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Tail asymptotics for Shepp-statistics of Brownian motion in \(\mathbb {R}^{d}\)

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Abstract

Let X(t), \(t\in \mathbb {R}\), be a d-dimensional vector-valued Brownian motion, d ≥ 1. For all \(\boldsymbol {b}\in \mathbb {R}^{d}\setminus (-\infty ,0]^{d}\) we derive exact asymptotics of

$$ \mathbb{P}\{\boldsymbol{X}(t+s)-\boldsymbol{X}(t) >u\boldsymbol{b}\text{ for some } t\in[0,T],\ s\in[0,1]\} \quad\text{as } u\to\infty, $$

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 → 0 or \(T\to \infty \). Results for high level 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.

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Acknowledgments

The authors would like to thank two anonymous referees for their valuable comments and suggestions to improve the quality of the paper. Support from SNSF Grant no. 200021-175752/1 is kindly acknowledged.

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Correspondence to Dmitry Korshunov.

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Appendix: Quadratic programming problems

Appendix: Quadratic programming problems

The next result is known and formulated for instance in Dębicki et al. (2018).

Lemma A.1

Let Σ bea positive definite matrix of sized × dwithinverse Σ− 1.If\(\boldsymbol {b} \in \mathbb {R}^{d} \setminus (-\infty , 0]^{d} \),then the quadratic programming problem πΣ(b) formulatedin Eq. 3 has a unique solution\(\widetilde {\boldsymbol {b}}\)andthere exists a unique non-empty index set\(I\subseteq \{1{\ldots } d\}\)withmdelementssuch that

$$ \begin{array}{@{}rcl@{}} \widetilde{\boldsymbol{b}}_{I} = \boldsymbol{b}_{I} , \text{ and if } J := \{ 1 {\ldots} d\}\setminus I \not = \emptyset, \text{ then } \widetilde{\boldsymbol{b}}_{J} &=& {\Sigma}_{JI} {\Sigma}_{II}^{-1} \boldsymbol{b}_{I} \geq \boldsymbol{b}_{J}, ~~{\Sigma}_{II}^{-1} \boldsymbol{b}_{I}>\boldsymbol{0}_I, \end{array} $$
(32)
$$ \begin{array}{@{}rcl@{}} \min_{\boldsymbol{x} \geq \boldsymbol{b}}\boldsymbol{x}^{\top} {{\Sigma}^{-1}}\boldsymbol{x}= \widetilde{\boldsymbol{b}}^{\top} {{\Sigma}^{-1}} \widetilde{\boldsymbol{b}} &=& \boldsymbol{b}_{I}^{\top} {\Sigma}_{II}^{-1}\boldsymbol{b}_{I}>0. \end{array} $$
(33)

Furthermore, for any\(\boldsymbol {x}\in \mathbb {R}^{d}\)we have

$$ \boldsymbol{x}^{\top} {{\Sigma}^{-1}} \widetilde{\boldsymbol{b}}= \boldsymbol{x}_I^{\top} {\Sigma}_{II}^{-1} \widetilde{\boldsymbol{b}}_I= \boldsymbol{x}_I^{\top} {\Sigma}_{II}^{-1}\boldsymbol{b}_I $$
(34)

and if\(\boldsymbol {b}= c \boldsymbol {1}, c\in (0,\infty )\), then 2 ≤|I|≤kandJis empty if Σ− 1b > 0.

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

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Keywords

  • Shepp-statistics
  • Vector-valued Brownian motion
  • High level excursion probability
  • Uniform double-sum method
  • Markov property
  • Quadratic programming problem

AMS 2000 Subject Classifications

  • Primary—60G70
  • Secondary—60G15