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An autoregressive process of order one with independent and identically distributed random variables as innovations is an affine process when the innovation exponential moments are finite. Its one-step coefficients are $A(\theta )=a\theta$ and $B(\theta )=b\theta +\psi (\theta )$, where $\psi$ is the innovation cumulant-generating function.

For an independent binomial distribution count $N\sim \mathrm{Binomial}(n,p)$, the aggregate can also be represented as ${\sum }_{j=1}^n{B}_j{X}_j$, with independent parameter-$p$ Bernoulli random variables $B_j$ and independent claim sizes. Its cumulant-generating function is $n\log (1-p+p{M}_X(t))$. Its third cumulant is $n(p{m}_3-3p^2{m}_1{m}_2+2p^3{m}_1^3)$ in severity moments. Constant positive claims $d$ give $np(1-p)(1-2p)d^3$, which is negative when $p>1/2$. Thus positive claim sizes do not by themselves force a positive third central moment.

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.

The $n$th cumulant is the coefficient of $t^n/n!$ in the cumulant-generating function. The first two cumulants are the expected value and variance; the third is the third central moment.

For a random sum of independent claims, conditioning on the count gives $M_S(t)=\mathbb{E}[M_X(t{)}^N]$ and therefore the displayed cumulant-generating function. Write $a_j={\kappa }_{N,j}$ and $b_j={\kappa }_{X,j}$. The chain rule gives ${\kappa }_{S,1}=a_1{b}_1$, ${\kappa }_{S,2}=a_1{b}_2+a_2{b}_1^2$, and ${\kappa }_{S,3}=a_1{b}_3+3a_2{b}_1{b}_2+a_3{b}_1^3$. In raw severity moments $m_j=\mathbb{E}{X}^j$, the third cumulant is $a_1{m}_3+3(a_2-a_1)m_1{m}_2+(a_3-3a_2+2a_1)m_1^3$. Positive exponential moments justify two-sided differentiation near zero; finite ordinary moments alone suffice for the corresponding left derivatives when claims are positive.

An Edgeworth expansion approximates a probability density function by a normal distribution multiplied by Hermite polynomials encoding higher cumulants. For a centered variable of variance ${\sigma }^2$ and third cumulant ${\kappa }_3$,The correction follows from inverse-transforming the cubic term of the cumulant-generating function. It is an asymptotic approximation near the bulk of the distribution; truncation can give a negative result in far tails.

An extended-real convex function $\Lambda :{\mathbb{R}}^d\to \mathbb{R}\cup \{+\infty \}$ is essentially smooth if its effective domain has nonempty interior, it is differentiable throughout that interior, and $\Vert \nabla \Lambda ({\theta }_n)\Vert \to \infty$ whenever interior points converge to a finite boundary point of the domain. A differentiable convex function finite on all of ${\mathbb{R}}^d$ satisfies the boundary condition vacuously. The cumulant-generating function $\log (r/(r-\theta ))$ on $\theta <r$ has derivative $(r-\theta {)}^{-1}$ and is essentially smooth. Restricting its domain to $\theta <\lambda <r$ destroys this property. This distinction matters in the lower bounds of the Gärtner–Ellis theorem.

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 $X_N\in {\mathbb{R}}^d$ 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.

The Hoeffding lemma controls the entire centered moment-generating function using only an interval width. For an essentially bounded real random variable $X$ with finite deterministic bounds $a\le X\le b$ almost surely, every real $\lambda$ satisfiesEquivalently, $\mathbb{E}{e}^{\lambda (X-\mathbb{E}X)}\le e^{{\lambda }^2(b-a{)}^2/8}$. This single-random variable statement needs no independence assumption; independence enters when applying it to the factors for a sum in the Hoeffding inequality. Both signs of $\lambda$ are allowed, and equality holds at $\lambda =0$ or when $a=b$.^[ref]

In the convention $\log \mathbb{E}{e}^{\lambda Y}\le {\lambda }^2{\sigma }^2/2$ for a centered sub-Gaussian random variable $Y$, the Hoeffding lemma supplies ${\sigma }^2=(b-a{)}^2/4$. This upper parameter need not equal the actual variance. For a Bernoulli random variable $B$ of success probability $p$,The actual variance is $p(1-p)$, whereas the supplied parameter is $1/4$ for every $p$. This explains part of the conservatism visible in Hoeffding sample size for Bernoulli frequencies.

