$\lim X_n = 0$ iff $b > 0$

Question

Probability with Martingales:

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It looks like $$\lim \exp\{aS_n - bn\} = 0$$ if $b > 0$ because

$$\lim aS_n - bn = -\infty \tag{*}$$

but how to prove $(*)$?

Answer 1

First, rewrite the exponenent as \begin{align*} \exp \left( n\left( a\frac{S_n}{n} - b \right) \right) \end{align*} By the Law of Large Numbers, we have $a S_n/n- b \to a \mathbb{E}(\xi_1) - b$ a.s..
So $a S_n -bn \to - \infty$ a.s. if and ony if $b>a \mathbb{E}(\xi_1) =0$, and thus \begin{align*} \exp( a S_n - b n ) \to 0 \text{ iff } b > 0. \end{align*}