论文标题
在较大的时间尺度上，亚稳态密度依赖的马尔可夫链的强高斯近似
Strong Gaussian approximation of metastable density-dependent Markov chains on large time scales
论文作者
论文摘要
密度依赖的马尔可夫链构成了人口动态中的一系列连续时间马尔可夫链。在任何固定的时间窗口[0，t]上，当比例参数k> 0很大时，此类链被ode的溶液（流体极限）很好地近似，而高斯波动叠加在其上。在本文中，我们量化了该高斯近似值保持精确的时间，在流体极限收敛到指数稳定的平衡点的情况下，在轨迹上均匀地保持在轨迹上。我们使用著名的koml {ó} s-major-tusn {á} dy theorem进行随机步行，基于库尔兹的构造，在密度依赖性链和近似高斯过程之间提供了新的耦合。我们表明，在轻度假设下，要达到阈值$ε$（k）<< 1所需的时间t（k）至少在order exp（v k $ε$（k））中，对于某些常数v> 0。这值得注意的是，高斯近似值会产生有关中等时间范围的正确差异性。我们还向逻辑出生和死亡过程的高斯近似呈现，以生存为条件，并估计了对流行病的成本进行建模的数量。
Density-dependent Markov chains form an important class of continuous-time Markov chains in population dynamics. On any fixed time window [0, T ], when the scale parameter K > 0 is large such chains are well approximated by the solution of an ODE (the fluid limit), with Gaussian fluctuations superimposed upon it. In this paper we quantify the period of time during which this Gaussian approximation remains precise, uniformly on the trajectory, in the case where the fluid limit converges to an exponentially stable equilibrium point. We provide a new coupling between the density-dependent chain and the approximating Gaussian process, based on a construction of Kurtz using the celebrated Koml{ó}s-Major-Tusn{á}dy theorem for random walks. We show that under mild hypotheses the time T(K) necessary for the strong approximation error to reach a threshold $ε$(K)<<1 is at least of order exp(V K $ε$(K)), for some constant V > 0. This notably entails that the Gaussian approximation yields the correct asymptotics regarding the time scales of moderate deviations. We also present applications to the Gaussian approximation of the logistic birth-and-death process conditioned to survive, and to the estimation of a quantity modeling the cost of an epidemic.
