Limiting distribution for the maximal standardized increment of a random walk

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• 作者：Zakhar Kabluchko^a ; ^{sachar.k@gmx.net" class="auth_mail} ; ^{zakhar.kabluchko@uni-ulm.de" class="auth_mail} ; Yizao Wang^b ; ^{yizao.wang@uc.edu" class="auth_mail}
• 关键词：60G50 ; 60G70 ; 60F10 ; 60F05
• 刊名：Stochastic Processes and their Applications
• 出版年：September 2014
• 年：2014
• 卷：124
• 期：9
• 页码：2824-2867
• 全文大小：736 K

文摘

Let X₁,X₂,…$X_1,X_2,\dots$ be independent identically distributed (i.i.d.) random variables with EX_k=0$\mathbb{E}{X}_k=0$, $\mathrm{V\; ar}{X}_k=1$. Suppose that 蠁(t)鈮攍ogEe^tX_k<∞ for all t>−蟽₀$t>-{蟽}_0$ and some 蟽₀>0${蟽}_0>0$. Let S_k=X₁+鈰?X_k and S₀=0$S_0=0$. We are interested in the limiting distribution of the multiscale scan statistic   We prove that for an appropriate normalizing sequence a_n$a_n$, the random variable $M_n^2-a_n$ converges to the Gumbel extreme-value law exp{−e^−cx}$exp\{-e^{-cx}\}$. The behavior of $M_n$ depends strongly on the distribution of the X_k$X_k$’s. We distinguish between four cases. In the superlogarithmic   case we assume that 蠁(t)<t²/2$蠁\left(t\right)<t^2/2$ for every t>0$t>0$. In this case, we show that the main contribution to $M_n$ comes from the intervals (i,j)$(i,j)$ having length l鈮攋−i of order a(logn)^p$a{(logn)}^p$, a>0$a>0$, where aa1bbdb5badb954f925d7778" title="Click to view the MathML source">p=q/(q−2)$p=q/(q-2)$ and q∈{3,4,…}$q\in \{3,4,\dots \}$ is the order of the first non-vanishing cumulant of X₁$X_1$ (not counting the variance). In the logarithmic   case we assume that the function 蠄(t)鈮?蠁(t)/t² attains its maximum m_∗>1$m_{\ast }>1$ at some unique point t=t_∗∈(0,∞)$t=t_{\ast }\in (0,\infty )$. In this case, we show that the main contribution to $M_n$ comes from the intervals (i,j)$(i,j)$ of length $d_{\ast }logn+a\sqrt{logn}$, a∈R$a\in \mathbb{R}$, where d_∗=1/蠁(t_∗)>0$d_{\ast }=1/蠁\left(t_{\ast }\right)>0$. In the sublogarithmic   case we assume that the tail of X_k$X_k$ is heavier than exp{−x^2−蔚}$exp\{-x^{2-蔚}\}$, for some 蔚>0$蔚>0$. In this case, the main contribution to $M_n$ comes from the intervals of length o(logn)$o(logn)$ and in fact, under regularity assumptions, from the intervals of length 1. In the remaining, fourth case, the X_k$X_k$’s are Gaussian  . This case has been studied earlier in the literature. The main contribution comes from intervals of length alogn$alogn$, a>0$a>0$. We argue that our results cover most interesting distributions with light tails. The proofs are based on the precise asymptotic estimates for large and moderate deviation probabilities for sums of i.i.d. random variables due to Cramér, Bahadur, Ranga Rao, Petrov and others, and a careful extreme value analysis of the random field of standardized increments by the double sum method.