Let $S_n$ denote a simple random walk with i.i.d. increments $X_i$ such that $P(X_1 = 0) = P(X_1=1) = 1/2$, i.e. $$S_0 = 0, \ S_n = X_1 + \dots + X_n.$$

The behavior of $S_n$ as $n \to \infty$ is clear, namely $S_n /n \to 1/2$ a.s. Now, let $q < 1$ and consider the random sum

$$ \sum_{k=0}^{n-1} q^{S_k}. $$

What can we deduce about the asymptotic behavior of this sum as $n$ tends to infinity? Any ideas how to approach this or references to work that have examined such sums?

Let $S_n$ denote a simple random walk with i.i.d. increments $X_i$ such that $P(X_1 = 0) = P(X_1=1) = 1/2$, i.e. $$S_0 = 0, \ S_n = X_1 + \dots + X_n.$$

The behavior of $S_n$ as $n \to \infty$ is clear, namely $S_n /n \to 1/2$ a.s. Now, let $q < 1$ and consider the random sum

$$ \sum_{k=0}^{n-1} q^{S_k}. $$

What can we deduce about the asymptotic behavior of this sum as $n$ tends to infinity? Any ideas how to approach this or references to work that have examined such sums?

Edit: From the answers below, we have $$ \sum_{k=0}^{n-1} q^{S_k} \to \sum_{k=0}^\infty q^{S_k}.$$

Now, for instance $$ \frac{q^{S_n}}{q^{n/2}} \to 1 \quad a.s.$$

My question is, whether we can find a similar sequence for $\sum_{k=0}^{n-1} q^{S_k}$ that describes its growth behavior, i.e. a function $f(n)$ (ideally a deterministic function) such that $$\frac{ \sum_{k=0}^{n-1} q^{S_k}}{f(n)} \to 1 $$ almost surely as $n \to \infty$?