Let $(\Omega,\mathcal{F},(\mathcal{F}_s)_{s=0}^{\infty},\mathbb{P})$ be a filtered probability space (time is discrete). Let $\{A_t\}$ and $\{B_t\}$ be martingales on $\mathcal{F},\mathbb{P}$ (but they are not assumed independent). The two martingales are assumed $\mathbb{P}$ almost surely strictly positive. What can be said about $\liminf_{K \rightarrow \infty} \mathbb{E}[ \left( \frac{B_{t+K}}{B_t} \frac{A_t}{A_{t+K}} \right)^{\overline{n}} | \mathcal{F}_t ]$ and about $\limsup_{K \rightarrow \infty} \mathbb{E}[ \left( \frac{B_{t+K}}{B_t} \frac{A_t}{A_{t+K}} \right)^{\overline{n}} | \mathcal{F}_t ]$, when $\overline{n}=1$. What about the same quantities when $\overline{n}=2$? Heuristically, my question is really: Under what conditions might these quantities be: Either (a) zero, (b) infinity, (c) they have a common value (the limit) which is strictly between zero and infinity?
As a slightly separate point, when might we be able to say (in other words, what extra conditions are required) that
$\lim_{K \rightarrow \infty} \frac{\mathbb{E}[ \frac{B_{t+K}}{A_{t+K}} | \mathcal{F}_{t+1} ]}{\mathbb{E}[ \frac{B_{t+K}}{A_{t+K}} | \mathcal{F}_t ]}$ equals unity.
I think the answer to this last question might be only when $B_t = A_t$, for all $t$.
This is not a homework question - so the answer might be that these circumstances are unknowable without strong assumptions on the nature of $\{A_t\}$ and $\{B_t\}$ (I am a researcher - but not in math).