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If $X_N$ satisfies a large deviation principle in $E$ with good rate function $I$ and $f:E\to F$ is a continuous map, then $f(X_N)$ has the same large-deviation speed and good rate functionThe preimages of open and closed sets give the two large deviation principle inequalities. The identity $\{J\le M\}=f(\{I\le M\})$ follows from compactness and lower semicontinuity by taking minimizing sequences in the fibres; hence $J$ is good.

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.

Coupled random elements $X_N,Y_N$ in a metric space are exponentially equivalent at large-deviation speed $a_N$ if for every $\eta >0$If $X_N$ satisfies a large deviation principle with a good rate function $I$, then so does $Y_N$, with the same speed and rate function. For the upper bound, $\{Y_N\in F\}$ is contained in $\{X_N\in F^{\eta }\}\cup \{d(X_N,Y_N)>\eta \}$, where $F^{\eta }$ is the closed $\eta$-neighborhood. Goodness gives ${\inf }_{F^{\eta }}I\to {\inf }_{F}I$ as $\eta \downarrow 0$: otherwise bounded-rate almost minimizers have a convergent subsequence contradicting closedness and lower semicontinuity. For an open-set lower bound, choose a ball about any $x\in G$ with a slightly larger ball still in $G$, subtract the superexponentially small error from the probability that $X_N$ belongs to the smaller ball, and shrink the ball. No independence assumption is required.

For a continuous $F$ and a family satisfying a large deviation principle at speed $L$, the displayed exponential tail condition extends Varadhan's lemma to $F$ unbounded above. Apply the bounded-above result to $\min (F,M)$ and split the full integral into $F\le M$ and its tail; the tail is negligible and the truncated variational suprema increase to ${\sup }_x(F(x)-I(x))$. A sufficient condition is ${\mathrm{limsup}}_L{L}^{-1}\log \mathbb{E}{e}^{\gamma LF(X_L)}<\infty$ for some $\gamma >1$, since the tail is at most $e^{-(\gamma -1)LM}\mathbb{E}{e}^{\gamma LF(X_L)}$.

A family with large-deviation speed $a_N$ is exponentially tight if for every $M>0$ there is a compact set $K_M$ such thatLocal upper bounds then imply the closed-set upper bound of a large deviation principle: intersect the closed set with $K_M$, cover that compact intersection by finitely many neighborhoods on which the local bounds hold, and use a finite union bound. The remaining Probability has exponent at most $-M$; let $M\to \infty$.

Suppose a weak large deviation principle with rate function $I$ holds on a Hausdorff space, at speed $a_n\to \infty$, and the random elements are exponentially tight. For a finite $r$, choose a compact set $K$ whose complement has upper exponential rate strictly below $-M$, with $M>r$. The open-set lower bound on $K^c$ implies ${\inf }_{K^c}I>M$, so $\{I\le r\}\subset K$. Lower semicontinuity makes this sublevel set closed, hence compact. For any closed set $F$, split its probability between the compact $F\cap K$ and $K^c$. The upper exponential rate is at most $\max \{-{\inf }_{F}I,-M\}$. Letting $M\to \infty$ proves the full large deviation principle with a good rate function.

For equal prior probabilities and independent and identically distributed observations on a finite alphabet, minimum sum of errors in a simple hypothesis test givesFor nonnegative $a,b$ and $0<s<1$, $\min (a,b)\le a^s{b}^{1-s}$. Applying this termwise and using the independence of random variables to factor the sum yieldsOptimizing $s$, with an infimum if needed, proves $p_n^{\ast }\le \frac{1}{2}{e}^{-nC(P,Q)}$. This short argument gives a bound for every $n$. The theorem identifying the optimal error exponent also needs the converse: no decision rule can achieve a strictly larger limiting exponent. A log-likelihood ratio is a sum of independent contributions, so large deviation principle supplies the relevant converse idea.

For a concrete nondegenerate small-noise model, letwhere $a$ is globally Lipschitz, $\Sigma$ is a constant invertible matrix and the time horizon $T$ is finite. The path laws on $C([0,T],{\mathbb{R}}^d)$ satisfy a large deviation principle of speed $1/\epsilon$. Their Freidlin--Wentzell action isfor absolutely continuous paths with $\varphi (0)=x_0$ and finite displayed integral, and $+\infty$ otherwise. Thus the initial condition and admissible path class are part of the rate, not optional qualifications.

