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
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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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\(I\subseteq \{1{\ldots } d\}\)withm ≤delementssuch that
Furthermore, for any\(\boldsymbol {x}\in \mathbb {R}^{d}\)we have
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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DOI: https://doi.org/10.1007/s10687-019-00357-z
Keywords
- Shepp-statistics
- Vector-valued Brownian motion
- High level excursion probability
- Uniform double-sum method
- Markov property
- Quadratic programming problem