Let $S_n$ be a centered random walk, i.e. increment mean $EX = 0$. For convenience let us say that $EX^2 < \infty$ and it only takes two values $\{a, -b\}$, $a,b > 0$. Considering objects such as $$ E ( e^{S_n}) \quad \text{ or } \quad E\left( \sum_{k = 1}^n e^{S_k}\right)$$ (or more generally $E(f(S_1, \dots, S_n))$, do these expectation always behave asymptotically the same as $$ E(e^{-S_n}), E\left( \sum_{k=1}^n e^{-S_k} \right), E (f(-S_1, \dots, -S_n)) \quad ?$$
More precisely, does $$ E (e^{S_n}) / E (e^{-S_n}) \to 1 $$ hold etc.?
I hope it is clear what my question is. Because the random walk is centered, it should not (so I suspected) make a difference whether we consider $S_n$ or $-S_n$ for large $n$, as the drift is normally the main criteria for the asymptotics of $S$. Is this true in general? And, can one show this rigorously?
It is true that all random walks with a second moment looks the same "from a distance", namely that they converge to a Brownian motion after proper rescaling. But you are asking for much more. Take the simple example of $$ \mathbb{P}(X_i = -1) = \frac23, \quad \mathbb{P}(X_i = 2) = \frac13. $$ Then $$ \mathbb{E}(e^{S_n}) = \left ( \frac23 e^{-1} + \frac13 e^2 \right )^n $$ while $$ \mathbb{E}(e^{-S_n}) = \left ( \frac23 e + \frac13 e^{-2} \right )^n, $$ so the ratio of the two is $$ \frac{\mathbb{E}(e^{S_n})}{\mathbb{E}(e^{-S_n})} = \left ( \frac{2e^{-1}+e^2}{2 e + e^{-2}} \right )^n, $$ which tends to $+ \infty$ exponentially fast.