State whether the following statements are true or false and also give the reason in support of your answer:
1. True. A finite-state discrete-time Markov chain always admits at least one stationary distribution because its transition matrix is stochastic and has eigenvalue 1; by the Brouwer–Perron–Frobenius argument there exists a probability left-eigenvector π with πP=π. Uniqueness and positivity require irreducibility and aperiodicity. 2. True. The Markov property states that for all n, P(X\_{n+1}\in A | X\_n, X\_{n-1}, …) = P(X\_{n+1}\in A | X\_n), i.e., conditional on the present, the future is independent of the past. 3. False. Independent increments do __________ ___ ______ ______ ___ _____.
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