Discrete Stochastic Processes and Optimal Filtering

4.6. Exercises for Chapter 4

Exercise 4.1.

Given a family of second order r.v. X,Y₁, …, Y_k, … we wish to estimate X starting from the Y_j and we state: .

Verify that .

(We say that the process , is a martingale with respect to the sequence of Y_K.)

Exercise 4.2.

Let be a sequence of independent r.v., of the second order, of law N(0,σ²) and let θ be a real constant.

We define a new sequence by if j ≥ 2.

1) Show that , the vector X^K = (X₁, …, X_K) is Gaussian.

2) Specify the mean, the matrix of covariance and the probability density of this vector.

3) Determine the best prediction in quadratic mean of X_k+P at instant K = 2, i.e. calculate E(X_2+P|X₁, X₂).

Solution 4.2.

1) Let us consider matrix belonging to M(K, K).

By stating U^K = (U₁, …U_K), we can write X^K = AU^K . The vector U^K being Gaussian ...