A short proof exposes why the constant is $1/8$. Put $Y=X-\mathbb{E}X$ and define exponential tilting byThe tilted probability measure has the same interval of possible values, and differentiating the cumulant-generating function gives ${\psi }^{\prime \prime }(\lambda )={\mathrm{Var}}_{{\mathbb{P}}_{\lambda }}(Y)$. Boundedness justifies these derivatives for every finite $\lambda$. Under any such tilted law, Popoviciu's inequality on variances gives ${\psi }^{\prime \prime }(\lambda )\le (b-a{)}^2/4$: the variance is no larger than the expected value of the squared distance from the interval midpoint, itself at most $(b-a{)}^2/4$. Since $\psi (0)={\psi }^{\prime }(0)=0$, the integral form of Taylor's theorem yieldsThis expression handles either sign of $\lambda$ without changing the argument.

The constant $1/8$ cannot be reduced uniformly over all essentially bounded real random variables. Equal endpoint probabilities at $a$ and $b$ give $Y=\pm (b-a)/2$, a scaled Rademacher random variable, andThus the sharpness is already forced near zero by the variance; the bound need not be equality at nonzero parameters for a particular probability distribution.

In the parametrization with density proportional to $y^{-3/2}\exp [-(\lambda /y+\varphi y)/2]$, the Inverse Gaussian distribution has cumulant-generating function $K(t)=\sqrt{\lambda \varphi }-\sqrt{\lambda (\varphi -2t)}$. For $n$ independent copies, the average has this same family with parameters $(n\varphi ,n\lambda )$, while the sum has parameters $(\varphi ,n^2\lambda )$. These assertions follow by multiplying moment-generating functions and scaling the density. The more usual mean parameter is $\mu =\sqrt{\lambda /\varphi }$, so the sum has mean $n\mu$ and shape $n^2\lambda$.

For a scalar discrete-time affine process, the conditional cumulant-generating function of ${\sum }_{j=1}^t{\theta }_j{X}_j$ is affine in $X_0$. Backward effective parameters $q_t={\theta }_t$, $q_j={\theta }_j+A(q_{j+1})$ give coefficients $A(q_1)$ and ${\sum }_{j}B(q_j)$.

For a sample mean from an independent identically distributed sample, let $K(t)$ be the cumulant-generating function of one observation and solve $K^{\prime }(t)=x$. Put $w=\mathrm{sgn}(t)\sqrt{2n(tx-K(t))}$ and $u=t\sqrt{n{K}^{\prime \prime }(t)}$. Under the smooth, nonlattice conditions for a saddlepoint density approximation, the corresponding approximation to the cumulative distribution function iswhere $\Phi$ and $\varphi$ are the standard normal distribution functions. The apparent singularity at $t=0$ is removable. Write $v=K^{\prime \prime }(0)$ and $k=K^{\prime \prime \prime }(0)$; Taylor expansion gives $w=t\sqrt{nv}(1+kt/(3v)+O(t^2))$ and $u=t\sqrt{nv}(1+kt/(2v)+O(t^2))$. Thus $1/w-1/u\to k/(6\sqrt{n}{v}^{3/2})$. The approximation requires suitable tail regularity and is not a universal formula for distributions without a moment-generating function.

Scale independent source count, service and buffer by the same factor $L$. For lookback horizon $n$, the source-average cumulative input has rate function $I_n$, and reaching buffer level $b$ requires input above $b+cn$. Optimizing this horizon gives the displayed exponent under uniform control of the infinite stationary supremum. For independent slots, $I_n(x)=n{I}_A(x/n)$; discrete time keeps $n$ an integer. Temporal dependence instead enters through the cumulative-input cumulant-generating function.

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.

Write $S_L=X^{\oplus L}$. Its Poisson distribution has mean $L\lambda$, since the probability-generating functions of the independent summands multiply. DefineThe scaled cumulant-generating function isA Taylor expansion of the exponential, valid for each fixed $\theta$ because $\beta <1$, givesThe remainder is $O(L^{(\beta -1)/2})$. The limit is finite and differentiable on all of $\mathbb{R}$, hence satisfies the essential-smoothness hypotheses of the Gärtner–Ellis theorem. Its Legendre-Fenchel transform isThis proves the Poisson moderate deviation principle with large-deviation speed $L^{\beta }$ and a good rate function.

