Cramér's theorem

For independent and identically distributed random variables with a moment-generating function finite on an open interval containing zero, the empirical mean satisfies a large deviation principle with rate function given by the Legendre transform of a cumulant-generating function. The exponential-moment hypothesis is part of this theorem; mere integrability does not guarantee this full large-deviation principle.

For sample means of Poisson random variables of mean $\lambda >0$, the cumulant-generating function is $\lambda (e^{\theta }-1)$. Its Legendre-Fenchel transform is the displayed good rate function, with $0\log 0=0$, and is infinite for $x<0$. Near its zero, $I_P(\lambda +h)=h^2/(2\lambda )+O(h^3)$. This local quadratic explains the Poisson moderate deviation principle.

For independent and identically distributed exponential random variables of rate $\lambda >0$, the sample mean has large-deviation speed $n$ and the displayed good rate function, infinite for $x\le 0$. The cumulant-generating function is $\log [\lambda /(\lambda -\theta )]$ for $\theta <\lambda$. Its Legendre-Fenchel transform has maximizing parameter $\theta =\lambda -1/x$. For $x\le 0$, sending $\theta \to -\infty$ gives infinite cost. The cost has its unique zero at $1/\lambda$ and diverges at both ends of its positive domain.

For independent unit-rate exponential random variables, the cumulant-generating function is $-\log (1-\theta )$ for $\theta <1$. Its Legendre transform of a cumulant-generating function is $I(x)=x-1-\log x$ for $x>0$ and infinite otherwise. The Cramér theorem gives the upper-tail logarithmic rate $-I(a)$ for $a>1$, and zero for $0\le a\le 1$. At the mean threshold the infima on both the open and closed upper half-lines are zero; below the mean the strong law of large numbers applies.

For independent identically distributed real variables with everywhere finite cumulant-generating function, let $\mu$ be the mean. If $a\ge \mu$ and $\mathbb{P}(X_1>a)>0$, then $\mathbb{P}(S_n/n\in [a,a+\epsilon )\mid S_n/n\ge a)\to 1$ for each $\epsilon >0$. The Legendre transform of a cumulant-generating function is finite at $a$, and a positive exponential parameter gives strictly greater cost at $a+\epsilon$. A Chernoff bound for the farther tail divided by the Cramér theorem lower estimate at $a$ decays exponentially. The support condition both ensures positive conditioning probabilities and avoids a vacuous conditioning event.

Exponential tilting by $\theta$ replaces a law $P$ byUnder the tilted law, the mean is ${\psi }^{\prime }(\theta )$ whenever the derivative exists.

Suppose $\varphi (\lambda )=\mathbb{E}{e}^{\lambda X}<\infty$ for all $\lambda \ge 0$. Under the exponential tilting $d{\mu }^{\lambda }=e^{\lambda x}d\mu /\varphi (\lambda )$, $X$ is an integrable random variable whenever $\lambda >0$: its negative exponential tail is bounded, and its positive contribution is dominated by a larger positive exponential moment. Differentiation under the integral sign gives the displayed mean. At the endpoint $\lambda =0$, the positive part of $X$ is integrable, but the negative part need not be. The right derivative ${\varphi }_{+}^{\prime }(0)=\mathbb{E}X$ is valid as an extended value, possibly $-\infty$; a finite mean at zero requires an extra negative-tail integrability assumption.