A question about convergence with probability one

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The discussion centers on the convergence of two sequences of random variables, Xn and Yn, where Xn converges to X with probability 1 and Yn converges to Y with probability 1. It is established that the joint convergence of (Xn, Yn) to (X, Y) also occurs with probability 1, as this is equivalent to the individual convergences of Xn to X and Yn to Y. The mathematical reasoning provided utilizes the properties of probability measures to confirm this conclusion.

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ziyanlan
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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.
 
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ziyanlan said:
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 [tex]P(A \cap B) = P(A)+P(B) -P(A \cup B) =1[/tex].
 

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