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Let $\kappa =2(1-H)$ and fix $a>0$. Starting with part (b), the contraction principle for large deviations under the map $q\mapsto q/a$ gives an LDP for $r(X)/(aN)$ with speed $N^{\kappa }$ and rate function $q\mapsto J(aq)$.

Alternatively, replace $N$ by $aN$ in parts (a) and (b), using the real-scale justification in part (a). The same family has large-deviation speed $(aN{)}^{\kappa }$ and rate function $J(q)$. When expressed at speed $N^{\kappa }$, that rate function is $a^{\kappa }J(q)$. The two descriptions of the same large deviation principle must agree:For clarity, uniqueness here is a general property of lower semicontinuous rate functions: at a point $x$, the shrinking-neighborhood logarithmic bounds recover $I(x)$, since ${\inf }_{B(x,\eta )}I\to I(x)$ as $\eta \downarrow 0$. Thus two rate functions for the same family and speed cannot differ.

Taking the base argument $q=1$ and replacing $a$ by any positive queue size givesFor $q<0$ the rate function is infinite. In the usual nondegenerate case $J(1)$ is finite, so the formula also holds at $q=0$. This is the self-similar Gaussian workload rate; the exponent comes from the Hurst parameter, while the constant comes from the path rate function and the drift and noise scales.

For the amount of work $A_j(0,t]$ offered by one flow in a time interval of length $t$, its effective bandwidth at parameter $\theta >0$ and timescale $t>0$ isIts units are work per unit time. The parameter $\theta$ measures the price placed on bursts: a stringent rare-overflow target generally requires a larger value of $\theta$ than a loose target. The timescale $t$ describes the duration of the congestion episode. When the limit exists, writeThis is the long-time effective bandwidth. It includes a burstiness cost, rather than just the mean offered rate. For a finite-time flow with the required moments, ${\lim }_{\theta \downarrow 0}{\alpha }_j(\theta ,t)=\mathbb{E}{A}_j(0,t]/t$, and effective bandwidth is increasing in $\theta$. If $A_j(0,t]/t$ is bounded, its limit as $\theta \to \infty$ is the essential supremum of that interval's input rate, the peak-rate limit. The analogous long-time mean limit requires the usual interchange or regularity assumptions.

For independent flows their cumulant-generating functions add, soThe Chernoff bound therefore gives, for any $\theta >0$ in the common domain,Optimizing over $\theta$ produces the relevant Legendre-Fenchel transform. The Cramér theorem states that sample means of independent and identically distributed inputs whose moment-generating function is finite near zero have a large deviation principle, with rate function equal to the Legendre-Fenchel transform of their cumulant-generating function. More generally, the Gärtner–Ellis theorem applies to a convergent scaled log moment-generating function when zero lies inside its domain and the limit is lower semicontinuous and essentially smooth. It gives the same transform as a good rate function. A sample-path version, together with the contraction principle for large deviations for the queue workload, translates input deviations into overflow deviations. The compact-time and infinite-horizon continuity/tail hypotheses must be checked for the chosen traffic model.

A particularly useful rigorous queue bound holds for inputs with stationary increments and independent increments, such that $\mathbb{E}{e}^{\theta A(t)}=e^{t\Lambda (\theta )}$. If $\Lambda (\theta )\le C\theta$, thenis a nonnegative supermartingale, since its conditional increment factor is $e^{(t-s)(\Lambda (\theta )-C\theta )}\le 1$. The maximal inequality for a nonnegative supermartingale on $[0,T]$, followed by $T\to \infty$, gives the continuous-time exponential workload boundHere $W$ is the stationary queue workload of the stable infinite-buffer queue, represented by input from the past. This directly bounds the supremum over time; applying a one-time Chernoff bound alone would not establish it.

For a nonzero Compound Poisson process with bounded positive packet sizes and mean input rate less than $C$, a positive root ${\theta }_{\ast }$ of $\Lambda (\theta )=C\theta$ exists and is unique. Here the root theorem can be made explicit:The upper bound was just proved. For the lower bound, use exponential tilting with density $e^{{\theta }_{\ast }(A(t)-Ct)}$ at time $t$. Under the tilted law the net input has positive mean drift ${\Lambda }^{\prime }({\theta }_{\ast })-C>0$, by strict convexity and the positive-root equation. The strong law of large numbers then makes the first crossing time ${\tau }_b$ finite almost surely under that law. Its overshoot is at most $s_{\max }$. Changing measure at ${\tau }_b$, first on $\{{\tau }_b\le T\}$ and then letting $T\to \infty$, givesDividing the logarithmic bounds by $b$ proves ${\lim }_{b\to \infty }{b}^{-1}\log \mathbb{P}(W\ge b)=-{\theta }_{\ast }$.

