Supremum of the Euclidean Norms of the Multidimensional Wiener Process and Brownian Bridge: Sharp Asymptotics of Probabilities of Large Deviations

Abstract

For T > 0, we prove theorems concerning sharp asymptotics of the probabilities$$ \mathbf{P}\left\{\underset{t\in \left[0,T\right]}{\sup}\sum \limits_{j=1}^n{w}_j^2(t)>{u}^2\right\},\kern1em \mathbf{P}\left\{\underset{t\in \left[0,T\right]}{\sup}\sum \limits_{j=1}^n{w}_{j0,T}^2(t)>{u}^2\right\}, $$as u → ∞, where wj(t), j = 1, . . . , n, are independent Wiener processes and wj0,T (t), j = 1, . . . , 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.