Limiting Value Sum $c^X$ for Uniform $RV$

Question

I have $n$ draws $X_i$ from a Uniform r.v. on $[0,1]$ and want to find the limit of $\frac{1}{n} \sum_{i=1}^n c^{X_i}$ where $c$ is a known constant in $[0,1]$. My thought is to transform this summation into a function of $\frac{1}{n} \sum_{i=1}^n X_i$ and use the continuous mapping theorem, but have not had success finding the right transformation.

Answer 1

If $X_i$ are i.i.d., then by Law of Large Numbers the limit of $\frac{\sum_{i=1}^n c^{X_i}}{n}$ is just $E[c^{X_1}] = M_{X_1}(\ln(c))$ almost surely, where $M_{X_1}(t) = E[e^{t X_1}]$ is the moment generating function of $X_1$. It is known, that if $X_1 \sim U[0;1]$, that $M_{X_1}(t) = \frac{e^t - 1}{t}$. Thus $\frac{\sum_{i=1}^n c^{X_i}}{n}$ converges to $\frac{c-1}{\ln(c)}$ almost surely.