ^2(|x-y|)= E(G(x)-G(y))^2$. Let $f$ be a function with $Ef^{2}(η)<\ff$, where $η=N(0,1)$. When $σ^2$ is regularly varying at zero and \[ \lim_{h\to 0}{h^2\over σ^2(h)}= 0\qquad {and}\qquad \lim_{h\to 0}{σ^2(h)\over h}= 0 \quad {but} \quad ({d^{2}\over ds^2}σ^2(s))^{j_0} \] is locally integrable for some integer $j_0\ge 1$, and satisfies some additional regularity conditions, \bea && \int_a^bf(\frac{G(x+h)-G(x)}{σ(h)}) dx \label{abst}\nn &&\qquad = \sum_{j=0}^{j_0} (h/σ(h))^{j} {E(H_{j}(η) f(η))\over\sq…

关于带漂移的双参数时变泊松随机场

On Two Parameter Time-Changed Poisson Random Fields with Drifts

本文研究了二元 Lévy 过程与逆子ordinator的复合，推导了其色散与自相关矩阵，并引入了带漂移的双参数时变 Poisson 随机场及其分布的分数阶微分方程。

The tail of the length of an excursion in a trap of random size

Consider a random walk with a drift to the right on $\{0,\ldots,k\}$ where $k$ is random and geometrically distributed. We show that the tail $P[T>t]$ of the length $T$ of an excursion from $0$ decreases up to constants like $t^{-\varrho}$ for some $\varrho>0$ but is not regularly varying. We compute the oscillations of $t^\varrho\,P[T>t]$ as $t\to\infty$ explicitly.

The principle of a single big jump: discrete and continuous time modulated random walks with heavy-tailed increments

We consider a modulated process S which, conditional on a background process X, has independent increments. Assuming that S drifts to -infinity and that its increments (jumps) are heavy-tailed (in a sense made precise in the paper), we exhibit natural conditions under which the asymptotics of the tail distribution of the overall maximum of S can be computed. We present results in discrete and in continuous time. In particular, in the absence of modulation, the process S in continuous time reduces to a Levy process with heavy-tailed Levy measure. A central point of the paper is that we make ful…

正象限中带漂移的随机游走

Random walks with drift in the positive quadrant

研究正象限中带漂移的随机游走，构造正调和函数并分析退出时间的渐进行为，同时应用局部极限定理研究特定格子游走的渐进行为。

Angular asymptotics for multi-dimensional non-homogeneous random walks with asymptotically zero drift

We study the first exit time $τ$ from an arbitrary cone with apex at the origin by a non-homogeneous random walk (Markov chain) on $\Z^d$ ($d \geq 2$) with mean drift that is asymptotically zero. Specifically, if the mean drift at $\bx \in \Z^d$ is of magnitude $O(\| \bx\|^{-1})$, we show that $τ<\infty$ a.s. for any cone. On the other hand, for an appropriate drift field with mean drifts of magnitude $\| \bx\|^{-β}$, $β\in (0,1)$, we prove that our random walk has a limiting (random) direction and so eventually remains in an arbitrarily narrow cone. The conditions imposed on the random walk a…

加性布朗噪声下随机微分方程局部线性化方法的误差分布

Error Distribution of the Local Linearization Method for Stochastic Differential Equations with Additive Brownian Noise

该研究证明加性布朗噪声随机微分方程局部线性化格式离散误差的泛函稳定极限定理，明确误差结构与归一化因子，给出收敛条件及极限过程的驱动项与系数构成。

Convex hulls of planar random walks with drift

Denote by $L_n$ the length of the perimeter of the convex hull of $n$ steps of a planar random walk whose increments have finite second moment and non-zero mean. Snyder and Steele showed that $n^{-1} L_n$ converges almost surely to a deterministic limit, and proved an upper bound on the variance $Var [ L_n] = O(n)$. We show that $n^{-1} Var [L_n]$ converges and give a simple expression for the limit, which is non-zero for walks outside a certain degenerate class. This answers a question of Snyder and Steele. Furthermore, we prove a central limit theorem for $L_n$ in the non-degenerate case.

Ordered random walks with heavy tails

This note continues paper of Denisov and Wachtel (2010), where we have constructed a $k$-dimensional random walk conditioned to stay in the Weyl chamber of type $A$. The construction was done under the assumption that the original random walk has $k-1$ moments. In this note we continue the study of killed random walks in the Weyl chamber, and assume that the tail of increments is regularly varying of index $α<k-1$. It appears that the asymptotic behaviour of random walks is different in this case. We determine the asymptotic behaviour of the exit time, and, using thisinformation, construct a c…

Cutpoints of non-homogeneous random walks

We give conditions under which near-critical stochastic processes on the half-line have infinitely many or finitely many cutpoints, generalizing existing results on nearest-neighbour random walks to adapted processes with bounded increments satisfying appropriate conditional increment moments conditions. We apply one of these results to deduce that a class of transient zero-drift Markov chains in $\mathbb{R}^d$, $d \geq 2$, possess infinitely many separating annuli, generalizing previous results on spatially homogeneous random walks.

带漂移随机游动凸包的迭代对数律

Iterated-logarithm laws for convex hulls of random walks with drift

本文针对具有非零漂移的多步多维随机游动，基于 Strassen 泛函迭代对数律和新型 0-1 律，建立了其凸包内蕴体积的迭代对数律，并显式计算了平面情况下的常数，同时推广至质心过程。

Weak quenched limit theorems for a random walk in a sparse random environment

We study the quenched behaviour of a perturbed version of the simple symmetric random walk on the set of integers. The random walker moves symmetrically with an exception of some randomly chosen sites where we impose a random drift. We show that if the gaps between the marked sites are i.i.d. and regularly varying with a sufficiently small index, then there is no strong quenched limit laws for the position of the random walker. As a consequence we study the quenched limit laws in the context of weak convergence of random measures.

On the exit time from a cone for random walks with drift

We compute the exponential decay of the probability that a given multi-dimensional random walk stays in a convex cone up to time $n$, as $n$ goes to infinity. We show that the latter equals the minimum, on the dual cone, of the Laplace transform of the random walk increments. As an example, our results find applications in the counting of walks in orthants, a classical domain in enumerative combinatorics.

标准指数族随机游走的均匀Lorden型超冲矩界：小漂移和指数修正

Uniform Lorden-type bounds for overshoot moments for standard exponential families: small drift and an exponential correction

本文研究了标准指数族随机游走的超冲矩界，在小漂移情况下提出均匀Lorden型界，并通过指数修正改进了残余寿命界。

带有漂移的金字塔内随机游走：存活概率的收敛速率

Random walks with drift inside a pyramid: convergence rate for the survival probability

本文研究金字塔内多维随机游走的存活概率收敛速率，证明当漂移在锥内部时，其指数收敛速率可通过随机游走增量Laplace变换的min-max计算得出。

漂移多子分数布朗运动在离散观测下的精确极大似然推断

Exact maximum likelihood inference for drifted multi-sub-fractional Brownian motion at discrete observation

本文针对离散观测的漂移多子分数布朗运动，证明了完整有限样本似然理论的存在，给出显式极大似然估计量、精确水平的区间与检验，并验证了强相合性、渐近正态性及模拟效果。