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Limiting distribution for the maximal standardized increment of a random walk

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  • Kabluchko, Zakhar
  • Wang, Yizao

Abstract

Let X1,X2,… be independent identically distributed (i.i.d.) random variables with EXk=0, V arXk=1. Suppose that φ(t)≔logEetXk<∞ for all t>−σ0 and some σ0>0. Let Sk=X1+⋯+Xk and S0=0. We are interested in the limiting distribution of the multiscale scan statisticMn=max0≤i0. In this case, we show that the main contribution to Mn comes from the intervals (i,j) having length l≔j−i of order a(logn)p, a>0, where p=q/(q−2) and q∈{3,4,…} is the order of the first non-vanishing cumulant of X1 (not counting the variance). In the logarithmic case we assume that the function ψ(t)≔2φ(t)/t2 attains its maximum m∗>1 at some unique point t=t∗∈(0,∞). In this case, we show that the main contribution to Mn comes from the intervals (i,j) of length d∗logn+alogn, a∈R, where d∗=1/φ(t∗)>0. In the sublogarithmic case we assume that the tail of Xk is heavier than exp{−x2−ε}, for some ε>0. In this case, the main contribution to Mn comes from the intervals of length o(logn) and in fact, under regularity assumptions, from the intervals of length 1. In the remaining, fourth case, the Xk’s are Gaussian. This case has been studied earlier in the literature. The main contribution comes from intervals of length alogn, 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.

Suggested Citation

  • Kabluchko, Zakhar & Wang, Yizao, 2014. "Limiting distribution for the maximal standardized increment of a random walk," Stochastic Processes and their Applications, Elsevier, vol. 124(9), pages 2824-2867.
  • Handle: RePEc:eee:spapps:v:124:y:2014:i:9:p:2824-2867
    DOI: 10.1016/j.spa.2014.03.015
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    References listed on IDEAS

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    1. Kabluchko, Zakhar, 2011. "Extremes of the standardized Gaussian noise," Stochastic Processes and their Applications, Elsevier, vol. 121(3), pages 515-533, March.
    2. Lanzinger, H. & Stadtmüller, U., 2000. "Maxima of increments of partial sums for certain subexponential distributions," Stochastic Processes and their Applications, Elsevier, vol. 86(2), pages 307-322, April.
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    Cited by:

    1. Hashorva, Enkelejd, 2018. "Representations of max-stable processes via exponential tilting," Stochastic Processes and their Applications, Elsevier, vol. 128(9), pages 2952-2978.
    2. Krzysztof Dȩbicki & Enkelejd Hashorva, 2020. "Approximation of Supremum of Max-Stable Stationary Processes & Pickands Constants," Journal of Theoretical Probability, Springer, vol. 33(1), pages 444-464, March.
    3. Alfredas Račkauskas & Martin Wendler, 2020. "Convergence of U-processes in Hölder spaces with application to robust detection of a changed segment," Statistical Papers, Springer, vol. 61(4), pages 1409-1435, August.

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