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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.

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 a walk with increments $+1$ with probability $p$ and $-1$ with probability $1-p$, putThen $\exp (\theta S_n-n\psi (\theta ))$ is a martingale. Optional stopping at the first linear-boundary crossing and optimization over $\theta$ gives exponential maximal-deviation bounds governed by the Legendre transform of a cumulant-generating function.

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.

Let $Y_i$ be independent and identically distributed random variables with moment-generating function finite on an open interval about $0$. Putallowing $\Lambda =+\infty$ outside its finite domain. The Cramér theorem states that ${\bar{Y}}_n=n^{-1}{\sum }_{i=1}^n{Y}_i$ satisfies a large deviation principle with speed $n$ and good rate function $I$, the Legendre transform of a cumulant-generating function. Explicitly, for every closed set $F$ and open set $O$,Here $\log 0=-\infty$ and $\inf \varnothing =+\infty$. The rate function is lower semicontinuous and its finite sublevel sets are compact. The exponential-moment hypothesis is essential for this form of the Cramér theorem.

An effective bandwidth measures how much capacity a random traffic source requires when unusually large demands matter. A mean-rate description loses burstiness; a peak-rate description can waste capacity when peaks are rare. The parameter $s>0$ in an effective bandwidth controls the emphasis placed on such bursts through the moment-generating function:Here $X$ is the traffic in a chosen time unit. If $X(t)$ instead measures traffic over length $t$, the corresponding rate is $\alpha (s,t)=\log \mathbb{E}[e^{sX(t)}]/(st)$. With independent stationary increments, this is independent of $t$ whenever the moment-generating function exists. More generally both the time scale and dependence structure matter.

The Chernoff bound follows directly from the Markov inequality, since $e^{sX}$ is nonnegative and $X\ge b$ implies $e^{sX}\ge e^{sb}$:Optimize over the admissible positive values of $s$ to obtainFor independent sources $X_1,\dots ,X_N$, moment-generating functions multiply and their effective bandwidths add. Thus a total capacity $c$ has the one-time overflow bound $\mathbb{P}({\sum }_i{X}_i\ge c)\le \exp \{s({\sum }_i{\alpha }_i(s)-c)\}$. In particular, for independent and identically distributed random variables,This is the positive-parameter part of the Legendre transform of a cumulant-generating function. The size of its exponent describes how a service margin reduces the chance of a large aggregate demand. A buffer-overflow event involves traffic over many intervals; a one-time Chernoff bound alone is not automatically a bound on that event. A time-dependent model and, for example, bounds on the union of relevant interval events are then needed.

Assume a finite moment-generating function in a neighborhood of zero. Differentiating the cumulant-generating function gives $K(0)=0$, $K^{\prime }(0)=\mathbb{E}[X]$ and $K^{\prime \prime }(0)=\mathrm{Var}(X)$. Therefore the small-parameter limit is the mean traffic rate:The variance term shows how an effective bandwidth charges extra capacity for burstiness. Also $K$ is a convex function with $K(0)=0$, so $K(s)/s$ is nondecreasing for $s>0$ in its domain.

For the other limit, let $M=\mathrm{esssup}X$, the essential supremum. If $M<\infty$, $X\le M$ almost surely gives $\alpha (s)\le M$. For every $a<M$, $p_a=\mathbb{P}(X\ge a)>0$, andTaking a lower limit and then $a\uparrow M$ proves the large-parameter limit is peak traffic:If $X$ has no finite upper bound and all positive exponential moments exist, the same lower bound for arbitrarily large $a$ makes the limit infinite. If positive exponential moments cease to exist at a finite parameter, the effective bandwidth is infinite beyond that domain, and the infinite limit is understood in this extended sense. For example, a normal distribution with mean $m$ and variance $v$ has $\alpha (s)=m+sv/2$: its unbounded upper tail has no finite peak. Thus effective bandwidth interpolates between mean and peak demand while permitting independent sources to be aggregated by addition.

Since $X_1$ is uniform on $\{-1,1\}$,For $|x|<1$, the supremum in the Legendre transform of a cumulant-generating function is attained whereSubstitution gives the Rademacher large-deviation rate functionfor $|x|\le 1$, with $0\log 0=0$; it is $+\infty$ for $|x|>1$.

LetOn $A$, one has $T\le n$, $S_T\ge \mu T+\epsilon n$, and convexity of $\psi$ gives $\psi (\theta )-\mu \theta \ge 0$. Hence for $\theta >0$,The optional stopping theorem applies because $T$ is bounded, so $\mathbb{E}{Z}_T=1$. Therefore

Optimize over $\theta >0$ for the upper deviation and apply the same argument with $\theta <0$ to the lower deviation. Since the supremum of the linearly interpolated centered walk is attained at grid points, the Legendre transform of a cumulant-generating functionand the union bound give

Put $Y=g(X)-\mathbb{E}g(X)$ and $H(\lambda )=\log \mathbb{E}{e}^{\lambda Y}$. Apply the assumed Poisson log-Sobolev inequality to $f=e^{\lambda g}$. Since $|Dg|\le 1$ and $|e^a-1|\le \lambda e^{\lambda }$ for $|a|\le \lambda$,After division by $\mathbb{E}{e}^{\lambda g}$, the entropy bound becomesBecause $(H(\lambda )/\lambda {)}^{\prime }=(\lambda H^{\prime }-H)/{\lambda }^2$ and $H^{\prime }(0)=0$, integration from $0$ to $\lambda$ givesThe Chernoff bound and optimization over $\lambda \ge 0$ therefore yieldwhere ${\psi }^{\ast }$ is the Legendre transform of a cumulant-generating function, also called the Chernoff-Cramér transform.

For an integrable real random variable $X$, its cumulant-generating function is the extended-real convex functionwhere $\psi (\theta )=+\infty$ if the exponential moment diverges. Its Legendre transform of a cumulant-generating function is