I asked the following question one week ago at math.stackexchange but didn't receive a response, so I want to give it here another try: 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 $\|X_{n+1}−X_n\|≤c$. 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 $X_{n+1}=\max(0,X_n+\frac{2}{\sqrt{X_n}}+Y_{n+1})$, where the $Y_i$ are i.i.d. with probability distribution $P[Y_i=−1]=P[Y_i=1]=1/2$. This Markov chain behaves like a classical random walk if $X_n$ is very large, so $E[X_n]$ should behave like $O(\sqrt{n})$.