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We are interested in the asymptotic behavior of Markov chains on the set ofpositive integers for which, loosely speaking, large jumps are rare and occurat a rate that behaves like a negative power of the current state, and suchthat small positive and negative steps of the chain roughly compensate eachother. If $X_{n}$ is such a Markov chain started at $n$, we establish a limittheorem for $\frac{1}{n}X_{n}$ appropriately scaled in time, where the scalinglimit is given by a nonnegative self-similar Markov process. We also study theasymptotic behavior of the time needed by $X_{n}$ to reach some fixed finiteset. We identify three different regimes (roughly speaking the transient, therecurrent and the positive-recurrent regimes) in which $X_{n}$ exhibitsdifferent behavior. The present results extend those of Haas & Miermont whofocused on the case of non-increasing Markov chains. We further present anumber of applications to the study of Markov chains with asymptotically zerodrifts such as Bessel-type random walks, nonnegative self-similar Markovprocesses, invariance principles for random walks conditioned to stay positive,and exchangeable coalescence-fragmentation processes.

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