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1. True/False. For each of the following statements, write T (True) if the statement is necessarily true, F (False) if the statement could possibly be false, or U (unsure) if you are unsure of the answer. "You do not need to explain your answer" (a) If a Markov chain is transient (i.e. all states are transient), then it has no invariant measure. (b) A recurrent Markov chain may not have an invariant distribution. (c) If a Markov chain (Xn)nzo is irreducible and positive recurrent, that the distribution of Xn will converge to certain limiting distribution as n → ∞. (d) Let (Xk)k>1 be an i.i.d. sequence with distribution 1 1 P(X₁ = −2) = 2, P(X₁ = 0) = P(X₁ = 4) = and set W₁ = X₁ +...+ X₂. Then the sequence (Wn)n>1 is a martingale with respect to (Xk)k>1. (e) For the symmetric simple random walk (Sn)nzo on Z and any stopping time T with P(T < ∞) = 1, we always have E[ST] = E[So].

1. True/False. For each of the following statements, write T (True) if the statement is necessarily true, F (False) if the statement could possibly be false, or U (unsure) if you are unsure of the answer. "You do not need to explain your answer" (a) If a Markov chain is transient (i.e. all states are transient), then it has no invariant measure. (b) A recurrent Markov chain may not have an invariant distribution. (c) If a Markov chain (Xn)nzo is irreducible and positive recurrent, that the distribution of Xn will converge to certain limiting distribution as n → ∞. (d) Let (Xk)k>1 be an i.i.d. sequence with distribution 1 1 P(X₁ = −2) = 2, P(X₁ = 0) = P(X₁ = 4) = and set W₁ = X₁ +...+ X₂. Then the sequence (Wn)n>1 is a martingale with respect to (Xk)k>1. (e) For the symmetric simple random walk (Sn)nzo on Z and any stopping time T with P(T < ∞) = 1, we always have E[ST] = E[So].

MATLAB: An Introduction with Applications
MATLAB: An Introduction with Applications
6th Edition
ISBN:9781119256830
Author:Amos Gilat
Publisher:Amos Gilat
Chapter1: Starting With Matlab
Section: Chapter Questions
Problem 1P
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• General notation for Markov chains: P(A) is the probability of the event A when the Markov chain starts in state x, Pu(A) the probability when the initial state is random with distribution μ. Ty = min{n ≥ 1 : Xn = y} is the first time after 0 that the chain visits state y. px,y = Px(Ty < ∞) . Ny is the number of visits to state y after time 0. 1. True/False. For each of the following statements, write T (True) if the statement is necessarily true, F (False) if the statement could possibly be false, or U (unsure) if you are unsure of the answer. "You do not need to explain your answer" (a) If a Markov chain is transient (i.e. all states are transient), then it has no invariant measure. (b) A recurrent Markov chain may not have an invariant distribution. (c) If a Markov chain (Xn)nzo is irreducible and positive recurrent, that the distribution of Xn will converge to certain limiting distribution as n → ∞. (d) Let (Xk)k≥1 be an i.i.d. sequence with distribution P(X₁ = = −2) = 1 P(X₁ = 0) = P(X₁ = 4) 1 = 4 and set W₂ = X₁ + ... + Xn. Then the sequence (Wn)n≥1 is a martingale with respect to (Xk)k≥1. (e) For the symmetric simple random walk (Sn)nzo on Z and any stopping time T with P(T < ∞) = 1, we always have E[ST] = E[So].
Transcribed Image Text:• General notation for Markov chains: P(A) is the probability of the event A when the Markov chain starts in state x, Pu(A) the probability when the initial state is random with distribution μ. Ty = min{n ≥ 1 : Xn = y} is the first time after 0 that the chain visits state y. px,y = Px(Ty < ∞) . Ny is the number of visits to state y after time 0. 1. True/False. For each of the following statements, write T (True) if the statement is necessarily true, F (False) if the statement could possibly be false, or U (unsure) if you are unsure of the answer. "You do not need to explain your answer" (a) If a Markov chain is transient (i.e. all states are transient), then it has no invariant measure. (b) A recurrent Markov chain may not have an invariant distribution. (c) If a Markov chain (Xn)nzo is irreducible and positive recurrent, that the distribution of Xn will converge to certain limiting distribution as n → ∞. (d) Let (Xk)k≥1 be an i.i.d. sequence with distribution P(X₁ = = −2) = 1 P(X₁ = 0) = P(X₁ = 4) 1 = 4 and set W₂ = X₁ + ... + Xn. Then the sequence (Wn)n≥1 is a martingale with respect to (Xk)k≥1. (e) For the symmetric simple random walk (Sn)nzo on Z and any stopping time T with P(T < ∞) = 1, we always have E[ST] = E[So].
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