Stochastic Processes

Stochastic Processes

Stochastic Processes

Stochastic Processes ( Fall 2014 ) Spectral representations and ergodic theorems for stationary stochastic processes Stationary stochastic processes

A stochastic process X is strongly stationary if its fdds are invariant under time shifts, that is, for any (finite) n, for any t0 and for all t1, ..., tn ∈ T , (Xt1 , ..., Xtn) and (Xt1+t0 , ..., Xtn+t0) have the same distribution. A stochastic process X is weakly stationary if its mean function is constant and its covariance function is invariant under time shifts. That is, for all t ∈ T , E(...

Stochastic Processes

x 6∈S cxf(x) 2 < ε. Of course g has finite support since g(x) = 0 for x 6∈ S. u t Exercise 2. (*) Recall that the operator norm of an operator T : U → W is define ||T || = supu 6=0 {||Tu||/||u||}. Show the following: • The transition matrix P is a self-adjoint operator from L(V ) into itself. • The operator norm of P is at most 1. • ||P|| ≤ ||P ||. • ||P || = ||P ||. Solution (exercise 2). Sinc...

Stochastic Processes

• We use the formula for final grade: 25%assignment + 25%midterm + 50%final. • There will be 4 assignments. • Office hours: to be announced. The text book for this course is Probability and Random Processes by Grimmett and Stirzaker. It is NOT essential to purchase the textbook. This course note will be free to download to all students registered. We may not be able to cover all the materials i...