The action measures the squared control needed to move the deterministic trajectory away from $\dot{\varphi }=a(\varphi )$. For zero drift and constant scalar noise coefficient $\sigma >0$, the least-action path from $x_0$ to $y$ in time $T$ is the straight line, with cost $(y-x_0{)}^2/(2{\sigma }^{2}T)$, by the Cauchy-Schwarz inequality. The noiseless solution has action zero. Neighborhood probabilities have leading exponential cost $I_T/\epsilon$, in the precise open/closed-set sense of the large deviation principle.

For constant $\Sigma$, continuity of the additive-noise solution map under the uniform norm follows from Gronwall's inequality. The Schilder theorem for scaled Brownian motion and the contraction principle for large deviations explain the path-space rate. With state-dependent diffusion, the integrand uses the inverse covariance matrix $[\sigma (\varphi )\sigma (\varphi {)}^T{]}^{-1}$; regularity and nondegeneracy assumptions must be checked rather than using an inverse for a singular covariance.^[ref]

Let $X_N\in {\mathbb{R}}^d$ and $a_N\to \infty$. Define the scaled cumulant-generating functionSuppose its pointwise limit $\Lambda$ exists, is lower semicontinuous, has $0$ in the interior of its effective domain, and is essentially smooth: it is differentiable on that interior, which is nonempty, and its gradient norm diverges at every finite boundary point approached from the interior. Then $X_N$ satisfies a large deviation principle with good rate function given by the Legendre-Fenchel transformThe Chernoff bound supplies the upper estimate. The lower estimate uses exponential tilting at a parameter whose gradient selects the required mean; essential smoothness supplies the boundary approximation needed for the full lower bound. A finite boundary slope can leave an affine branch of ${\Lambda }^{\ast }$ outside this argument, so a separate lower-bound proof is then necessary.

Hugo Touchette develops connections between the large deviation principle and statistical mechanics. Rare fluctuations often have probabilities that decrease exponentially with system size or observation time; their rate function captures information beyond a local Gaussian approximation. His treatment uses this structure to derive variational descriptions of equilibrium and nonequilibrium states. Nonconcave entropies expose limits of ordinary Legendre transforms and help explain why different statistical ensembles need not be equivalent.^[ref]

A sequence has Hurstiness H when it satisfies a large deviation principle at the displayed speed with a good rate function having a unique zero and at least one strictly positive finite value. This definition concerns a sequence's exponential Probability scale; it does not assert a sample-path Hurst exponent or self-similarity. Larger H means slower rare-event decay. The positive finite value excludes the slower-speed degeneration of a large deviation principle.

A family satisfies this principle with speed $a_n\to \infty$ and rate function $I$ when the displayed variational identity holds for every bounded continuous real function $f$. A large deviation principle implies it by the Varadhan lemma. The lower bound restricts the expectation to a neighborhood of a chosen point; the upper bound partitions the bounded range of $f$ into small intervals and applies the closed-set probability bounds.

For nonnegative $X_n$, the identity also givesFor $\mu >0$, truncate $f(s)=-\mu s$ below at $-\mu R$. The resulting bounded-function identity applies, while the difference of the two exponential expectations is at most $e^{-\mu R{a}_n}$. Choosing $R$ sufficiently large makes this error negligible on the claimed exponential scale. The case $\mu =0$ is immediate. This extension identifies moment-decay rates of a passive scalar from fluctuations of its logarithmic stretching.

After diffusion arrest of Gaussian packet compression, define the logarithmic area increase $A_t=\frac{1}{2}\log [\det J(t)/\det J(0)]$. It is nonnegative. Assume $A_t/t$ has a large deviation principle with convex rate function $\mathcal{I}$, minimized at $\lambda >0$. The exact amplitude relation $f(t)=f(0)e^{-A_t}$ and the Laplace principle for probability measures give the displayed moment decay rate. The rate function refers to area, so rare width fluctuations are included rather than discarded without justification. For differentiable strictly convex $\mathcal{I}$, the interior minimizer satisfies ${\mathcal{I}}^{\prime }(s_{\mu })=-\mu$. If ${\mu }_{\ast }=-{\mathcal{I}}^{\prime }(0)$ is finite, then ${\gamma }_{\mu }=\mathcal{I}(0)$ for $\mu \ge {\mu }_{\ast }$: rare nearly unstretched packets dominate high moments. Below the threshold, $d({\gamma }_{\mu }/\mu )/d\mu =-\mathcal{I}(s_{\mu })/{\mu }^2<0$. These statements concern a positive Gaussian packet with flow-only averaging; a homogeneous signed random initial scalar requires a different averaging kernel. Arbitrary random $C(t)$ alone does not imply this statistical model.

