A Note on the Upper Bound for I.I.D. Large Deviations

Abstract

Let $\bar{X}_n$ denote the mean of an i.i.d. sequence of random vectors $X_1,X_2,X_3,\ldots$ taking values in $\mathbf{R}^d$. If $\lambda$ denotes the convex conjugate of the logarithm of the moment generating function for $X_1$, then $\lim\sup\frac{1}{n}\log P(\bar{X}_n \in C) \leq -\inf\{\lambda(\nu): \nu \in C\}$ when $C \subset \mathbf{R}^d$ is closed and the moment generating function for $X_1$ is finite in a neighborhood of the origin. An example is given in which this upper bound fails for a certain closed set in $\mathbf{R}^3$ and the moment generating function for $X_1$ is not finite in a neighborhood of the origin. An example is also given in which this upper bound is valid for all closed sets but the moment generating function for $X_1$ is not finite in a neighborhood of the origin.