ziyanlan said:Suppose I have two sequences of r.v.s Xn and Yn. Xn converges to X with probability 1, and Yn converges to Y with probability 1. Does (Xn, Yn) converges to (X, Y) with probability 1? Is there a reference to confirm or negate this?
Thanks a lot.
Convergence with probability one, also known as almost sure convergence, is a type of convergence in probability theory where a sequence of random variables approaches a fixed value with a probability of one. This means that the probability of the sequence not converging to the fixed value is zero.
The main difference between almost sure convergence and convergence in probability is the certainty of the convergence. In almost sure convergence, there is a certainty of the sequence approaching a fixed value, while in convergence in probability, there is only a high probability that the sequence will approach the fixed value.
Convergence with probability one and convergence in distribution are two different types of convergence in probability theory. Convergence with probability one focuses on the behavior of a sequence of random variables, while convergence in distribution focuses on the behavior of the distribution of these random variables.
Some examples of sequences that exhibit convergence with probability one are the sample mean of a sequence of independent and identically distributed random variables, the maximum value in a sequence of random variables, and the result of repeatedly rolling a fair die and taking the average of the outcomes.
Convergence with probability one is important in probability theory because it provides a stronger notion of convergence compared to other types of convergence. It guarantees that the sequence of random variables will approach a fixed value with a probability of one, which makes it a useful tool in many applications, including statistics, economics, and physics.