Since e x t x t as t t each term in the sum thus r x

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Since E [ X t - X t 0 ] 0 as t t 0 , each term in the sum 0 . Thus R X ( t, s ) is continuous at t = s = t 0 . 2 Example 1 (Homogeneous Poisson counting process). R X ( t, s ) = λ min( t, s ) + λ 2 ts. This is continuous at every point at every t = s = t 0 for every t 0 R + . So the process is mean-square continuous. But recall the sample path is a series of jumps: any realization is discontinuous with probability 1. 2
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ECE 6010: Lecture 7 – Analytical Properties of Random Processes 2 Example 2 Let X t be a Gaussian random process with R X ( t, s ) = σ 2 min( t, s ) + μ 2 ts. This is also mean-square continuous. It can be shown that this process is also continuous with probability 1. (This process is called the Wiener process ; we will have more to say about it later.) It models random diffusion or Brownian motion. 2 For a W.S.S. process, we have the following: A W.S.S. r.p. X t is mean-square continu- ous if and only if R X ( τ ) is continuous at τ = 0 . This follows since R X ( t, s ) = R X ( t - s ) , and lim ( t,s ) ( t 0 ,t 0 ) R X ( t, s ) = lim τ 0 R X ( τ ) . Differentiation Recall that f : T R is differentiable at t 0 T if lim t t 0 f ( t ) - f ( t 0 ) t - t 0 = f 0 ( t 0 ) exists. Similarly, a r.p. { X t , t T } is mean-square differentiable at t 0 if X 0 t 0 = lim t t 0 X t - X t 0 t - t 0 exists in the mean-square sense, that is, E [ X 0 t 0 - X t - X t 0 t - t 0 2 ] 0 as t t 0 . If X t is mean-square differentiable at every t 0 T , then X 0 t defines another random pro- cess on the underlying probability space ( ω, F , P ) . Suppose Y t is a second-order random process, and lim t t 0 Y t = Z (m.s.). Then: 1. E [ Z 2 ] < ; 2. and if E [ X 2 ] < then lim t t 0 E [ Y t X ] = E [ ZX ] . Proof 1. Z = Y t + Z - Y t . Then since ( a + b ) 2 4 a 2 + 4 b 2 , we have Z 2 4 Y 2 t + 4( Z - Y t ) 2 and E [ Z 2 ] 4 E [ Y 2 t ] + 4 E [( Z - Y t ) 2 ] But each of these are < for t sufficiently close to t 0 (by mean-square conver- gence.) So E [ Z 2 ] < for all t .
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  • Fall '08
  • Stites,M
  • Probability theory, Xt

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