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Suppose $(S_n)_n$ is a sequence of real random variables. Denote their cumulant generating functions by $K_n(t) = \log\mathbb{E}\left[\mathrm{e}^{t S_n}\right]$, and assume that each $K_n$ is finite for all $t$. Suppose also that for all $t$ the limit $K(t) = \lim_n \frac{1}{n}K_n(t)$ exists and is finite.

Is it true that when $K$ is strictly convex then $\frac{1}{n}S_n$ satisfies a large deviation principle with rate $K^\star$, the Legendre transform of $K$? That is, is it true that $\mathbb{P}[S_n \geq n a] = \mathrm{e}^{-n K^\star(a)+o(n)}$ for all $a > \lim_n\frac{1}{n}\mathbb{E}[S_n]$?

Note 1: The case of sums of iid is the case that $K=\frac{1}{n}K_n$ for all $n$, in which case the statement is true.
Note 2: The assumptions on the convergence of $\frac{1}{n}K_n$ imply that the limit $\lim_n\frac{1}{n}\mathbb{E}[S_n]$ also exists. Note 3: The upper bound $\mathbb{P}[S_n \geq n a] \leq \mathrm{e}^{-n K^\star(a)+o(n)}$ follows easily from the Chernoff bound.

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I believe you are missing an $n$ in your definition of $K_n(t)$, that is $K_n(t)=\log E(e^{tnS_n})$. I assume in the sequel that this is what you meant.

If $S_n$ satisfies the large deviations principle with a non-convex rate function, then clearly the rate function is not $K^*$. So your question can be rephrased as "does there exist $S_n$ that satisfies both your assumptions and the LDP with non-convex rate function?"

Here is an example: take $S_n=0$ with probability $1/2$ and $S_n=n$ with probability $1/2$. Then $K_n(t)=[\log ( (e^{tn}+1)/2)]$. Thus, $K(t)=t$ for $t\geq 0$ and $K(t)=0$ for $t<0$. You get $K^*(a)=\infty$ for $a<0$, $K^*(a)=0$ for $a\in [0,1]$ and $K^*(a)=\infty$ for $a>1$. But $P(S_n\in (0,1))=0$ which contradicts a LDP with rate $K^*$ (the lower bound fails). In fact, you get a LDP with rate function $I$ satisfying $I(a)=\infty$ if $a\notin \{0,1\}$ and $I(0)=I(1)=0$.

Edit: with the original statement of the OP, it is $S_n/n$ that satisfies the LDP, not $S_n$, and then the same example works with the case $S_n=1$ replaced by $S_n=n$.

Edit2: The question keeps changing in edits, so the answer above does not address the (new) strict convexity requirements.

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  • $\begingroup$ Thanks, Ofer! Note that I did mean my definition of $K_n$. I am thinking of $S_n$ as something that scales with $n$ like a sum of iid. The prime example is indeed $S_n=X_1+\cdots+X_n$ for iid $X_n$. In this case the statement is true as I've written it. I think :). Your comment on convexity is a good point. For this to be true $K$ would have to be strictly convex, at least at an interval. $\endgroup$
    – Vladimir
    Commented Sep 3, 2022 at 1:50
  • $\begingroup$ But $S_n$ does not satisfy a LDP with your definition, $S_n/n$ does. So either you keep your definition and then say that $S_n/n$ satisfies a LDP, or you use the standard notion that I used (compare with Dembo-Zeitouni's book). If you use your notion, just modify my example with $S_n=n$ instead of $S_n=1$. $\endgroup$
    – ofer zeitouni
    Commented Sep 3, 2022 at 4:43
  • $\begingroup$ Thanks again, Ofer! I changed it to $S_n/n$, and I also added the requirement that $K$ is strictly convex. Note that $K$ is always convex as a pointwise limit of convex functions, and thus $K^\star$ is also always convex. When $K$ is strictly convex then so is $K^\star$. I think. $\endgroup$
    – Vladimir
    Commented Sep 3, 2022 at 14:05
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    $\begingroup$ $K$ strictly convex should be enough, just follow the lower bound proof in gartner ellis. But I feel that changing the question in retrospect is not great way for math overflow (and I believe against the guidelines), since latecomers to the answer in a year will find the answers given irrelevant and will not follow the context. For this reason, and because the process is time consuming, I will stop here. $\endgroup$
    – ofer zeitouni
    Commented Sep 4, 2022 at 18:25
  • $\begingroup$ Thanks a lot, Ofer! This is very helpful. My question was indeed missing an important qualifier, but it does sound like the idea is correct in spirit. $\endgroup$
    – Vladimir
    Commented Sep 4, 2022 at 23:42

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