Here are two code line to run in R:
prod(rnorm(100, mean=1, sd=0)) # (1)
prod(rnorm(100, mean=1, sd=0.2)) # (2)
$prod(..)$ returns the product of a sequence. The sequence it given by $rnorm(n, mean, sd)$. This function $rnorm(...)$ return n values randomly drawn from a normal distribution with $mean$ and $sd$.
The only things that differ between the two lines in the $sd$. Line (1) is totally deterministic as $sd=0$ and therefore it always returns the value $1$. Line (2) is not determinstic and almost always (in 90% of the cases I'd say!) return a value that is lower than $1$.
- Why is it so?
- What is the expected value of line (2)?
- In what percentage of the cases does line(2) returns values greater than $1$
If the sequence $(x_k)$ is i.i.d. and $P[x_1=0]=0$, when $n$ is large, $\left|\prod\limits_{k=1}^nx_k\right|=\exp(n\xi_\sigma+o(n))$ almost surely, where $\xi_\sigma=E[\log |x_1|]$. This follows from the law of large numbers applied to the i.i.d. sequence $(\log |x_k|)$, plus the identity $\left|\prod\limits_{k=1}^nx_k\right|=\exp\left(\sum\limits_{k=1}^n\log |x_k|\right)$.
In your case (2), one assumes that each $x_k$ is a normal random variable $(1,\sigma^2)$ with $\sigma=0.2$, then $\xi_\sigma=E[\log|1+\sigma z|]$ where $z$ is standard normal. What happens is that $\xi_\sigma\lt0$ when $\sigma=0.2$ hence $\prod\limits_{k=1}^nx_k$ converges to zero almost surely.
We checked numerically that $\xi_\sigma\lt0$ when $\sigma=0.2$. Note that the function logarithm is concave, hence $E[\log u]\lt\log E[u]$ for every nondegenerate positive random variable $u$ (this is Jensen inequality, with a strict inequality sign when the distribution of $u$ is not Dirac). But this only shows that $\xi_\sigma$ is "not too positive" when $\sigma$ is "not too large", not that $\xi_\sigma\lt0$ always.
As a word of caution against some too simple arguments, note finally that, for some larger values of $\sigma$, indeed $\xi_\sigma\gt0$, the transition occuring between $\sigma=1.55$ and $\sigma=1.56$ (and surely the zero of the function $\sigma\mapsto E[\log|1+\sigma z|]$ has a name...).