Let {Z} be Gaussian white noise, i.e. {Zt} is a sequence of i.i.d. normal r.v.s each with mean zero and variance 1. Define Xt = [Zt₁ if t is even; (Z²-₁-1)/√√2, if t is odd If {X} and {Y} are uncorrelated stationary sequences, i.e., if X, and Y, are uncorrelated for every r and s, show that {Xt + Yt} is stationary with autocovariance function equal to the sum of the autocovariance functions of {Xt} and {Y}.

College Algebra (MindTap Course List)
12th Edition
ISBN:9781305652231
Author:R. David Gustafson, Jeff Hughes
Publisher:R. David Gustafson, Jeff Hughes
Chapter8: Sequences, Series, And Probability
Section8.7: Probability
Problem 39E: Assume that the probability that an airplane engine will fail during a torture test is 12and that...
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Let {Z} be Gaussian white noise, i.e. {Z} is a sequence of i.i.d. normal r.v.s each with mean zero
and variance 1. Define
Xt
=
{
if t is even;
(Z²-₁-1)/√√2, if t is odd
Zt,
If {Xt} and {Y} are uncorrelated stationary sequences, i.e., if X, and Y, are uncorrelated for every
r and s, show that {Xt + Yt} is stationary with autocovariance function equal to the sum of the
autocovariance functions of {Xt} and {Yt}.
Transcribed Image Text:Let {Z} be Gaussian white noise, i.e. {Z} is a sequence of i.i.d. normal r.v.s each with mean zero and variance 1. Define Xt = { if t is even; (Z²-₁-1)/√√2, if t is odd Zt, If {Xt} and {Y} are uncorrelated stationary sequences, i.e., if X, and Y, are uncorrelated for every r and s, show that {Xt + Yt} is stationary with autocovariance function equal to the sum of the autocovariance functions of {Xt} and {Yt}.
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