For an arbitrary open set $B$, the lower bound uses ${\inf }_B{I}_{\beta }$ and the upper bound uses ${\inf }_{\overline{B}}{I}_{\beta }$. These infima agree: every point of the closure is a limit of points of $B$ and $I_{\beta }$ is finite and continuous. Thus the logarithmic probabilities of open sets for a continuous rate function give the stated logarithmic limit for every open $B$, not just intervals. The empty-set case follows from the usual infinite-value conventions.

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.

Where the moment-generating function $M_Y(t)=\mathbb{E}{e}^{tY}$ is finite in a neighborhood of zero, define the cumulant-generating function ${\kappa }_Y(t)=\log M_Y(t)$ and the $r$th cumulant by ${\kappa }_r(Y)={\kappa }_Y^{(r)}(0)$. Differentiation givessince $M_Y(0)=1$. Thus the first two cumulants are the mean and variance.

Conditional on $N=n$, independence gives $\mathbb{E}(e^{tT}\mid N=n)=M_V(t{)}^n$. Summing over the Poisson distribution yieldsHence the compound Poisson cumulants giveFor the moment statements, finite second moments suffice even when a moment-generating function does not exist near zero: conditioning gives $\mathrm{Var}T=\mathbb{E}N\mathrm{Var}V+\mathrm{Var}N(\mathbb{E}V{)}^2=\nu \mathbb{E}{V}^2$. This distinguishes the raw second moment from the variance of an individual claim.

For a single flood, the law of total expectation and law of total variance similarly giveCondition on the number of floods and apply the same laws once more. All floods have these moments and are independent of $M$, so the annual total satisfiesThe additional $\nu {\lambda }^2{\mu }^2$ term is the variation in the number of entire flood clusters; it would be lost by treating individual claims as an ordinary Poisson stream.

For the annual count $N={\sum }_{i=1}^M{N}_i$, its probability generating function isThis is the nested Poisson flood count. Differentiating, or conditioning on $M$, gives $\mathbb{E}N=\nu \lambda$ and $\mathrm{Var}N=\nu \lambda (1+\lambda )$. Thus $N$ is not Poisson when $\nu ,\lambda >0$, since its variance exceeds its mean. If either parameter is zero it is identically zero, a degenerate Poisson distribution.

Let $K_i$ be the number of valid claims in flood $i$. Independent validity marking gives $K_i\mid N_i\sim \mathrm{Binomial}(N_i,1-p)$. Its generating function isso Poisson thinning gives $\boxed{K_i\sim \mathrm{Poisson}(\lambda (1-p))}$. With validity independent also of claim amount, each claim contributes expected paid amount $(1-p)\mu$. Therefore the annual expected payment is $\boxed{\nu \lambda (1-p)\mu }$. The independent-mark assumption is needed for this multiplication; a size-dependent invalidity rule would require the joint expected value of amount and validity instead.

Let $K(t)=\log \mathbb{E}{e}^{tY}$ be the one-observation cumulant-generating function, finite on an open interval containing zero. At a value $y$, find the real tilt $\hat{t}$ solvingIf the distribution is nondegenerate then $K^{\prime \prime }(t)>0$ in the interior, so the solution is unique whenever $y$ is in the range of $K^{\prime }$. The leading saddlepoint density approximation to the sample mean isTo derive it, use exponential tilting:Under this tilted distribution the mean is $K^{\prime }(t)$ and the variance is $K^{\prime \prime }(t)$, by differentiating its cumulant function $K(t+u)-K(t)$. For $S_n={\sum }_i{Y}_i$, changing measure gives the exact density identityAt $t=\hat{t}$, $ny$ is the tilted mean. A local normal approximation there gives $f_{S_n,\hat{t}}(ny)\simeq (2\pi n{K}^{\prime \prime }(\hat{t}){)}^{-1/2}$. The Jacobian determinant for $S_n=n\bar{Y}$ contributes the factor $n$, yielding the boxed formula.

This approximation retains the exponential large-deviation factor, while the curvature is evaluated at the tilt appropriate to each $y$. At the population mean, $\hat{t}=0$, so it reduces to the usual central normal approximation. Under suitable smooth nonlattice transform-inversion conditions, its relative error at fixed interior $y$ is $O(n^{-1})$; existence of a few moments alone does not establish this density accuracy. Boundary values and absent exponential moments need separate treatment. The leading density need not integrate exactly to one, so it can be divided by its integral when a normalized approximation is desired.