  In this paper, we analyze a multivariate counterpart of the generalized
weighted Kolmogorov-Smirnov statistic, which is the supremum of weighted
locally stationary chi-square process over non-compact interval. The
boundedness and the exact tail asymptotic behavior of the statistics are
derived.
  We illustrate our findings by two examples where the statistic is defined by
Brownian bridge and fractional Brownian motion respectively.
