间断观测的Kalman滤波：弱收敛于平稳分布
Kalman Filtering with Intermittent Observations: Weak Convergence to a Stationary Distribution
摘要：
本文研究了当观测值的到达用Bernoulli i.i.d.描述时，Kalman滤波中随机代数Riccati方程（RARE）的渐近性态。过程我们将RARE建模为一个保序的强次线性随机动力系统（RDS）。在一个充分条件下，随机有界性，并使用保序，强次线性RDS的极限集二分法的结果，我们建立了渐近性质的RARE：序列的随机预测误差协方差矩阵弱收敛到一个唯一的不变分布，其支持表现出分形行为。特别是，这种弱收敛在广泛的条件下，甚至当观测到达率低于平均稳定的临界概率。我们应用弱Feller属性的马尔可夫过程管理的RARE的特征支持的限制不变分布的拓扑封闭的可数点集，这在一般情况下，是不密集的一组半正定矩阵。我们使用的支持不变分布和样本路径的几乎必然遍历的明确表征，很容易计算的不变分布的时刻。一个一维的例子说明了支撑是具有自相似性质的非负实数的一个断裂子集。
英文摘要：
The paper studies the asymptotic behavior of Random Algebraic Riccati Equations (RARE) arising in Kalman filtering when the arrival of the observations is described by a Bernoulli i.i.d. process. We model the RARE as an order-preserving, strongly sublinear random dynamical system (RDS). Under a sufficient condition, stochastic boundedness, and using a limit-set dichotomy result for order-preserving, strongly sublinear RDS, we establish the asymptotic properties of the RARE: the sequence of random prediction error covariance matrices converges weakly to a unique invariant distribution, whose support exhibits fractal behavior. In particular, this weak convergence holds under broad conditions and even when the observations arrival rate is below the critical probability for mean stability. We apply the weak-Feller property of the Markov process governing the RARE to characterize the support of the limiting invariant distribution as the topological closure of a countable set of points, which, in general, is not dense in the set of positive semi-definite matrices. We use the explicit characterization of the support of the invariant distribution and the almost sure ergodicity of the sample paths to easily compute the moments of the invariant distribution. A one dimensional example illustrates that the support is a fractured subset of the non-negative reals with self-similarity properties.
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