Probabilities of Large Deviations for Random Vectors

Integral large deviation limit theorems for random vectors taking values in a k-dimensional Euclidean space were investigated in (Borovkov, Rogozin, 1965), (Vilkauskas, 1965), (W. Richter, 1957), (Bahr, 1967), (W.-D. Richter, 1978), (Rozovskii, 1982) and other papers. Asymptotic formulas for special classes of sets, as shown in (Osipov, 1982), (W.-D. Richter, 1982), (Saulis, 1983), can be essentially simplified. It was established in (Aleskevicienė, 1983), (Svetulevicienė, 1981), (Saulis, 1984, 1987) that in theorems of large deviations for convex Borel sets, it suffices to study a multidimensional analog of the Cramer — Petrov series at the closest to the origin point of the set.