Thus the familiar admission rule compares ${\sum }_j{a}_j(\theta )$ with $C$, choosing $\theta$ from the desired buffer-tail exponent. For more general inputs, a logarithmic root theorem requires its own light-tail and renewal or sample-path hypotheses; the displayed supermartingale upper bound needs only the moment and increment assumptions stated above.

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.

Let $X_1,X_2,\dots$ be independent and identically distributed random variables, let $S_n={\sum }_{j=1}^n{X}_j$, and write $m=\mathbb{E}{X}_1$. Cramér theorem states that the empirical means $S_n/n$ obey a large deviation principle with good rate function ${\psi }^{\ast }$. In particular, for $x\ge m$,with the usual extended-real interpretation; the analogous lower-tail formula holds for $x\le m$.

Let a centered Gaussian process $Z$ satisfy $Z(a\cdot )\stackrel{d}{=}{a}^{H}Z$ with $0<H<1$, and suppose $Z/\sqrt{L}$ has a large deviation principle in the sublinear path space for fluid queues, with speed $L$ and good rate function $I$. For $\delta ,\sigma >0$, put $r={\sup }_{t\ge 0}(\sigma Z(t)-\delta t)$ and $\kappa =2(1-H)$. Self-similarity gives $Z(N\cdot )/N\stackrel{d}{=}Z/\sqrt{N^{\kappa }}$. Time substitution and the contraction principle for large deviations therefore give an LDP for $r/N$ with speed $N^{\kappa }$ andThe same family $r/(aN)$ has rate function $J(aq)$ by contraction, and $a^{\kappa }J(q)$ by replacing $N$ with $aN$ and expressing the speed again as $N^{\kappa }$. Uniqueness of a rate function gives $J(aq)=a^{\kappa }J(q)$, hence $J(q)=q^{\kappa }J(1)$ for $q>0$, with $J(0)=0$.

Let $S$ be a fixed sum of independent exponential random variables with smallest rate $\lambda >0$, and let $Z_N$ be the sum of $N-k$ independent rate-$r$ exponential random variables, independent of $S$, with fixed $k$ and $r>\lambda$. At large-deviation speed $N$, $S/N$ has rate function $\lambda y$ for $y\ge 0$; the sample mean has rate function $K_r(z)=rz-1-\log (rz)$ for $z>0$. The product large-deviation principle and contraction principle for large deviations giveIndeed the derivative of the quantity minimized with respect to $z$ is $r-\lambda -1/z$. Its minimizer is $\min (x,(r-\lambda {)}^{-1})$. Beyond that transition, a single slow exponential random variable carries the extra excursion; the rate function becomes affine.

A steady state does not by itself give every fluctuating observable an entropy production symmetry. If the dimensionless total entropy production ${\Sigma }_t$ has the same-law finite relation and a large deviation principle with rate function $I$ for ${\Sigma }_t/t$, then, on its common domain,The first expression denotes leading exponential scaling; the second follows by taking the scaled logarithm of the probability ratio. A finite-state, autonomous Markov process with time-even states, positive bidirectional rates and a positive stationary distribution supplies a standard setting for a finite total-entropy production relation. The stochastic system entropy is then bounded, so its endpoint contribution does not change the long-time rate function of bath entropy production. ^[ref]

Unbounded state spaces can have endpoint fluctuations on the scale relevant to a large deviation principle. Dropping system entropy, changing the measured observable to heat, or changing the reverse preparation therefore requires a fresh argument. The exact finite integral fluctuation theorem, a same-law finite relation, and an asymptotic rate function symmetry are distinct claims.

If the independent exponential random variables $T_i$ have rates $i\lambda$, thenwhere the $E_i$ are independent exponential random variables of rate $\lambda$. To see this, the first of $N$ independent exponential clocks rings at rate $N\lambda$; after it rings, the memoryless property leaves $N-1$ independent rate-$\lambda$ clocks. The successive order statistic gaps are therefore independent with rates $N\lambda ,(N-1)\lambda ,\dots ,\lambda$. Their sum has the same distribution as $\sum T_i$. In particular, for $x>0$, the tail at $Nx$ has logarithm $-\lambda Nx+o(N)$, since $1-(1-e^{-\lambda Nx}{)}^N\sim N{e}^{-\lambda Nx}$. The sum divided by $N$ has a large deviation principle with speed $N$ and rate function $\lambda x$ on $x\ge 0$, infinity otherwise.

If random elements satisfy a large deviation principle at speed $a_n\to \infty$ with rate function I, then a bounded continuous real function f satisfies the displayed exponential-integral identity. For the upper bound, partition its bounded range into finitely many closed intervals of width epsilon and apply the closed-set upper bound to their preimages. The largest exponential term controls the finite sum. For the lower bound, restrict to an open neighborhood where f is within epsilon of its value at a chosen point and use the open-set lower bound. Let epsilon tend to zero and optimize over the point. Boundedness avoids a separate exponential tail assumption.