Sample Mean of a Product Sequence

You have two independent sequences of IID random variables: $X_1, X_2, \ldots$ with mean $\mu_X$ and variance $\sigma_X^2$, and $Y_1, Y_2, \ldots$ with mean $\mu_Y$ and variance $\sigma_Y^2$. The two sequences are independent of each other. Define $Z_i = X_i Y_i$. Does the sample mean $\frac{Z_1 + Z_2 + \cdots + Z_n}{n}$ converge to a limiting distribution as $n \to \infty$? If it does, state the sum of the mean and variance of that limiting distribution. If it does not converge, answer $-100$. Use $\mu_X = 5$, $\mu_Y = 6$, $\sigma_X^2 = 25$, $\sigma_Y^2 = 36$.