A sequence $(Z_n)$ satisfies a Large deviation principle with speed $n$ and Rate function $I$ when probabilities of measurable sets $A$ decay exponentially between the bounds

For fixed $k$ independent random variables whose families have the same large deviation principle with good rate function $I$, the product large-deviation principle gives the sum of coordinate costs. Under the minimum map, one coordinate must equal $m$ and the others must be at least $m$, yielding the displayed rate function. If $I$ is convex with unique zero $\mu$, then $J(m)=I(m)$ for $m\le \mu$ and $J(m)=kI(m)$ for $m\ge \mu$.

Let $X$ have the positive-support geometric distribution $\mathbb{P}(X=k)=p(1-p{)}^{k-1}$ with $0<p<1$. As $L\to \infty$, $X/L$ satisfies a large deviation principle of speed $L$ and good rate functionFor every real threshold $x$, the exact upper-tail formula isThe maximum with zero matters at $x=0$ and at thresholds below the first lattice point; the uncapped exponent would otherwise give a Probability greater than one.

For positive $x$, dividing the logarithm by $L$ gives the limit $-ax$. These tails bound closed sets through their leftmost nonnegative point. For an open neighborhood of a positive $x$, choose integers $k_L$ with $k_L/L\to x$; their individual masses have the same logarithmic limit. Neighborhoods of zero have Probability tending to one, supplying rate zero there. Negative-only sets have Probability zero, and the rate sublevel sets are compact intervals.

For example, with $p=1/2$, $L=100$ and $x=1/5$, the event is $X\ge 20$ and has Probability $2^{-19}$. Meanwhile $X/L$ converges in Probability to zero. The nontrivial rate describes the rare excursions away from that degenerate weak limit.

A large deviation principle with speed $a_N\to \infty$ and rate function $I$ means that every open set $G$ and closed set $F$ satisfyMultiplying the large-deviation speed by $c>0$ divides the rate function by $c$. The speed must match the rare-event scale: a fixed exponential random variable divided by $N^p$ has speed $N^p$, whereas a sum of $N$ independent random variables with a normal distribution, divided by $N^p$, has speed $N^{2p-1}$ when $p>1$.

If a large deviation principle has a finite continuous rate function, then the infima on any open set and its closure agree. Approximate every closure point by points of the set and use continuity. The open-set lower and closed-set upper bounds consequently match, giving the displayed logarithmic limit, including the empty-set convention. A single-point event need not obey such an equality.

Let $X_i$ be independent and identically distributed with mean $\mu$, variance ${\sigma }^2\in (0,\infty )$, and a moment-generating function finite on an open interval about zero. Choose $b_n>0$ with $b_n\to \infty$ and $b_n/\sqrt{n}\to 0$. The moderate deviation principle foris a large deviation principle of speed $b_n^2$ and good rate function $I(x)=x^2/(2{\sigma }^2)$.

To see the Gaussian quadratic rate, center one summand and expand its logarithmic moment-generating function at zero. For every fixed real $\theta$,The limiting function is finite and differentiable everywhere, so the Gärtner–Ellis theorem applies; its Legendre-Fenchel transform is the displayed rate. This proof states an actual sufficient exponential-moment hypothesis rather than leaving the assumptions implicit.

For $b_n=n^{1/4}$, a fixed positive $Z_n=x$ corresponds to a sample-mean displacement of order $n^{-1/4}$, between the ordinary $n^{-1/2}$ central-limit scale and a fixed order-one displacement. Its tail has logarithmic Probability of order $-\sqrt{n}{x}^2/(2{\sigma }^2)$. The scaling is intermediate, but the rate is still quadratic under these hypotheses.

Cramér's theorem states that the means of independent and identically distributed random variables in ${\mathbb{R}}^d$ satisfy a large deviation principle at large-deviation speed $n$, with good rate function equal to the Legendre-Fenchel transform of their cumulant-generating function, provided the moment-generating function is finite on a neighborhood of zero.

Here the exponential distribution hasFor $x>0$, maximize $\theta x-\Lambda (\theta )$. Its derivative is $x-(\lambda -\theta {)}^{-1}$ and its second derivative is negative, so the maximizing parameter is ${\theta }_x=\lambda -x^{-1}$. Therefore the rate function of an exponential sample mean isFor $x\le 0$, letting $\theta \to -\infty$ in the same supremum gives $+\infty$, including the logarithmic divergence when $x=0$. The moment-generating function is finite near zero, so Cramér's theorem supplies the full large deviation principle.

The rate function is nonnegative, with its unique minimum at $\mu ={\lambda }^{-1}$; $I^{\prime \prime }(x)=x^{-2}>0$. It tends to infinity both as $x\downarrow 0$ and as $x\to \infty$. Its finite sublevel sets are closed bounded intervals contained in $(0,\infty )$, hence compact. Thus it is a good rate function.