Heavy-tails in Kalman filtering with packet losses: confidence bounds vs second moment stability

Abstract

In this paper, we study the existence of a steady state distribution and its tail behaviour for the estimation error arising from Kalman filtering for unstable scalar systems. Although a large body of literature has studied the problem of Kalman filtering with packet losses in terms of analysis of the second moment, no study has addressed the actual distributionof the estimation error. By drawing results from Renewal Theory, in this work we show that under the assumption that packet loss probability is smaller than unity, and the system is on average contractive, a stationary distribution always exists and is heavy-tailed, i.e. its absolute moments beyond a certain order do not exist. We also show that under additional technical assumptions, the steady state distribution of the Kalman prediction error has an asymptotic power-law tail, i.e. $P[|e| \gt s] \sim s^{-\alpha}$, as $s \rightarrow \infty$, where $\alpha$ can be explicitly computed. We further explore how to optimally select the sampling period assumingexponential decay of packet loss probability with respect to the sampling period. In order to minimize the expected value of second moment or the confidence bounds, we illustrate thatin general a larger sampling period will need to be chosen in the latter case as a result of the heavy tail behaviour.