I'm trying to show that a positive $n$-dimensional stochastic process $X_t = (x_1(t), \cdots, x_n(t))$ is nice in that it's well-behaved and controlled (in the sense that the process doesn't grow too quickly, has properties that are bounded, etc.) The stochastic process represents the sizes $n$ interacting units.
One thing I was able to prove is that:
$$\limsup_{t\rightarrow \infty} \frac{1}{t}E\left[ \log x_i(t)\right] < b_i$$
for some constant b (could be positive or negative depending on parameter values). I'm having a difficult time explaining in words what this property tells me about the process, if anything at all. It would be nicer to have been able to show that $\limsup_{t\rightarrow \infty} \frac{1}{t}E\left[ x_i(t)\right] < b_i$, but I have yet to find a way to prove it and have only been able to prove it with the $\log$ term. In the latter case, when $b_i<0$, this tells me that the expected size of the unit decays to 0, and if $b_i > 0$, the expected size of the unit can grow. However, now I have $\log x_i$ instead of just $x_i$ so I'm not sure if those properties are still true (in words)