Intuitive Difference Between Weak And Strong Convergence in Probability

[IniquiTrance]

I've seen numerous rigourous/conceptual explanations of the difference between convergence in probability (weak), and strong, almost sure convergence.

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

$$\lim_{n \rightarrow \infty} \mathbb{P}\left\{|X_{n}-X|<\epsilon\right\}=1$$

or:$$\lim_{n \rightarrow \infty} \mathbb{P}\left\{\omega:|X_{n}(\omega)-X(\omega)|>\epsilon\right\}=0$$

While strong convergence means:

$$\mathbb{P}\left\{\lim_{n \rightarrow \infty} X_{n}=X\right\}=1$$

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 $|X_{n}-X|$ 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!

 

[SW VandeCarr]

I've seen numerous rigourous/conceptual explanations of the difference between convergence in probability (weak), and strong, almost sure convergence.

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

$$\lim_{n \rightarrow \infty} \mathbb{P}\left\{|X_{n}-X|<\epsilon\right\}=1$$

While strong convergence means:

$$\mathbb{P}\left\{\lim_{n \rightarrow \infty} X_{n}=X\right\}=1$$

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?
Thanks!

My intuitive understanding is that strong convergence of a probability is analogous to sampling real numbers from the interval [0,1], say by Dedekind cuts. Every real number has a uniform probability of zero of being 'drawn' but the sum (density) of probabilities over the interval [0,1] is 1. Even though the probability of any real number is zero, a real number is always chosen by a Dedekind cut on the interval.

Weak convergence is analogous to the probability density of an event under an infinite continuous distribution. An infinite distribution cannot be uniform. Given that some events have a non zero probability density, then for every event, there can be an event with a smaller non zero probability density. Note that the closed interval [0,1] is finitely bounded by, and includes, 0 and 1 while the range of the Gaussian pdf is not finitely bounded.

BTW my intuition based on this example may be too restrictive or even wrong; in which case I'm sure someone will jump in and correct me. I responded because your post has gone unanswered for a while. Essentially, my understanding of strong convergence of a probability is defined in terms of a sample space while almost everywhere convergence is defined in terms of a pdf.

 

[bpet]

... 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.

One important difference is that the strong limit need not even exist when the weak one does. A neat example is given on the Wikipedia with an archer doing target practice: if X(n)=1 is a hit and X(n)=0 is a miss, then the probability of missing decreases as they practice (weak convergence to X=1) but there is always a non-zero chance of missing (no strong convergence).