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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.

Write $A_n$ for work arriving in slot $n$ and $c$ for the service rate of a queue. For a work-conserving single-server queue, the Lindley recursion is $Q_{n+1}=(Q_n+A_{n+1}-c{)}^{+}$. For stable independent and identically distributed random variables with mean $\mu <c$, its stationary workload supremum has the formDifferent scaling limits of a queue concern different quantities growing large; they need not share either a probability speed or a likely path to overflow.

For large-buffer queue scaling, fix the input law and $c>\mu$ and let the buffer size of a queue $b\to \infty$. Assume finite exponential moments around a positive root ${\theta }_{\ast }$ of ${\Lambda }_A(\theta )=c\theta$. An overflow accumulated in $n\simeq tb$ slots requires an average input about $c+1/t$, so Cramér's theorem assigns logarithmic cost $bt{I}_A(c+1/t)$. Optimizing the duration gives the Cramér-Lundberg workload exponentFor an interior root with ${\Lambda }_A^{\prime }({\theta }_{\ast })>c$, Fenchel–Young inequality verifies the minimization: the lower bound is ${\theta }_{\ast }$ because $I_A(x)\ge {\theta }_{\ast }x-{\Lambda }_A({\theta }_{\ast })$, and equality occurs at $t=[{\Lambda }_A^{\prime }({\theta }_{\ast })-c{]}^{-1}$. The exponential workload bound from cumulative arrival moments gives the upper exponent for every $\theta <{\theta }_{\ast }$; an exponentially tilted positive-drift path or the duration calculation gives the matching lower exponent. In effective bandwidth notation, ${\Lambda }_A({\theta }_{\ast })/{\theta }_{\ast }=c$. This regime measures long accumulation into a large buffer with a fixed positive stability margin.

For many-source queue scaling, superpose $L$ independent copies of an input process, give the server capacity $Lc$, and give it buffer size $Lb$. For each lookback horizon $n$, the average of the $L$ source cumulative arrivals has a large deviation principle with rate function $I_n$. Overflow at that horizon costs $L{I}_n(b+cn)$. Optimizing the lookback horizon suggestsFor independent slots, $I_n(x)=n{I}_A(x/n)$, giving $V(b)={\inf }_{n\ge 1}n{I}_A(c+b/n)$. The integer horizon is important in a slotted system. For temporally dependent sources, use the cumulative-input cumulant-generating function, rather than multiplying a one-slot formula by $n$. The heuristic needs uniform control of the tail of the stationary supremum to justify optimizing an unbounded set of horizons; stability and suitable exponential-moment estimates provide this in the usual models. The growing quantity here is the number of independent sources, while per-source capacity and buffer remain fixed.

For moderate-deviation queue scaling, consider a bufferless queue with aggregate Poisson input of mean $L\lambda$ and capacity $L\lambda +C{L}^{(1+\beta )/2}$, $0<\beta <1$. The spare capacity is much larger than the ordinary central limit theorem scale $\sqrt{L}$ but much smaller than $L$. The Poisson sample-mean rate function near its minimum has expansionWith $h=C{L}^{(\beta -1)/2}$, the cost $L{I}_P(\lambda +h)$ is asymptotic to $C^2{L}^{\beta }/(2\lambda )$. Hence Poisson moderate deviations give $\log \mathbb{P}(\mathrm{overflow})\sim -C^2{L}^{\beta }/(2\lambda )$. The quadratic cost comes from a shrinking neighborhood of the mean, rather than from a fixed macroscopic overload. The regime is useful for a large aggregate whose service margin lies between its typical fluctuation scale and its mean scale.

For a heavy-traffic limit of a queue, instead hold the input law fixed with mean $\mu$ and positive finite variance ${\sigma }^2$, and let $c_{\epsilon }=\mu +\epsilon d$ with $d>0$ and $\epsilon \downarrow 0$. In $t/{\epsilon }^2$ slots the net input has mean about $-dt/\epsilon$ and random fluctuation of size $\sigma \sqrt{t}/\epsilon$. Thus the functional central limit theorem, followed by the Skorokhod reflection map on the half-line, suggestswhere $R$ is reflected Brownian motion with negative drift $-d$ and variance parameter ${\sigma }^2$. The stationary law of negatively drifted reflected Brownian motion has tail $\mathbb{P}(R_{\infty }>b)=e^{-2db/{\sigma }^2}$. When passing the stationary distribution through the limit is justified, this predictsFor dependent input satisfying a suitable invariance principle, replace ${\sigma }^2$ by its long-run variance of a stationary process. This regime is appropriate near critical load and on long time scales; a functional transient limit alone does not automatically justify the stationary-limit interchange.

