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If a good rate function has unique zero mu, every closed set missing mu has strictly positive infimum rate, possibly infinity. Otherwise a sequence of rates tending to zero lies eventually in the compact unit sublevel set; a subsequence converges to a point of the closed set, and lower semicontinuity makes that point another zero. Compactness also ensures existence of a zero whenever the infimum is zero.

A sublevel set which is compact. A good rate function has this property for every finite level.

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.

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.

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.

Let $A_i$ be independent random variables with normal distribution $N(\mu ,{\sigma }^2)$, ${\sigma }^2>0$, and let $B$ be an independent exponential random variable of rate $\lambda >0$. At speed $L$, $(B+{\sum }_{i=1}^L{A}_i)/L$ has the good rate functionThe product large-deviation principle gives the joint cost $\lambda b+(z-\mu {)}^2/(2{\sigma }^2)$ for $b\ge 0$, and the contraction principle for large deviations gives its infimal convolution under $b+z=x$. The minimum occurs at $b=(x-\mu -\lambda {\sigma }^2{)}^{+}$. The fixed exponential perturbation changes the far upper tail even though it vanishes in probability after division by $L$; a fixed Gaussian random variable divided by $L$ instead has exponential equivalence with zero at speed $L$.

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.

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.

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 $m\ge \mu$.

For a geometric random variable $X$ with $\mathbb{P}(X=k)=p(1-p{)}^{k-1}$, $0<p<1$, the family $X/L$ has large-deviation speed $L$ and good rate function $I(x)=ax$ for $x\ge 0$, infinite otherwise, where $a=-\log (1-p)$. The exact tail $\mathbb{P}(X/L\ge x)=(1-p{)}^{\lceil Lx\rceil -1}$ gives the upper bound. A lattice point tending to any positive $x$ supplies the open-neighborhood lower bound; neighborhoods of zero have probability tending to one. Finite sublevel sets are compact intervals.

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.

For fixed $k$, independence and the product large-deviation principle give the vector $(Y_n(1),\dots ,Y_n(k))$ the good rate function ${\sum }_{j=1}^{k}I(y_j)$. Since the minimum is a continuous map, the contraction principle for large deviations givesIn every vector in this fibre, at least one coordinate equals $m$ and all others are at least $m$. The large-deviation rate of a minimum of independent copies therefore reduces the infimum toIf $0<m<\mu ={\lambda }^{-1}$, the other coordinates can equal $\mu$ and cost zero. If $m\ge \mu$, the rate function is increasing on $[\mu ,\infty )$, so every coordinate costs at least $I(m)$ and equality is achieved when all coordinates equal $m$. For $m\le 0$ the cost is infinite. ThusAt the threshold both expressions are zero. Goodness follows from the contraction principle for large deviations. A small minimum requires only one atypically small mean; a large minimum requires all $k$ means to be atypically large, which explains the two costs.

Varadhan's lemma states that if $U_n$ has a large deviation principle with good rate function $K$ and large-deviation speed $a_n$, then for every bounded continuous real function $h$,For a continuous function unbounded above, the same conclusion holds under the exponential tail extension of Varadhan's lemma, for example when the weighted contribution from $h>M$ is superexponentially negligible as $M\to \infty$.

The scalar clipping map $g(m)=\min (b,\max (a,m))$ is continuous. The rate function under scalar clipping has endpoint costs ${\inf }_{m\le a}J(m)$ and ${\inf }_{m\ge b}J(m)$. Since $a<\mu <b$ and the costs decrease towards $\mu$ from the left and increase to its right, both infima are attained at the corresponding endpoints. Thus the good rate function of $Z_n$ equals $J(z)$ on $[a,b]$ and is infinite outside. The function $\log z$ is bounded and continuous on that compact interval, soOn $[a,\mu ]$, the derivative of $\log z-I(z)$ is $2/z-\lambda >0$, so the maximum there is $\log \mu$. On $[\mu ,b]$, the derivative is $(k+1)/z-k\lambda$, with negative second derivative. Its unconstrained maximizer isIt lies in $[\mu ,b]$ whenever $k\ge (\lambda b-1{)}^{-1}$. The right-hand objective increases initially at $\mu$, so this maximizer dominates the whole left-hand interval. Substitution gives the limit for high moments of a clipped minimum of exponential sample meansThe moment here is the expectation of the $n$th power. If the printed notation were instead read as $(\mathbb{E}{Z}_n{)}^n$, its logarithmic limit would be $\log \mu =-\log \lambda$, since boundedness and convergence in probability give $\mathbb{E}{Z}_n\to \mu$.

A large deviation principle for random elements $X^L$ in a Hausdorff space $E$, at large-deviation speed $a_L\to \infty$, has a lower semicontinuous rate function $I:E\to [0,\infty ]$. For every open set $G$ and every closed set $F$,Equivalently, for every Borel set $A$, the lower and upper bounds use its interior and closure, respectively. The conventions are $\inf \varnothing =+\infty$ and $\log 0=-\infty$. A good rate function has compact sublevel sets $\{I\le r\}$ for every finite $r$. Here take $a_L=L$; the different speeds in the moderate deviation principle are stated explicitly when used.

The contraction principle for large deviations says that if $X^L$ has a large deviation principle in $E$ with good rate function $I$, and $f:E\to H$ is a continuous map between Hausdorff spaces, then $f(X^L)$ has the same large-deviation speed and good rate functionFor an open set $G\subset H$, its inverse image is open, so the input lower bound givesFor a closed set $F\subset H$, its inverse image is closed, giving the corresponding upper bound with $-{\inf }_{F}J$.

It remains to prove goodness, including the endpoint issue in the infimum. Each fibre $f^{-1}(\{y\})$ is closed. If $J(y)<\infty$, compact sublevel sets and lower semicontinuity make $I$ attain its infimum on that fibre: intersect it with the compact set $\{I\le J(y)+1\}$ and use the closed nested sublevels tending down to $J(y)$. Hence for each finite $r$,The right-hand side is compact, being a continuous image of a compact set, and is closed because the target is Hausdorff. Thus $J$ is lower semicontinuous and good, completing the proof.

Use the standard Hausdorff state-space convention, so compact sets are closed. Let $F$ be any closed set and choose $K_M$ from exponential tightness. The weak large deviation principle applies to the compact set $F\cap K_M$, whileThe principle of the largest exponential term givesLet $M\to \infty$. This is the full closed-set upper bound; the open-set lower bound was already present.

For goodness, fix $r<\infty$ and choose $M>r$. The complement $K_M^c$ is open. Its lower bound and its exponential-tightness upper estimate implyThus ${\inf }_{K_M^c}I\ge M$ and $\{I\le r\}\subset K_M$. Lower semicontinuity makes this sublevel set closed, so it is a closed subset of a compact set and therefore compact. This proves that exponential tightness upgrades a weak large deviation principle to the full principle with the same good rate function.