Self-similar scaling limits of Markov chains on the positive integers

Understand

We are interested in the asymptotic behavior of Markov chains on the set of positive integers for which, loosely speaking, large jumps are rare and occur at a rate that behaves like a negative power of the current state, and such that small positive and negative steps of the chain roughly compensate each other.

• If $X_{n}$ is such a Markov chain started at $n$, we establish a limit theorem for $\frac{1}{n}X_{n}$ appropriately scaled in time, where the scaling limit is given by a nonnegative self-similar Markov process.
• We also study the asymptotic behavior of the time needed by $X_{n}$ to reach some fixed finite set.
• We identify three different regimes (roughly speaking the transient, the recurrent and the positive-recurrent regimes) in which $X_{n}$ exhibits different behavior.