A different large-service regime holds the input law fixed. With Poisson distribution of mean $\lambda$ and bufferless capacity $c\to \infty$, let $m=\lfloor c\rfloor +1$. The single-point lower bound $\mathbb{P}(X=m)$ and the optimized Chernoff bound for $X\ge m$, using Stirling's formula, give the Poisson large-service overflow asymptoticHere the service rate grows while the input mean does not. A quadratic moderate-deviation approximation would describe the wrong part of the Poisson tail.

To choose a regime, compare the spare capacity, typical fluctuations, number of sources and buffer size before selecting a formula. Use the large-buffer exponent for a fixed stable input and a long overload episode; use the many-source variational cost when independent flows and resources grow together; use moderate deviations for an intermediate safety margin; use the heavy-traffic diffusion when the stability margin tends to zero; and use the fixed-input Poisson tail when service alone becomes very large. Check temporal dependence and exponential-moment assumptions. For heavy-tailed subexponential distributions, a single exceptionally large arrival can dominate overflow, so a finite exponential workload root and a Brownian approximation to the far tail need not apply. These comparisons explain why the same nominal queueing theory system can have different useful asymptotic descriptions.

A large molecular Schmidt number $\mathrm{Sc}=\nu /\kappa$ separates the velocity's smallest smooth length scale from the scalar's smaller diffusive length scale. Follow a Lagrangian trajectory $X(t)$ and measure displacement $x$ from it. Within a sufficiently small scalar packet, Taylor expansion of the smooth velocity field givesThe moving frame removes uniform sweeping, leaving a linear random velocity gradient tensor. Thus the local scalar equation is the advection-diffusion equation with velocity $Cx$, and averaging over realizations of $C$ gives the corresponding ensemble average. The approximation requires the packet to lie within the velocity's locally smooth region. Its thin, diffusive direction can satisfy this condition even when its long direction eventually becomes too large for one global linearization; applications to a spatially varying flow then use local packet descriptions. The statistical conclusions below refer to the linear-flow model and the usual mixing assumptions, not to every conceivable trace-free random matrix process.

Write $Q=\int \chi dA$ and assume sufficiently rapid decay to discard boundary terms. Integration by parts givesFor $J_{ij}=\int x_i{x}_j\chi dA$, integrate the advective term once and the diffusive term twice. Since ${\partial }_k(x_i{x}_j)={\delta }_{ki}{x}_j+{\delta }_{kj}{x}_i$ and $\Delta (x_i{x}_j)=2{\delta }_{ij}$, one obtainsThese are mass conservation and the evolution of the unnormalized second moment tensor. The first moment obeys $\dot{M}=CM$, so a centered packet stays centered.

For a positive Gaussian scalar packet in a linear incompressible flow, $B$ must be a positive-definite matrix. Diagonalizing its symmetric matrix and applying the Gaussian integral givesThe factor $Q^2$ in the last identity matters because $J$ has not been divided by the scalar mass. Substitution into the advection-diffusion equation also givesThese identities agree with the second moment equation and prove that the Gaussian family is preserved. For an initially negative packet the same analysis applies after replacing $\chi$ by $-\chi$ and taking its positive mass.

To see the roles of stretching and diffusion explicitly, let the deformation gradient $F$ solve $\dot{F}=CF$, $F(0)=I$. Incompressibility gives $\det F=1$. Differentiating $F^{-1}J{F}^{-T}$ integrates the moment equation:Without diffusion, $J=FJ(0)F^T$ and $\det J$ is constant. In a statistically stationary, mixing flow with positive Lyapunov exponent $\lambda$, the two stretching exponents are $\lambda$ and $-\lambda$. Writing the ordered eigenvalues of $J$ as $e^{2{\rho }_1},e^{2{\rho }_2}$, their typical initially nondiffusive evolution isThe fluctuation scale presumes the usual central limit theorem for accumulated strain; it is not implied by incompressibility alone. During this stage the packet becomes long and thin but keeps its area and peak concentration.

