Variance of a normalized exponential weight

Statement

Let $X$ be a bounded real random variable and define the normalized weight $W_t=e^{tX}/M(t)$ , where $M(t)=\mathbb E e^{tX}$ . Then $\mathbb EW_t=1$ , and its variance under the original measure is

\operatorname{Var}(W_t)=\frac{M(2t)}{M(t)^2}-1.

If $X$ is not almost surely constant, this variance is strictly increasing for $t>0$ .

Proof and limits

The formula follows by integrating $W_t^2$ . With $g=\log M$ , the derivative of $g(2t)-2g(t)$ is $2(g'(2t)-g'(t))>0$ , since the tilted variance identity gives $g''>0$ . The variance of $W_t$ is different from the tilted variance of $X$ . Monotonicity alone does not imply divergence as $t\to\infty$ : for a Bernoulli variable with success probability $p\in(0,1)$ , the limit is $1/p-1$ .