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