E. Ostrovsky, L. Sirota
We construct an example of a continuous centered random process with light tails of finite-dimensional distribution but with heavy tail of maximum distribution.
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We bring in this short report (counter -) examples on order to show a difference between tails behavior and Grand Lebesgue Spaces norms for some classes of random variables.
Full negative regression dependence: counterexamples for tail dependence and negative association
本文通过构造最小状态空间上的精确反例，证明完全负回归相依既不蕴含尾部相依也不蕴含负关联，并解决了相关开放问题。
Given a fixed graph H, what is the (exponentially small) probability that the number X_H of copies of H in the binomial random graph G_{n,p} is at least twice its mean? Studied intensively since the mid 1990s, this so-called infamous upper tail problem remains a challenging testbed for concentration inequalities. In 2011 DeMarco and Kahn formulated an intriguing conjecture about the exponential rate of decay of \Pr(X_H \ge (1+ε) \E X_H) for fixed ε>0. We show that this upper tail conjecture is false, by exhibiting an infinite family of graphs violating the conjectured bound.
A strictly stationary sequence of random variables is constructed with the following properties: (i) the random variables take the values -1 and +1 with probability 1/2 each, (ii) every five of the random variables are independent, (iii) the sequence is "causal" in a certain sense, (iv) the sequence has a trivial double tail sigma-field, and (v) regardless of the normalization used, the partial sums do not converge to a (nondegenerate) normal law. The example has some features in common with a recent construction (for an arbitrary fixed positive integer N), by Alexander Pruss and the author, o…
We derive in this article the asymptotic behavior as well as non-asymptotical estimates of tail of distribution for self-normalized sums of random variables (r.v.) under natural classical norming. We investigate also the case of non-standard random norming function and the tail asymptotic for the maximum distribution for self-normalized statistics. We do not suppose the independence or identical distributionness of considered random variables, but we assume the existence and sufficient smoothness of its density. We show also the exactness of our conditions imposed on the considered random vari…
Finite-length Analysis on Tail probability for Markov Chain and Application to Simple Hypothesis Testing
本文利用信息几何推导有限长度Markov链样本均值尾概率及累积量生成函数的界，并据此建立简单假设检验中第二类错误最小概率的Hoeffding型界，还给出中心极限定理的新证明。
We characterize the tail behavior of the distribution of the PageRank of a uniformly chosen vertex in a directed preferential attachment graph and show that it decays as a power law with an explicit exponent that is described in terms of the model parameters. Interestingly, this power law is heavier than the tail of the limiting in-degree distribution, which goes against the commonly accepted {\em power law hypothesis}. This deviation from the power law hypothesis points at the structural differences between the inbound neighborhoods of typical vertices in a preferential attachment graph versu…
Tail asymptotics and precise large deviations for some Poisson cluster processes
本文研究了再生Poisson聚类过程和Hawkes过程的尾渐进行为，利用正则变化性假设和Karamata定理推导出精确的大偏差结果。
The G-normal distribution was introduced by Peng [2007] as the limiting distribution in the central limit theorem for sublinear expectation spaces. Equivalently, it can be interpreted as the solution to a stochastic control problem where we have a sequence of random variables, whose variances can be chosen based on all past information. In this note we study the tail behavior of the G-normal distribution through analyzing a nonlinear heat equation. Asymptotic results are provided so that the tail "probabilities" can be easily evaluated with high accuracy. This study also has a significant impa…
We consider the spatial stochastic model of single-tier downlink cellular networks, where the wireless base stations are deployed according to a general stationary point process on the Euclidean plane with general i.i.d. propagation effects. Recently, Ganti & Haenggi (2016) consider the same general cellular network model and, as one of many significant results, derive the tail asymptotics of the signal-to-interference ratio (SIR) distribution. However, they do not mention any conditions under which the result holds. In this paper, we compensate their result for the lack of the condition and e…
We study sample-path large deviations for Lévy processes and random walks with heavy-tailed jump-size distributions that are of Weibull type. Our main results include an extended form of an LDP (large deviations principle) in the $J_1$ topology, and a full LDP in the $M_1'$ topology. The rate function can be represented as the solution to a quasi-variational problem. The sharpness and applicability of these results are illustrated by a counterexample proving the nonexistence of a full LDP in the $J_1$ topology, and by an application to a first passage problem.
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。
We consider a spatial stochastic model of wireless cellular networks, where the base stations (BSs) are deployed according to a simple and stationary point process on $\mathbb{R}^d$, $d\ge2$. In this model, we investigate tail asymptotics of the distribution of signal-to-interference ratio (SIR), which is a key quantity in wireless communications. In the case where the path-loss function representing signal attenuation is unbounded at the origin, we derive the exact tail asymptotics of the SIR distribution under an appropriate sufficient condition. While we show that widely-used models based o…
In this paper we study the Poisson Hypothesis, which is a device to analyze approximately the behavior of large queueing networks. We prove it in some simple limiting cases. We show in particular that the corresponding dynamical system, defined by the non-linear Markov process, has a line of fixed points which are global attractors. To do this we derive the corresponding non-linear equation and we explore its self-averaging properties. We also argue that in cases of havy-tail service times the PH can be violated.
In high dimensional percolation at parameter $p < p_c$, the one-arm probability $π_p(n)$ is known to decay exponentially on scale $(p_c - p)^{-1/2}$. We show the same statement for the ratio $π_p(n) / π_{p_c}(n)$, establishing a form of a hypothesis of scaling theory. As part of our study, we provide sharp estimates (with matching upper and lower bounds) for several quantities of interest at the critical probability $p_c$. These include the tail behavior of volumes of, and chemical distances within, spanning clusters, along with the scaling of the two-point function at "mesoscopic distance" fr…
Type-II Error Bounds for Test Supermartingales from Lower-Tail Hypotheses
本文研究检验超鞅下尾概率假设对序贯检验II型误差的影响，通过主不等式和Legendre变换导出多种误差界，并涵盖经验Bernstein变体。
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