Let $\{X_n\}$ be a stochastic process which is strictly positive, i.e. $X_n > 0$ almost surely for all $n$. It then follows that $\{Z_n = \log(X_n) \}$ is a well-defined stochastic process as well.
The concavity of $\log$ implies that if $\{X_n\}$ is a super-martingale, then $\{Z_n\}$ is a super-martingale as well. Are there conditions under which we can infer that $\{X_n\}$ is a super-martingale given that $\{Z_n\}$ is a super-martingale? This question is particularly relevant when $\{X_n\}$ is a multiplicative random walk, as the process $\{Z_n\}$ is an additive random walk, and so admits a much simpler analysis.
Alternatively, a pointer to a reference on finding super-martingales in non-negative multiplicative random walks would be much appreciated.
Any and all help would be appreciated!