Large deviations for sequences that are not sums of iid 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.
 A: 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.
