Product of random variables

Question

Say that I have a process that produces these values $X_1, X_2, \ldots, X_n$ as follows:

1. We have a constant $c$.
2. We have a random number generator that gives us $\epsilon$ following a bell distribution with a zero mean ($\mu = 0$) and some standard deviation ($\sigma$).
3. For any $i$, $X_i = c + \epsilon$.

I guess the above would imply that the mean of $X_1, ..., X_n$ is asymptotically going to converge at $c$. Is this right?

My quesetieon is about their products. What about their products? E.g. $X_1 X_2 \ldots X_n$. Anything that can be said? E.g.

1. Can we say that $X_1X_2\ldots X_n = Z^n$?
2. Perhaps with some bounds? E.g. sandwitch the answer?

My goal is to solve the product $X_1X_2 \ldots X_n$ more simply.