= Uniform consistency of an empirical moment-generating function
{title2=$\sup_{0\le s\le t}|\widehat m_n(s)-m(s)|\to0$}
If $X\ge0$ and $\mathbb E e^{tX}<\infty$ for some finite $t\ge0$, the estimator $\widehat m_n(s)=n^{-1}\sum_i e^{sX_i}$ satisfies $\sup_{s\in[0,t]}|\widehat m_n(s)-\mathbb E e^{sX}|\to0$ almost surely. The <dominated convergence theorem> makes the population <moment-generating function> uniformly continuous on this interval. Parameter subintervals give <function brackets> whose endpoints are ordered exponentials and whose widths are the corresponding increments of the population mean. Apply the <uniform strong law from finite L1 bracketing>. Continuity in the parameter makes the supremum measurable.
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