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.

中文翻译：

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ℝd$ \ mathbb {R} ^ {d} $中布朗运动的谢普统计的尾部渐近性

让X（吨），\（T \在\ mathbb {R} \） ，是d维矢量值布朗运动，d ≥1.对于所有\（\ boldsymbol {B} \在\ mathbb {R} ^ {d} \ setminus（-\ infty，0] ^ {d} \）我们得出$$ \ mathbb {P} \ {\ boldsymbol {X}（t + s）-\ boldsymbol {X}（ t）> u \ boldsymbol {b} \ text {for}} \ t \ in [0，T]，\ s \ in [0,1] \} \ quad \ text {as} u \ to \ infty，$$这是X的Shepp统计量的矢量值类似物的尾部分布的渐近行为；我们不仅涵盖固定时间水平T > 0的情况，而且还涵盖T →0或\（T \ to \ infty \）。向量值过程的高级别偏移概率的结果在文献中很少见，目前没有适用于我们问题的可用方法。我们的证明利用了矢量值布朗运动的一些分布特性，并且是由二次编程问题导致的。作为副产品，我们为向量值布朗运动的“至高”得出了一个新的不等式。