Recurrence times are insensitive to distributions. Let XO, X1, X2, ... be drawn i.i.d. p(x), X E X = {1, 2,...,m}, and let N be the waiting time to the next occurrence of XO. Thus N = min n{Xn = XO). (a) Show that EN = m. (b) Show that E log N≤ H (X). (c) (Optional) Prove part (a) for (Xi) stationary and ergodic.
Recurrence times are insensitive to distributions. Let XO, X1, X2, ... be drawn i.i.d. p(x), X E X = {1, 2,...,m}, and let N be the waiting time to the next occurrence of XO. Thus N = min n{Xn = XO). (a) Show that EN = m. (b) Show that E log N≤ H (X). (c) (Optional) Prove part (a) for (Xi) stationary and ergodic.
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter1: Fundamental Concepts Of Algebra
Section1.2: Exponents And Radicals
Problem 90E
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![Recurrence times are insensitive to
distributions. Let XO, X1, X2, ... be drawn
i.i.d. - p(x), x eX = {1, 2,...,m}, and let N
be the waiting time to the next
occurrence of XO. Thus N = min n{Xn =
XO). (a) Show that EN = m. (b) Show that
E log N< H (X). (c) (Optional) Prove part
(a) for {Xi} stationary and ergodic.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F46f84ab4-034f-40ee-bc92-f9e4c180d53b%2F2a8a394f-3e7c-42b3-84c2-b8c6c8da2a58%2Fo0t1dyad_processed.png&w=3840&q=75)
Transcribed Image Text:Recurrence times are insensitive to
distributions. Let XO, X1, X2, ... be drawn
i.i.d. - p(x), x eX = {1, 2,...,m}, and let N
be the waiting time to the next
occurrence of XO. Thus N = min n{Xn =
XO). (a) Show that EN = m. (b) Show that
E log N< H (X). (c) (Optional) Prove part
(a) for {Xi} stationary and ergodic.
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