论文标题
部分可观测时空混沌系统的无模型预测
On Convergence of General Truncation-Augmentation Schemes for Approximating Stationary Distributions of Markov Chains
论文作者
论文摘要
在对马尔可夫链条和过程的分析中，有时用“截断”有界状态空间替换无界状态空间很方便。当进行这种替换时，人们经常想知道截短链或过程的平衡行为是否接近未截断系统的平衡行为。例如，当考虑用于计算无限状态空间上固定分布的数值方法时，这些问题自然会出现。在本文中，我们研究了一般的截断 - 启发方案，其中过渡矩阵或内核的地形截断的“西北角”任意随机化（或增强）。在存在涉及强制功能的Lyapunov条件下，我们表明，只要选择截断作为Lyapunov函数的巨型集合，这些方案通常是在可数的状态空间中收敛的。对于$ \ mathbb z _+$上随机单调的马尔可夫链，我们证明我们始终可以选择截断集为$ \ {0,1，...，...，n \} $的形式。然后，我们为弱连续的马尔可夫链提供了足够的条件，在该链中，一般的截断 - 启发方案在连续状态空间中弱化。最后，我们简要讨论了理论的扩展到连续时间马尔可夫跳跃过程。
In the analysis of Markov chains and processes, it is sometimes convenient to replace an unbounded state space with a "truncated" bounded state space. When such a replacement is made, one often wants to know whether the equilibrium behavior of the truncated chain or process is close to that of the untruncated system. For example, such questions arise naturally when considering numerical methods for computing stationary distributions on unbounded state space. In this paper, we study general truncation-augmentation schemes, in which the substochastic truncated "northwest corner" of the transition matrix or kernel is stochasticized (or augmented) arbitrarily. In the presence of a Lyapunov condition involving a coercive function, we show that such schemes are generally convergent in countable state space, provided that the truncation is chosen as a sublevel set of the Lyapunov function. For stochastically monotone Markov chains on $\mathbb Z_+$, we prove that we can always choose the truncation sets to be of the form $\{0,1,...,n\}$. We then provide sufficient conditions for weakly continuous Markov chains under which general truncation-augmentation schemes converge weakly in continuous state space. Finally, we briefly discuss the extension of the theory to continuous time Markov jump processes.
