If Xn→X and Yn→Y almost surely, does (Xn,Yn)→(X,Y) almost surely?

[ziyanlan]

Suppose I have two sequences of r.v.s Xn and Yn. Xn converges to X with probability 1, and Yn converges to Y with probability 1. Does (Xn, Yn) converges to (X, Y) with probability 1? Is there a reference to confirm or negate this?

Thanks a lot.

 

[disregardthat]

Suppose I have two sequences of r.v.s Xn and Yn. Xn converges to X with probability 1, and Yn converges to Y with probability 1. Does (Xn, Yn) converges to (X, Y) with probability 1? Is there a reference to confirm or negate this?

Thanks a lot.

I don't know what context this is in, but my answer would be that (Xn,Yn) converges to (X,Y) is equivalent to the statement that Xn converges to X and Yn converges to Y (with respect to any topology). Hence, if we treat these events as A and B respectively, you know P(A) = 1, P(B) = 1, hence $$P(A \cap B) = P(A)+P(B) -P(A \cup B) =1$$.