In the instantaneous principal-axis basis, write $\tilde{C}$ for the rotated velocity gradient tensor. Differentiation of an eigenvalue givesThe formulas hold through a repeated eigenvalue by continuity of their sum. The compressed physical variance is $e^{2{\rho }_2}/Q$. Its contraction is arrested when it is of order $\kappa /\lambda$, the square of the Batchelor scalar microscale ${\ell }_B\sim \sqrt{\kappa /\lambda }$. If its initial length is $L\gg {\ell }_B$, the crossover occurs atThereafter ${\rho }_2$ has stationary fluctuations associated with the strain and the diffusive floor, while ${\rho }_1$ continues to grow typically at rate $\lambda$. Under sufficiently fast mixing, the central part of the distribution of ${\rho }_1$ is approximately Gaussian with mean $\lambda t+O(1)$ and variance $D_{\rho }t+o(t)$ for a strain-dependent coefficient $D_{\rho }$. This central approximation does not determine far tails, which control high concentration moments; those require a full large deviation principle. The two logarithmic widths need not be independent. The area now increases and $f$ decreases typically as $e^{-\lambda t}$. This is diffusion arrest of Gaussian packet compression.

For example, a constant principal strain $C=\mathrm{diag}(\lambda ,-\lambda )$ gives the exact physical variancesThe thin variance approaches its diffusive floor, the long variance grows exponentially, and the mass-to-area relation gives the quoted typical amplitude decay. A random strain additionally makes rare histories important for concentration moments.

For a precise statistical formulation define the logarithmic area increaseThe Gaussian identities give the exact relation $f(t)=f(0)e^{-H_t}$; henceAssume the long-time strain statistics give a large deviation principle for $H_t/t$ on $s\ge 0$, with convex good rate function $I(s)$, $I(\lambda )=0$. At exponential accuracy the probability of $H_t/t$ near $s$ is $e^{-tI(s)}$. Combining that cost with the amplitude factor and applying the Laplace principle for probability measures gives the large-deviation decay rates of a Gaussian scalar packetThe initial amplitude only changes a time-independent prefactor. One can see the minimization directly by dividing the possible area-growth rates into small intervals: each contributes its probability cost times $e^{-\mu ts}$, and the interval with least total exponential cost dominates. An interval around a minimizing $s$ gives the matching lower bound. The nonnegative area increase bounds the exponential weight by one, so unusually large positive growth cannot produce a divergent weighted tail. Using the area rather than only ${\rho }_1$ also avoids discarding possibly important fluctuations of the compressed direction.

Suppose $I$ is strictly convex and differentiable, with its minimum at $\lambda >0$ and finite negative right derivative at zero. DefineFor $0<\mu <{\mu }_{\ast }$ the minimizing rate $s_{\mu }$ is positive and uniquely solvesFor $\mu \ge {\mu }_{\ast }$, convexity puts the minimum at the endpoint $s=0$, givingHigh moments are dominated by rare histories with little net area growth, in which a packet keeps a substantial concentration. Their probability cost is $I(0)$, independent of the moment order. A finite threshold is conditional: if $I^{\prime }(0+)=-\infty$, the minimizer can approach zero without reaching it at any finite $\mu$.

Below the threshold, differentiation of the minimizing expression cancels the terms involving $s_{\mu }^{\prime }$ because $I^{\prime }(s_{\mu })+\mu =0$. Thus ${\gamma }_{\mu }^{\prime }=s_{\mu }$, andStrict negativity uses a nondegenerate rate function with a unique zero at $\lambda$: for positive $\mu$, $s_{\mu }<\lambda$ and $I(s_{\mu })>0$. For deterministic strain the ratio can instead be constant. Near the typical rate, if $I(s)\simeq (s-\lambda {)}^2/(2D)$ with $D>0$, then ${\gamma }_{\mu }\simeq \lambda \mu -D{\mu }^2/2$ for small $\mu$. If this quadratic expression holds all the way to zero, it gives the illustrative full resultThe quadratic form is an example, not a universal law of turbulent strain.

A thin scalar packet is stretched until molecular diffusion broadens its compressed direction; conservation of its scalar mass then converts area growth into dilution. Ensemble averages sample different stretching histories, producing different decay rates for different moments and saturation when weakly stretched histories dominate. The present moment calculation is specifically for the centered Gaussian packet and averaging over the flow; signed random initial fields with overlapping packets can also have cancellations and require their initial-field statistics.

