Heads Proportion I

A coin is flipped indefinitely under the following rules: The probability of landing a heads on the $i$th flip is a random variable $P_i$ supported on $[0,1]$ with mean $\mu$ and variance $\sigma^2$, independent of other $P_j$, $i \neq j$. Let $H_n$ be the number of heads occurring in the first $n$ flips. If the limit exists, compute

$$\displaystyle \lim_{n \rightarrow \infty} \dfrac{H_n}{n}$$

Report the answer when $P_1,P_2,\dots \sim \unif{0}{1}$ IID. If this limit doesn't exist, report $-1$.