On the sequence of Bayesian estimates of normal random walk process

Abstract

Let ( μ t ) ∞ t=0 be a k-variate ( k⩾1) normal random walk process with successive increments being independently distributed as normal N( δ, R), and μ 0 being distributed as normal N(0, V 0). Let X t have normal distribution N( μ t , Σ) when μ t is given, t = 1, 2,….Then the conditional distribution of μ t given X 1, X 2,…, X t is shown to be normal N( U t , V t ) where U t 's and V t 's satisfy some recursive relations. It is found that there exists a positive definite matrix V and a constant θ, 0 < θ < 1, such that, for all t⩾1, |R 1 2 (V −1 t−V −1R 1 2 |<θ t|R 1 2 (V −1 0−V −1)R 1 2 | where the norm |·| means that | A| is the largest eigenvalue of a positive definite matrix A. Thus, V t approaches to V as t approaches to infinity. Under the quadratic loss, the Bayesian estimate of μ t is U t and the process { U t } ∞ t = 0, U 0=0, is proved to have independent successive increments with normal N( θ, V t − V t+1 + R) distribution. In particular, when V 0 = V then V t = V for all t and { U t } ∞ t=0 is the same as { μ t } ∞ t=0 except that U 0 = 0 and μ 0 is random.