Convergence Rates of Large Deviation Probabilities in the Multidimensional Case

Let $\{W_n\}_{n=1,2,\cdots}$ denote a sequence of $k$-dimensional random vectors on a probability space $(\Omega, \mathscr{A}, P)$. Using moment-generating function techniques sufficient conditions are given for the existence of limits $\rho(A) = \lim_{n\rightarrow\infty} \lbrack P(W_n \not\in k_n A)\rbrack^{1/k_n}$ for certain subsets $A \subset R^k$, where $\{k_n\}_{n=1,2,\cdots}$ is a divergent sequence of positive real numbers. The results are multivariate analogs of well-known large deviation theorems on the real line.