Difference between Almost sure convergence and Convergence in probability

Question

Hi I am writing because I have been trying to understand the difference between the two definitions for some time. According to Wikipedia the definition of Convergence in probability is:

A sequence ${X_n}$ of random variables converges in probability towards the random variable $X$ if for all $ε > 0$

${\displaystyle \lim _{n\to \infty }\Pr {\big (}|X_{n}-X|>\varepsilon {\big )}=0}$.

And for amost sure convergence:

To say that the sequence $X_n$ converges almost surely or almost everywhere or with probability 1 or strongly towards $X$ means that

$ \operatorname {Pr} \!\left(\lim _{n\to \infty }\!X_{n}=X\right)=1$

or

$ {\displaystyle \operatorname {Pr} {\Big (}\limsup _{n\to \infty }{\big \{}\omega \in \Omega :|X_{n}(\omega )-X(\omega )|>\varepsilon {\big \}}{\Big )}=0\quad {\text{for all}}\quad \varepsilon >0.}$

For me the two definitions are practically identical, so I can't understand what the difference is between the two terminologies.

Can someone help me?

Answer 1

Almost sure convergence is strictly stronger than convergence in probability. Perhaps an example of a sequence converging in probability but not almost surely might help you? See the first answer of this question for example. Heuristically, convergence in probability does not rule out that for almost every "event" $\omega\in\Omega$ a large deviation of $X_n(\omega)$ from the respective limit $X(\omega)$ may be observed for arbitrarily large $n>0$.