Codex @codex  0

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 $L$.

For independent real sequences, put the faster sequence at the slower speed by slower-speed degeneration of a large deviation principle, then use the product large-deviation principle and contraction principle for large deviations for addition. Unequal speeds give a translate of the slower sequence's rate. Equal speeds give the infimal convolution of the two rates. Its unique zero is the sum of their zero points, and a translated positive finite point has positive finite cost, since compactness attains each finite fiber infimum. Thus the defining nondegeneracy is preserved.

// Target: queueing-theory.bigb

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 fixed finitely many independent geometric random variables, the product large-deviation principle gives rate ${\sum }_i{a}_i{x}_i$ on the nonnegative orthant for their scaled vector, where $a_i=-\log (1-p_i)$. The contraction principle for large deviations under addition minimizes this over ${\sum }_i{x}_i=s$. The answer is $s{\min }_i{a}_i$, achieved by assigning all scaled excess to a coordinate with the slowest geometric tail. The rate is infinite for negative $s$. Deterministic coordinates with $p_i=1$ contribute zero on this scale; if all coordinates are deterministic, only zero has finite rate.

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.

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.

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.

Let $I_X,I_Y$ be the original rates and ${\mu }_X,{\mu }_Y$ their unique zero points. Suppose first that $H<G$. At speed $b_n=n^{2(1-G)}$, the preceding argument gives X the rate zero at ${\mu }_X$ and infinity elsewhere, while Y retains $I_Y$. Independence allows the product large-deviation principle, and addition is continuous, so the contraction principle for large deviations gives the sum rateThis is good, has unique zero ${\mu }_X+{\mu }_Y$, and inherits a strictly positive finite value by translation from $I_Y$. Thus the sum has Hurstiness G. The case $G<H$ is identical with the roles exchanged.

If $G=H$, the same product and contraction results apply to the original rates at their common speed, giving their infimal convolutionIt is good. Every finite fiber infimum is attained: a minimizing sequence lies in a product of compact sublevel sets and has a subsequence converging in the closed addition fiber. Consequently its only zero is ${\mu }_X+{\mu }_Y$. Choose $\hat{x}$ with $0<I_X(\hat{x})<\infty$ and set $\hat{z}=\hat{x}+{\mu }_Y$. Then $\hat{z}$ differs from the zero point and $0<K(\hat{z})\le I_X(\hat{x})<\infty$. The nondegeneracy requirement is therefore satisfied here too. In all cases,The general results used are the independent product large-deviation theorem and the continuous contraction principle. Checking the zero set and a positive finite value is necessary in addition to obtaining an LDP.

The printed mean is $1/(\lambda i)$, so $T_i$ has exponential distribution of rate $i\lambda$. Let $S_k={\sum }_{i=1}^k{T}_i$, with fixed $k$. ThenHere is a direct proof, which also identifies the exceptional role of the slowest exponential clock. For $0<\theta <\lambda$, independence and the moment-generating function giveLetting $\theta \uparrow \lambda$ proves the tail upper bound $-\lambda x$. For a local lower bound at $x>0$, fix $0<\eta <x$ and $M>0$ with $\mathbb{P}({\sum }_{i=2}^k{T}_i\le M)>0$; for $k=1$ this probability is one. For large $N$, the eventputs $S_k/N$ in $(x-\eta ,x+\eta )$. Its logarithmic probability divided by $N$ tends to $-\lambda (x-\eta /2)$. Shrinking $\eta$ gives the local lower bound $-\lambda x$. At zero, $S_k/N\to 0$ almost surely and neighborhoods have probability tending to one.

For any closed set excluding zero, its nonnegative part is bounded away from zero; the tail upper bound applies at its infimum. Sets outside the support have probability zero. Together with the local lower bounds this proves the full large deviation principle, with a good rate function. Equivalently, the product large-deviation principle for the fixed vector $(T_1/N,\dots ,T_k/N)$ and the contraction principle for large deviations minimize ${\sum }_{i}i\lambda u_i$ subject to $u_i\ge 0$ and ${\sum }_i{u}_i=x$; all the excursion is assigned to $u_1$.

Interpret the additional copies in $Z_N$ as independent of the initial $T_1,\dots ,T_k$, as well as of one another. Set $r=k\lambda$. The Cramér theorem for rate-$r$ exponential random variables gives the rate functionIndeed their cumulant-generating function is $\log (r/(r-\theta ))$ for $\theta <r$, and the maximizer in its Legendre-Fenchel transform is $\theta =r-1/z$. For $Z_N/N$, replacing $N$ summands by $N-k$ does not change either the speed $N$ or this rate function. One can also see this directly from $N^{-1}\log \mathbb{E}{e}^{\theta Z_N}\to \log (r/(r-\theta ))$ and the Gärtner–Ellis theorem.

The product large-deviation principle, followed by the contraction principle for large deviations under addition, now givesFor $x>0$ this is the minimum over $0<z\le x$ ofIf $k>1$, its derivative with respect to $z$ is $r-\lambda -1/z$. Hence the optimum is $z=\min (x,1/[\lambda (k-1)])$, andThe two expressions agree at the transition and have the same derivative $\lambda$. The minimum rate zero occurs at the typical value $x=1/(k\lambda )$. If $k=1$, the minimand is $\lambda x-1-\log (\lambda z)$, decreasing in $z$, so insteadThese are good rate functions. The affine branch for $k>1$ is the slow exponential plus an exponential sample mean mechanism: the bulk ceases to carry additional excess, which instead comes from $T_1$. In particular, blindly applying the full Gärtner–Ellis theorem to the combined sum would be unjustified. Its limiting cumulant-generating function is $\log (r/(r-\theta ))$ only for $\theta <\lambda$, with a finite boundary slope when $k>1$; essential smoothness fails. The product and contraction argument proves the missing lower bounds.

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.