Probability, Random Variables, and Random Processes: Theory and Signal Processing Applications

7.10 DOOB–MEYER DECOMPOSITION

Next, we describe another decomposition of random sequence X[k] that turns out to be a reformulation of the previous innovations decomposition. This decomposition is also additive as in (7.230), but the component sequences have properties differing from those of XMS[k] and V[k], although they are related to each other as demonstrated later. By examining the difference X[k]−X[k−1], we find that one component of the decomposed X[k] is a martingale sequence.

Theorem 7.20 (Doob–Meyer). Random sequence X[k] with for can be expressed uniquely as the sum of two sequences as follows:

(7.235)

where XDM[k] is an MS predictable sequence and E[k] is a martingale.

Note that the Doob-Meyer (DM) predictable part XDM[k] differs from XMS[k] of the innovations-based decomposition. (Although we refer to (7.235) as the DM decomposition, it is more precisely known as the Doob decomposition for discrete-time sequences. Doob-Meyer is the corresponding decomposition for continuous-time processes.)

Proof. (Larson and Shubert, 1979). We begin by using the following notation to define a sequence of increments:

(7.236)

The DM predictable part of the increment sequence given ...