I'm interested in the following problem: We have got a time-discrete Markov chain $(X_n)$ with state space $S=\mathbb{R}_+^d$. The transition kernel is discrete in the sense, that for each $s \in S$ there is a finite subset $S' \subset S$, such that $P[X_{n+1} \in S' | X_n = s] = 1$. Furthermore, we can assume that there exists a constant $c$, such that for all $n$ we have $\|X_{n+1}-X_n\| \leq c$. We also know, that there is a drift away from the origin, which tends to zero as $\|s\|$ increases.
I want to show, that $E[\|X_n\|]$ is in $o(n)$ (i.e., it behaves in a sublinear way). Could you give me some ideas how to show this statement?
To illustrate the question I give a concrete example: Assume $X_0 = 0$ and $X_{n+1} = \max(0, X_n + \frac{2}{1 + \sqrt{X_{n}}} + Y_{n+1})$, where $P[Y_i = -1] = P[Y_i = 1] = 1/2$. This Markov chain behaves like a classical random walk if $X_n$ is very large, so $E[X_n]$ should behave like $O(\sqrt{n})$.