On the Convexity of a Rate Function

Consider a sequence of independent, identically distributed random variables from a common distribution Q. Suppose that there are k rare events associated with Q, for example events involving the k sample moments. Then the dominant rate at which the probability of the intersection of the k events converges to zero can be expressed as the relative entropy of a certain distribution and Q. Considering the intersection of the k events as a function of α in a k-dimensional Euclidean space, we shall show that the dominant rate of convergence is convex in α . This result is the consequence of the convex property of a certain function associated with the dominant rate of convergence which shall be shown.