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1.
Let FF be a distribution function with negative mean and regularly varying right tail. Under a mild smoothness condition we derive higher order asymptotic expansions for the tail distribution of the maxima of the random walk generated by FF. The expansion is based on an expansion for the right Wiener–Hopf factor which we derive first. An application to ruin probabilities is developed.  相似文献   

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Let X and Y be two nonnegative and dependent random variables following a generalized Farlie-Gumbel-Morgenstern distribution. In this short note, we study the impact of a dependence structure of X and Y on the tail behavior of XY. We quantify the impact as the limit, as x, of the quotient of Pr(XY>x) and Pr(XY>x), where X and Y are independent random variables identically distributed as X and Y, respectively. We obtain an explicit expression for this limit when X is regularly varying or rapidly varying tailed.  相似文献   

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Let I(F) be the distribution function (d.f.) of the maximum of a random walk whose i.i.d. increments have the common d.f. F and a negative mean. We derive a recursive sequence of embedded random walks whose underlying d.f.'s Fk converge to the d.f. of the first ladder variable and satisfy FF1F2 on [0,∞) and I(F)=I(F1)=I(F2)=. Using these random walks we obtain improved upper bounds for the difference of I(F) and the d.f. of the maximum of the random walk after finitely many steps.  相似文献   

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Let \(X(t), t\in \mathcal {T}\) be a centered Gaussian random field with variance function σ 2(?) that attains its maximum at the unique point \(t_{0}\in \mathcal {T}\), and let \(M(\mathcal {T})=\sup _{t\in \mathcal {T}} X(t)\). For \(\mathcal {T}\) a compact subset of ?, the current literature explains the asymptotic tail behaviour of \(M(\mathcal {T})\) under some regularity conditions including that 1 ? σ(t) has a polynomial decrease to 0 as tt 0. In this contribution we consider more general case that 1 ? σ(t) is regularly varying at t 0. We extend our analysis to Gaussian random fields defined on some compact set \(\mathcal {T}\subset \mathbb {R}^{2}\), deriving the exact tail asymptotics of \(M(\mathcal {T})\) for the class of Gaussian random fields with variance and correlation functions being regularly varying at t 0. A crucial novel element is the analysis of families of Gaussian random fields that do not possess locally additive dependence structures, which leads to qualitatively new types of asymptotics.  相似文献   

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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 Zn(A) be the number of particles of generation n located in A. We show central limit theorems for the counting measure Zn(·) with appropriate normalization.  相似文献   

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一类具有红灯环境下随机游动的极限性态   总被引:1,自引:0,他引:1  
就某类红灯随机环境下随机游动{Xn}n∈Z ,建立相应的Markov-双链{ηn}n∈Z ={(Xn,Tn)}n∈Z ;并给出在该红灯环境下,{Xn}n∈Z 的发展受红灯环境影响的关系式;以及由此关系式得出:由于红灯环境的影响减缓了{Xn}n∈Z 的发展速度.  相似文献   

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Let X1,X2,…X1,X2, be independent variables, each having a normal distribution with negative mean −β<0β<0 and variance 1. We consider the partial sums Sn=X1+?+XnSn=X1+?+Xn, with S0=0S0=0, and refer to the process {Sn:n≥0}{Sn:n0} as the Gaussian random walk. This paper is concerned with the cumulants of the maximum Mβ=max{Sn:n≥0}Mβ=max{Sn:n0}.  相似文献   

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We show that the scaling limit exists and is invariant under dilations and rotations. We give some tools that might be useful to show universality.  相似文献   

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We use the known convergence of loop-erased random walk to radial SLE(2) to give a new proof that the scaling limit of loop-erased random walk excursion in the upper half-plane is chordal SLE(2). Our proof relies on a version of Wilson’s algorithm for weighted graphs which is used together with a Beurling-type estimate for random walk excursion. We also establish and use the convergence of the radial SLE path to the chordal SLE path as the bulk point tends to a boundary point. In the final section we sketch how to extend our results to more general simply connected domains.  相似文献   

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We establish the local and so-called “extended” large deviation principles (see [1, 2]) for random walks whose jumps fail to satisfy Cramér’s condition but have distributions varying regularly at infinity.  相似文献   

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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.  相似文献   

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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.  相似文献   

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