Contents
What do you mean by convergence of random variables?
Definition. A sequence {Xn} of random variables converges in probability towards the random variable X if for all ε > 0. More explicitly, let Pn(ε) be the probability that Xn is outside the ball of radius ε centered at X.
What are different types of convergence?
There are four types of convergence that we will discuss in this section:
- Convergence in distribution,
- Convergence in probability,
- Convergence in mean,
- Almost sure convergence.
What happens when the sum of two random variables converges?
And if we have another sequence of random variables that converges to a certain number, b, which means that the probability distribution of Yn is heavily concentrated around b. In that case, then the probability distribution of the sum of the two random variables is heavily concentrated in the vicinity of a plus b. So what are we saying?
What is the purpose of convergence in probability?
One is to verify that the notion of convergence in probability is quite natural and that it has properties similar to the notion of convergence of sequences. And the second purpose is to get a little bit of practice with the formal definition of convergence in probability. So what is the statement saying?
What happens when the probability of Xn converges to 0?
The probability that something happens or something else is happening is less than or equal to the sum of their probabilities. And now, since Xn converges to a in probability, then by definition, we know that this quantity converges to 0 as n goes to infinity.
What happens when the sum of two sequences converges to 0?
Therefore, the sum of these two sequences also converges to 0. In essence, here we’re applying what we established earlier about convergence of numbers. If a sequence converges to 0 and another sequence converges to 0, then the sum of these sequences also converges to 0 as n goes to infinity.