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Tail asymptotics for Shepp-statistics of Brownian motion in ℝ d

Research output: Contribution to Journal/Magazine › Journal article › peer-review

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Tail asymptotics for Shepp-statistics of Brownian motion in ℝ d. / Korshunov, Dmitry; Wang, Longmin.
In: Extremes, Vol. 23, No. 1, 31.03.2020, p. 35-54.

Research output: Contribution to Journal/Magazine › Journal article › peer-review

Harvard

Korshunov, D & Wang, L 2020, 'Tail asymptotics for Shepp-statistics of Brownian motion in ℝ d', Extremes, vol. 23, no. 1, pp. 35-54. https://doi.org/10.1007/s10687-019-00357-z

APA

Korshunov, D., & Wang, L. (2020). Tail asymptotics for Shepp-statistics of Brownian motion in ℝ d. Extremes, 23(1), 35-54. https://doi.org/10.1007/s10687-019-00357-z

Vancouver

Korshunov D, Wang L. Tail asymptotics for Shepp-statistics of Brownian motion in ℝ d. Extremes. 2020 Mar 31;23(1):35-54. Epub 2019 Sept 11. doi: 10.1007/s10687-019-00357-z

Author

Korshunov, Dmitry ; Wang, Longmin. / Tail asymptotics for Shepp-statistics of Brownian motion in ℝ d. In: Extremes. 2020 ; Vol. 23, No. 1. pp. 35-54.

Bibtex

@article{e5701be7602f483e934eea6fca4aa078,
title = "Tail asymptotics for Shepp-statistics of Brownian motion in ℝ d",
abstract = "Let $X(t)$, $t\in R$, be a $d$-dimensional vector-valued Brownian motion, $d\ge 1$. For all $b\in R^d\setminus (-\infty,0]^d$ we derive exact asymptotics of \[ P(X(t+s)-X(t) >u b\mbox{ for some }t\in[0,T],\ s\in[0,1]}\mbox{as }u\to\infty, \] that is the asymptotical behavior of tail distribution of vector-valued analog of Shepp-statistics for $X$; we cover not only the case of a fixed time-horizon $T>0$ but also cases where $T\to 0$ or $T\to\infty$. Results for high excursion probabilities of vector-valued processes are rare in the literature, with currently no available approach suitable for our problem. Our proof exploits some distributional properties of vector-valued Brownian motion, and results from quadratic programming problems. As a by-product we derive a new inequality for the `supremum' of vector-valued Brownian motions.",
keywords = "Shepp-statistics, Vector-valued Brownian motion, High level excursion probability, Uniform double-sum method, Markov property, Quadratic programming problem",
author = "Dmitry Korshunov and Longmin Wang",
note = "The final publication is available at Springer via http://dx.doi.org/10.1007/s10687-019-00357-z",
year = "2020",
month = mar,
day = "31",
doi = "10.1007/s10687-019-00357-z",
language = "English",
volume = "23",
pages = "35--54",
journal = "Extremes",
issn = "1386-1999",
publisher = "Springer Netherlands",
number = "1",

}

RIS

TY - JOUR

T1 - Tail asymptotics for Shepp-statistics of Brownian motion in ℝ d

AU - Korshunov, Dmitry

AU - Wang, Longmin

N1 - The final publication is available at Springer via http://dx.doi.org/10.1007/s10687-019-00357-z

PY - 2020/3/31

Y1 - 2020/3/31

N2 - Let $X(t)$, $t\in R$, be a $d$-dimensional vector-valued Brownian motion, $d\ge 1$. For all $b\in R^d\setminus (-\infty,0]^d$ we derive exact asymptotics of \[ P(X(t+s)-X(t) >u b\mbox{ for some }t\in[0,T],\ s\in[0,1]}\mbox{as }u\to\infty, \] that is the asymptotical behavior of tail distribution of vector-valued analog of Shepp-statistics for $X$; we cover not only the case of a fixed time-horizon $T>0$ but also cases where $T\to 0$ or $T\to\infty$. Results for high excursion probabilities of vector-valued processes are rare in the literature, with currently no available approach suitable for our problem. Our proof exploits some distributional properties of vector-valued Brownian motion, and results from quadratic programming problems. As a by-product we derive a new inequality for the `supremum' of vector-valued Brownian motions.

AB - Let $X(t)$, $t\in R$, be a $d$-dimensional vector-valued Brownian motion, $d\ge 1$. For all $b\in R^d\setminus (-\infty,0]^d$ we derive exact asymptotics of \[ P(X(t+s)-X(t) >u b\mbox{ for some }t\in[0,T],\ s\in[0,1]}\mbox{as }u\to\infty, \] that is the asymptotical behavior of tail distribution of vector-valued analog of Shepp-statistics for $X$; we cover not only the case of a fixed time-horizon $T>0$ but also cases where $T\to 0$ or $T\to\infty$. Results for high excursion probabilities of vector-valued processes are rare in the literature, with currently no available approach suitable for our problem. Our proof exploits some distributional properties of vector-valued Brownian motion, and results from quadratic programming problems. As a by-product we derive a new inequality for the `supremum' of vector-valued Brownian motions.

KW - Shepp-statistics

KW - Vector-valued Brownian motion

KW - High level excursion probability

KW - Uniform double-sum method

KW - Markov property

KW - Quadratic programming problem

U2 - 10.1007/s10687-019-00357-z

DO - 10.1007/s10687-019-00357-z

M3 - Journal article

VL - 23

SP - 35

EP - 54

JO - Extremes

JF - Extremes

SN - 1386-1999

IS - 1

ER -