Basic example, assume that the expected daily return of the S&P 500 stock market index is 0, i.e., the return on any given day of the stock market is 0.0%. But, we also (generally) expect that when compounded over long periods of time--say 20 years--that the expected value of that compounded return (e.g., compounded daily returns each with expectation to be 0) is greater than 0. Is it possible for both of these assumptions to be true, i.e., a 0 expectation for each individual instance but an expected value of the compounded value to be greater than 0?
Assuming ${X_i}$~$\mathcal{\pi}(mean = 0,...)$ for all $i\in\mathbb{Z}^+$ ($\pi(\cdot)$ being some arbitrary distribution with mean = 0). Assume $Cov[X_{i},X_{j}]=0$ for every $i\ne{j}$ (i.e., $X$ is independent and identically distributed). I believe the compounded (net) return over $n>1$ periods is $$R_n=\prod_{i=1}^n{(1+X_i)}-1$$ Therefore, the question is: Is it possible for $$E[R_{n}]>0$$ when $E[X_{i}]=0$ for all $i$?
It would seem to me this would not be possible if you assumed the random variable came from a symmetric distribution, e.g., a normal distribution. However, if you assumed the random variable was generated by a distribution with positive skewness then this might be possible. Is this correct? Are there other features of a distribution that would make both of these assumptions hold true.