Supremum of the Euclidean norms of the multidimensional Wiener process and Brownian bridge: Sharp asymptotics of probabilities of large deviations

V. R. Fatalov

Fundamentalʹnaâ i prikladnaâ matematika, Tome 23 (2020) no. 1, pp. 219-257

Résumé

For $ T > 0 $, we prove theorems concerning sharp asymptotics of the probabilities $$ \mathbf P \biggl\{ \sup\limits_{t \in [0, T]} \sum\limits_{j=1}^n w_j^2(t) > u^2 \biggr \}, \mathbf P \biggl \{ \sup\limits_{t \in [0, T]} \sum\limits_{j=1}^n w_{j0,T}^2(t) > u^2 \biggr \}, $$ as $u \to \infty$, where $ w_j(t) $, $ j = 1, \dots, n$, are independent Wiener processes and $ w_{j0,T}(t) $, $ j = 1, \dots, n $, are independent Brownian bridges on the segment $ [0, T] $. Our research method is the double sum method for the Gaussian processes and fields. We also give an application of the obtained results to the statistical tests for the homogeneity hypothesis of $k$ one-dimensional samples.