吸收马尔可夫链的条件稳定律与稀有事件极限
Conditional Stable Laws and Rare-Event Limits for Absorbing Markov Chains
AI总结：
本文针对紧致度量空间上的吸收马尔可夫链，建立了条件极限定理，证明归一化观测点过程收敛于泊松随机测度，导出稳定律及非标准高斯极限，并给出条件中心极限定理、指数偏差界和收缩目标访问的泊松律。
AI中文摘要：
我们在紧致度量空间$M$上，针对吸收马尔可夫链，建立了逐点依赖于初始状态的条件极限定理。我们假设转移密度在$L^1(M,\rho)$中连续，且链不可约且非周期。对于可观测函数$f_\beta(x)=d_M(x,x_0)^{-\beta}$，其中适当的$x_0$满足$\rho(B_r(x_0))\sim C_d(x_0)r^d$，我们证明了归一化观测的点过程收敛于泊松随机测度。这产生了完全右偏的$\alpha$-稳定律，其中$\alpha=d/\beta\in(0,2)$，并且在边界值$\alpha=2$处，得到具有非标准归一化$\sqrt{n\log n}$的高斯极限。我们还建立了$L^2$可观测函数的条件中心极限定理，有界可观测函数的指数偏差界，以及访问收缩目标的条件泊松律。
英文摘要：
We establish conditional limit theorems, pointwise in the initial state, for absorbing Markov chains on a compact metric space $M$. We assume $L^1(M,ρ)$-continuous transition densities, irreducibility and aperiodicity. For the observable $f_β(x)=d_M(x,x_0)^{-β}$, with suitable $x_0$ satisfying $ρ(B_r(x_0))\sim C_d(x_0)r^d$, we prove that the point-process of normalised observations converges to a Poisson random measure. This yields totally right-skewed $α$-stable laws for $α=d/β\in(0,2)$ and, at the boundary value $α=2$, a Gaussian limit with the non-standard normalisation $\sqrt{n\log n}$. We also establish a conditional central limit theorem for $L^2$ observables, exponential deviation bounds for bounded observables and a conditional Poisson law for visits to shrinking targets.
