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 $S_n$ 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.

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.

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 $S_n$ 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.

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 $S_n$ 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.