# Thread: Almost sure convergence & convergence in probability

1. ## Almost sure convergence & convergence in probability

"Almost sure convergence" always implies "convergence in probability", but the converse is NOT true.

Thus, there exists a sequence of random variables Y_n such that Y_n->0 in probability, but Y_n does not converge to 0 almost surely.
I think this is possible if the Y's are independent, but still I can't think of an concrete example. What is of example of this happening?

Any help is appreciated!

[note: also under discussion in math help forum]

2. Check the following link .
https://www.stat.duke.edu/courses/Sp...ec/s05wk07.pdf
see page 6

3. Originally Posted by vinux
https://www.stat.duke.edu/courses/Sp...ec/s05wk07.pdf
see page 6
Hi, I cannot read the page as it says that: "There is a problem with this website's security certificate."

What is the simplest counterexample?

4. If you just need an example. Solution is not there.

I have uploaded the counter example part.

5. Let X_n be a sequence of independent random variables such that
P(X_n=0)=1-1/n and P(X_n=1)=1/n

Then X_n converges in probability to 0.

By Borel-Cantelli's lemma, since

∑ 1/n = ∞ (diverges),
n=1
X_n does NOT converge almost surely to 0.

Borel Cantelli Lemma:
Let A1,A2,A3,... be events.
(i) if ∑P(An)<∞, then P(An io)=0
(ii) if the A's are independent, and ∑P(An)=∞, then P(An io)=1
where P(An io) stands for the probability that an infinite number of the A's occurs.

I don't understand the argument in red, why does Borel Cantelli lemma implies that X_n does NOT converge almost surely to 0? Are we using part (i) or part (ii) of Borel Cantelli lemma?

Can someone please explain this in more detail?
Thank you!

6. Originally Posted by kingwinner
Let X_n be a sequence of independent random variables such that
P(X_n=0)=1-1/n and P(X_n=1)=1/n

Then X_n converges in probability to 0.

By Borel-Cantelli's lemma, since

∑ 1/n = ∞ (diverges),
n=1
X_n does NOT converge almost surely to 0.

Borel Cantelli Lemma:
Let A1,A2,A3,... be events.
(i) if ∑P(An)<∞, then P(An io)=0
(ii) if the A's are independent, and ∑P(An)=∞, then P(An io)=1
where P(An io) stands for the probability that an infinite number of the A's occurs.

I don't understand the argument in red, why does Borel Cantelli lemma implies that X_n does NOT converge almost surely to 0? Are we using part (i) or part (ii) of Borel Cantelli lemma?

Can someone please explain this in more detail?
Thank you!
since P(X_n=1)=1/n by the B-C Lemma we know that [X_n=1 i.o.]

if this is the case how could X_n converge almost surely to zero?

7. Originally Posted by Martingale
since P(X_n=1)=1/n by the B-C Lemma we know that [X_n=1 i.o.]

if this is the case how could X_n converge almost surely to zero?
By part (ii) of B-C lemma, since ∑ P(X_n=1)= ∑1/n = ∞, this implies that P(X_n=1 io)=1, but WHY does this imply that Xn does not converge almost surely to 0? I don't understand this.
Does it converge almost surely to any other value? If it does not converge almost surely to 0, then it converges almost surely to what value?

Thank you!

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