Finally, positive exponential decay is a statistical conclusion requiring the strain assumptions above, not a consequence of the local differential equation by itself. For $C\equiv 0$ and initially isotropic physical variance $L^2$, the exact solution has $J=Q(L^2+2\kappa t)I$ and $f=Q/[2\pi (L^2+2\kappa t)]$. Its moments decay algebraically as $t^{-\mu }$. Thus the general equations for the exponential rates are the rate-function minimization above; numerical rates and a finite saturation threshold require the long-time statistics of the flow.

Use the positive-integer geometric distribution, $\mathbb{P}(X=k)=p(1-p{)}^{k-1}$ for $k\ge 1$, initially with $0<p<1$. Put $a=-\log (1-p)>0$. The precise good rate function on $\mathbb{R}$ isThe large-deviation speed is $L\to \infty$. We prove both bounds of the large deviation principle for $X^L=X/L$.

For $x>0$, the geometric tail gives the exact expressionLet $F$ be a closed set. If it does not meet $[0,\infty )$ its probability is zero. If it contains zero, $\mathrm{limsup}{L}^{-1}\log \mathbb{P}(X^L\in F)\le 0=-{\inf }_{F}I$. Otherwise, if it meets the nonnegative half-line, its least possible nonnegative value $r$ is positive: a sequence in $F$ approaching zero would force zero into the closed set. Its event is contained in $\{X^L\ge r\}$, giving

For an open set $G$ containing $x>0$, choose positive integers $k_L$ with $k_L/L\to x$. Eventually $k_L/L\in G$, andIf $0\in G$, the event $X=1$ alone gives lower exponential rate zero; alternatively the probability of a neighborhood of zero tends to one. Optimizing over $x\in G$ proves the open-set lower bound. Sets containing no nonnegative point require only the trivial lower bound $-\infty$.

The rate function is lower semicontinuous at zero as well as elsewhere, and $\{I\le r\}=[0,r/a]$ for finite $r\ge 0$ is compact. Hence the large deviation principle has a good rate function. This establishes the large deviations of a scaled geometric random variable. For $p=1$, $X=1$ deterministically and the rate instead is zero at $x=0$ and infinite elsewhere; the displayed logarithmic coefficient must not be treated as an ordinary finite number in that case.

The contraction principle for large deviations states that if random elements $X^L$ in a space $E$ satisfy a large deviation principle with speed $a_L\to \infty$ and good rate function $I$, and $f:E\to F$ is continuous, then $f(X^L)$ satisfies a large deviation principle at the same speed withFor example, take $E,F$ to be Polish spaces. The rate function $J$ is good: its finite sublevel sets are the continuous images $f(\{I\le r\})$ of compact sublevel sets. Compactness and lower semicontinuity ensure that a finite fibre infimum relevant to such a sublevel set is attained. The probability bounds follow by taking preimages of open and closed sets.

With $n$ fixed, the asymptotic family is the scaled sumAn unscaled fixed random variable alone has no diverging parameter for a large deviation principle; this scaling is the continuation of part (a). Suppose first $0<p_i<1$ and put $a_i=-\log (1-p_i)$. Independence and the product large-deviation principle give the vector $(X_1/L,\dots ,X_n/L)$ rateAddition is continuous, so the contraction principle for large deviations givesFor $s\ge 0$ every feasible vector has cost at least $s{a}_{\ast }$, where $a_{\ast }={\min }_i{a}_i$. Equality holds by assigning all scaled excess to an index attaining $a_{\ast }$. For $s<0$ the constraint set is empty. ThusEquivalently, the coefficient is $-\log (1-p_{\ast })$ with $p_{\ast }={\min }_i{p}_i$. On the exponential scale the rare large sum is carried by a variable having the slowest geometric tail. Several equal slowest tails can change polynomial prefactors, but not this rate function. This is the large deviations of a fixed sum of geometric random variables.

As a direct check of the tail exponent, for each $0<\theta <a_{\ast }$ the moment-generating function of the sum is finite, so the Chernoff bound gives an upper rate at most $-\theta s$ for $s>0$. Letting $\theta \uparrow a_{\ast }$ gives $-a_{\ast }s$. For a lower neighborhood bound, keep all other variables at fixed finite values and let a slowest-tail variable take an integer near $Ls$; its probability has rate $-a_{\ast }s$. If some $p_i=1$, their deterministic contributions disappear after division by $L$, so minimize over the nondeterministic variables. If all $p_i=1$, the rate is zero only at zero and infinite elsewhere.

