I've seen numerous rigourous/conceptual explanations of the difference between convergence in probability (weak), and strong, almost sure convergence.(adsbygoogle = window.adsbygoogle || []).push({});

One explanation my prof gave was that convergence in probability entails:

[tex]\lim_{n \rightarrow \infty} \mathbb{P}\left\{|X_{n}-X|<\epsilon\right\}=1[/tex]

or:

[tex]\lim_{n \rightarrow \infty} \mathbb{P}\left\{\omega:|X_{n}(\omega)-X(\omega)|>\epsilon\right\}=0[/tex]

While strong convergence means:

[tex]\mathbb{P}\left\{\lim_{n \rightarrow \infty} X_{n}=X\right\}=1[/tex]

So my prof explains one difference is the limit is taken outside the probability for convergence in probability, while it is inside the probability for almost sure convergence.

Can anyone elaborate on this?

Also, the limits in both forms of convergence seem to imply the same thing.

How is it that saying the probability that [itex]|X_{n}-X|[/itex] is greater than some epsilon, goes to 0 for large n, implies that difference can jump above epsilon, infinitely many times?

And if it does, how is it that saying the probability of it staying below epsilon goes to 1, as n goes to infinity, implies that it CAN'T jump above epsilon EVER, after some n? (And is thus a somehow stronger form of convergence).

Thanks!

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# Intuitive Difference Between Weak And Strong Convergence in Probability

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