Sequence of random variables convergence in probability

1. How do you prove a sequence converges in probability?
2. How do you explain convergence in probability?
3. What is a sequence of random variables?

How do you prove a sequence converges in probability?

A sequence of random variables X1, X2, X3, ⋯ converges in probability to a random variable X, shown by Xn p→ X, if limn→∞P(|Xn−X|≥ϵ)=0, for all ϵ>0.

How do you explain convergence in probability?

The concept of convergence in probability is based on the following intuition: two random variables are "close to each other" if there is a high probability that their difference is very small.

What is a sequence of random variables?

In sum, a sequence of random variables is in fact a sequence of functions Xn:S→R. Example. Consider the following random experiment: A fair coin is tossed once. Here, the sample space has only two elements S=H,T.