We solve the problem of asymptotic behaviour of the renewal measure (Green function) generated by a transient Lamperti's Markov chain $X_n$ in $\mathbf R$, that is, when the drift of the chain tends to zero at infinity. Under this setting, the average time spent by $X_n$ in the interval $(x,x+1]$ is roughly speaking the reciprocal of the drift and tends to infinity as $x$ grows. For the first time we present a general approach relying in a diffusion approximation to prove renewal theorems for Markov chains. We apply a martingale type technique and show that the asymptotic behaviour of the renewal measure heavily depends on the rate at which the drift vanishes. The two main cases are distinguished, either the drift of the chain decreases as $1/x$ or much slower than that, say as $1/x^\alpha$ for some $\alpha\in(0,1)$. The intuition behind how the renewal measure behaves in these two cases is totally different. While in the first case $X_n^2/n$ converges weakly to a $\Gamma$-distribution and there is no law of large numbers available, in the second case a strong law of large numbers holds true for $X_n^{1+\alpha}/n$ and further normal approximation is available.

中文翻译：

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具有渐近零漂移的瞬态马尔可夫链的更新理论

我们解决了 $\mathbf R$ 中由瞬态 Lamperti 马尔可夫链 $X_n$ 生成的更新测度（格林函数）的渐近行为问题，即当链的漂移在无穷远处趋于零时。在这种设置下，$X_n$在$(x,x+1]$区间内花费的平均时间大致上是漂移的倒数，随着$x$的增长趋于无穷大。方法依赖于扩散近似来证明马尔可夫链的更新定理。我们应用鞅类型技术并表明更新度量的渐近行为在很大程度上取决于漂移消失的速率。区分两种主要情况，要么链的漂移以 $1/x$ 或比这更慢的速度减少，例如对于某些 $\alpha\in(0,1)$ 以 $1/x^\alpha$ 为例。更新措施在这两种情况下的表现背后的直觉是完全不同的。虽然在第一种情况下 $X_n^2/n$ 弱收敛到 $\Gamma$ 分布并且没有可用的大数定律，但在第二种情况下，强大数定律对 $X_n^{1 成立+\alpha}/n$ 和进一步的正态近似是可用的。