Drift of random walks and limit theorems (Q1921308)

The drift of a Markov chain \(X_n\) of \(\mathbb{R}\) is defined as the function \(d(x) = E[X_n-X_{n-1} \mid X_{n-1} = x]\). Analogously, its second order moment is defined as \(c_2(x) = E[(X_n-X_{n-1})^2 \mid X_{n-1}=x]\). If the function \(d\) (resp. \(c_2)\) is bounded and has a limit \(m_1\) (resp. \(m_2)\), as \(x\to\infty\), the Markov chain is said to be of type \(\text{S}_1\) (resp. \(\text{S}_2)\). It is proved that, for a large class of one-dimensional hypergroups, any random walk on the hypergroup, whose increments have a first moment (resp. a second moment), is of type \(\text{S}_1\) (resp. \(\text{S}_2)\). These properties \(\text{S}_1\) and \(\text{S}_2\), in turn can be used to prove some limit theorems for the Markov chain. Indeed, if \(X_n\) is a Markov chain of type \(\text{S}_1\), for which \(c_2\) is a bounded function, then one has \(\lim X_n/n=0\) p.s. if \(X_n\) is positive recurrent, and \(\lim X_n/n=m_1\) p.s. if \(X_n\) is null recurrent or transient. Analogously, some central limit theorems can be obtained for the quantity \(V_n=n^{-1/2}(X_n-\sum^n_{k=1} d(X_{k-1}))\). If \(X_n\) is positive recurrent with invariant measure \(\lambda\), then \(V_n\) converges in distribution towards a centered normal variable with veriance \(\int c_2(x)-d^2(x) \lambda (dx)\), while if \(X_n\) is null recurrent or transient, of type \(\text{S}_2\), then the same conclusion holds, with the variance being equal to \(m_2-m^2_1\).