This is a question about a strictly positive stochastic process in discrete-time.
Suppose $(X_t)$ is an $\mathcal{F}_t$ measurable stochastic process, defined on a probability space $(\Omega, \mathcal{F}, \mathbb{P})$, and assume $X_t > 0$ almost surely. Assume further $(X_t)$ satisfies for (integer) $n \in \lbrace-2,-1,1,2\rbrace$: $$\lim_{K \rightarrow \infty} \mathbb{E}[ X_{t+K}^{n} | \mathcal{F}_t ]$$ exists and for each $n \in \lbrace-2,-1,1,2\rbrace$:
$$\lim_{K \rightarrow \infty} \mathbb{E}[ X_{t+K}^{n} | \mathcal{F}_t ] < \infty$$. (in words, the first and second moments and inverse moments have finite limits).
Is this true? $$\lim_{K \rightarrow \infty} \mathbb{E}[ X_{t+K} | \mathcal{F}_t ] > 0$$ $$\lim_{K \rightarrow \infty} \mathbb{E}[ 1/(X_{t+K}) | \mathcal{F}_t ] > 0$$
Would it be true if I additionally assumed this: $$\sup_K \mathbb{E}[ X_{t+K}^2 | \mathcal{F}_t ] < \infty$$ $$\sup_K \mathbb{E}[ 1/(X_{t+K})^2 | \mathcal{F}_t ] < \infty$$