A standard form of Varadhan's integral theorem is as follows. Let ${\mu }_L$ be probability measures on a Polish space $E$ satisfying a large deviation principle with speed $L\to \infty$ and good rate function $I$. If $F:E\to \mathbb{R}$ is continuous and bounded above, thenThe usual bounded-continuous version follows immediately. If $F$ is unbounded above, the same conclusion holds under the additional exponential tail conditionThis is the exponential tail extension of Varadhan's lemma. A convenient sufficient hypothesis isAn integrability hypothesis is essential for general unbounded functions; the ordinary large deviation principle alone does not control arbitrarily large exponential weights.

For example, let $X^L=L$ with probability $e^{-L^2}$ and $X^L=0$ otherwise. It satisfies the large deviation principle at speed $L$ with rate zero at zero and infinite elsewhere. But for $F(x)=x^2$, $\mathbb{E}{e}^{LF(X^L)}\ge e^{L^3-L^2}$, whose logarithmic rate diverges, while $\sup (F-I)=0$. This shows why the tail hypothesis cannot simply be omitted.

For the lower bound, fix $x$ with $I(x)<\infty$ and $\epsilon >0$. By continuity choose an open set $G$ containing $x$ on which $F(y)\ge F(x)-\epsilon$. ThenApply the open-set lower bound of the large deviation principle:Let $\epsilon \downarrow 0$ and take the supremum over $x$ to obtain the required lower bound. This argument also applies when $F$ is not bounded above.

For the upper bound, suppose $F\le M$ and set $S={\sup }_E(F-I)$, which is finite. Choose a large $K$ and partition $[-K,M]$ into finitely many closed intervals $[a_j,a_j+\epsilon ]$ of length at most $\epsilon$. The preimagesare closed sets. The overlapping endpoints do not matter for an upper estimate. Split off the lower tail and bound each remaining exponential weight:For a finite sum of nonnegative terms the upper logarithmic rate is at most the largest of their upper logarithmic rates. Applying the closed-set upper bound givesFor $x\in C_j$, $a_j+\epsilon \le F(x)+\epsilon$, hence every inner term is at most $S+\epsilon$. Empty preimages contribute no term. Let $K\to \infty$ and then $\epsilon \downarrow 0$ to get the upper bound $S$. Together with the lower bound this proves the bounded-above theorem.

For the unbounded extension, put $F_M=\min (F,M)$. This is continuous and bounded above, so the proved result applies, withThe full integral is at most the integral with $F_M$ plus the integral on $\{F>M\}$. The upper logarithmic rate is therefore at most the larger of $S_M$ and the tail rate. The tail hypothesis and then $M\to \infty$ give the upper bound $S$; the earlier local argument gives the lower bound. If $S=+\infty$, that lower argument already yields the asserted infinite limit.

Finally, the sufficient exponential-moment condition really implies the required tail estimate: on $F>M$,After integration the upper logarithmic rate is bounded by a finite constant minus $(\gamma -1)M$, which tends to $-\infty$. This completes the statement and proof of Varadhan's lemma, including the hypotheses needed for the unbounded version.

At large-deviation speed $L$, the exponential distribution gives the large deviation principle on $\mathbb{R}$ with good rate functionIn particular, $\mathbb{P}(B/L\ge b)=e^{-\lambda Lb}$ for $b\ge 0$. Its finite sublevel sets are the compact sets $[0,r/\lambda ]$.

Write $S_L={\sum }_{i=1}^L{A}_i$ and $Z_L=S_L/L$. The cumulant-generating function of one normal random variable isCramér's theorem states that empirical means of independent and identically distributed random variables whose moment-generating function is finite on a neighborhood of zero satisfy a large deviation principle at the sample-size large-deviation speed, with good rate function equal to the Legendre-Fenchel transform of their cumulant-generating function. Here that givesFor ${\sigma }^2>0$, differentiation gives the maximizing parameter $\theta =(x-\mu )/{\sigma }^2$, and henceThis rate function tends to infinity as $|x|\to \infty$, so its finite sublevel sets are compact. Equivalently, the Gärtner–Ellis theorem applies because the limiting scaled cumulant-generating function is finite and differentiable everywhere; it is lower semicontinuous and essentially smooth, with no finite domain boundary at which to check steepness. If ${\sigma }^2=0$ is permitted, $Z_L=\mu$ deterministically and the rate function is zero at $\mu$ and infinity elsewhere.

