Question

Let X0,X1,... be a Markov chain whose state space is Z (the integers).

Recall the Markov property: P(Xn = in | X0 = i0,X1 = i1,...,Xn−1 = in−1) = P(Xn = in | Xn−1 = in−1), ∀n, ∀it. Does the following always hold: P(Xn ≥0|X0 ≥0,X1 ≥0,...,Xn−1 ≥0)=P(Xn ≥0|Xn−1 ≥0) ?

(Prove if “yes”, provide a counterexample if “no”)

Let Xo,Xi, be a Markov chain whose state space is Z (the integers). Recall the Markov property: P(X,-'n l Xo-io, Xi = 11, , xn-1-In-l)-P(X,-İn l X-1in-1), Vn, Vi. Does the following always hold: ( Prove if "yes", provide a counterexample if "no ")

Answer 1

The answer is yes

let us observe

$P(X_{m+1} \geq 0| X_{m} \geq 0,..., X_{0} \geq 0)$

in 20,in-120,...io20

P(X, -inX-1 by Markov Property bu Markov Propert in20,in-120

$= P(X_{n} \geq 0| X_{n-1} \geq 0)$

The sum is possible since the events are disjoint for fixed k and different i that is why the events are also disjoint for different combinations of
in and in-1

Hence the result