The product large-deviation principle states that independent families satisfying large deviation principles at the same speed with good rate functions $J$ and $I$ have joint good rate function $J(b)+I(z)$. The contraction principle for large deviations states that a continuous map $f$ sends such a principle to one with good rate function ${\inf }_{f(u)=x}I(u)$. Apply these results to $(B/L,S_L/L)$ and addition. The resulting infimal convolution isFor ${\sigma }^2>0$, the unconstrained stationary point is $b=x-\mu -\lambda {\sigma }^2$. Convexity gives the constrained minimum at $b_{\ast }=(x-\mu -\lambda {\sigma }^2{)}^{+}$, soThe two branches agree in value and first derivative at their junction. Both tails tend to infinity, giving a good rate function. This is the fixed exponential perturbation of a Gaussian empirical mean.

There is a reason to use the product large-deviation principle rather than invoke the supplied Gärtner–Ellis theorem without checking its hypotheses. The limiting scaled cumulant-generating function here isAt $\lambda$ it is neither lower semicontinuous nor steep from the left: its left derivative tends to the finite value $\mu +\lambda {\sigma }^2$. Its Legendre-Fenchel transform still equals $K$, but the stated Gärtner–Ellis theorem does not establish the full lower bound on the linear branch. The product large-deviation principle and contraction principle for large deviations do establish it. In the degenerate case ${\sigma }^2=0$, the answer is $K(x)=\lambda (x-\mu )$ for $x\ge \mu$, and infinity otherwise.

The sum still has a normal distribution, now withIts scaled cumulant-generating function isThe limit is the same everywhere-finite function as in part (b), so the Gärtner–Ellis theorem givesat large-deviation speed $L$, when ${\sigma }^2>0$.

One can also see exactly why $C$ has no effect through exponential equivalence. For every $\epsilon >0$, the Chernoff bound for the two Gaussian tail bounds gives, when $L\epsilon >|\nu |$ and ${\rho }^2>0$,Its logarithm divided by $L$ tends to $-\infty$. If ${\rho }^2=0$, this probability is eventually zero. The transfer result for exponential equivalence says that coupled families whose distance exceeds each fixed positive tolerance with superexponentially small probability share a large deviation principle with the same good rate function. Apply it to $(C+S_L)/L$ and $S_L/L$. This also covers ${\sigma }^2=0$, with the point-mass rate function from part (b).

Work in a Hausdorff space $E$, for example a metric space. A rate function is a lower semicontinuous map $I:E\to [0,\infty ]$. A good rate function additionally has compact finite sublevel sets $\{x:I(x)\le r\}$.

A large deviation principle at large-deviation speed $L$ requires, for every open set $G$ and closed set $F$,The conventions are $\log 0=-\infty$ and $\inf \varnothing =+\infty$. A weak large deviation principle imposes the same lower bound but only the upper bound for compact sets. Exponential tightness supplies compact sets $K_M$ with $\mathrm{limsup}{L}^{-1}\log \mathbb{P}(X_L\notin K_M)<-M$ for every $M\ge 0$. The strict inequality can equivalently be obtained from the common non-strict definition by choosing a larger containment level.

The indexing defect also affects the downstream claim. For the two inputs in part (b), now choose $\lambda <d<c$. The constant input has zero downstream queue workload, while the input with $a_0=c+1$ hasTheir printed distance is zero, so the literal post-slot downstream continuity assertion is false as well.

In the corrected weighted cumulative-input topology for a slotted queue, apply the finite-window proof of part (b) separately with services $d$ and $c$. Both have strictly negative long-window drift because $\lambda <d<c$. There are a neighborhood of $a$ and finite constants $C_d,C_c$ such thatThis proves continuity, and indeed local Lipschitz continuity, for the intended downstream function.

Suppose a family of random arrival paths $a^{(L)}$ has a large deviation principle in this corrected input space, at speed $L$ with good rate function $\mathcal{I}$. The contraction principle for large deviations then gives the downstream good rate functionIt is infinity for $r<0$. The path large deviation principle, its topology, and its subcritical domain must actually be established for the chosen stochastic model: a collection of finite-dimensional large deviation principles alone does not control infinitely long workload windows. Nor should upstream departures be assumed independent; the path contraction avoids that unjustified assumption.