When we intend to say a sequence of random variables converges, it is important to specify in what manner they converge. Here I will describe some of the different types of convergence. Recall a random variable is hosted in a probability space (Ω, 𝒜, ℙ).
Pointwise convergence.
For each point in the space Ω, a sequence of random variables
Xn will converge to a random variable X. Or in other words,
for every ω in Ω, the sequence Xn(ω)
approaches X(ω).
Although point-wise convergence is intuitive, it is not very useful for probability theory. Because equality in probability theory is almost always defined almost surely, the strict equality is too strong a requirement to meet. In practice, point-wise convergence is not often used.
Almost sure convergence.
This is a way of saying that the probability that the sequence
Xn approaches X is almost certain, or has a probability
equal to one. In math speak, we can write this as
ℙ(Xn approaches X) = 1.
This mode of convergence can be written in another way as well, which is analogous to the definition of point-wise convergence. Let F be a set in 𝒜 such that ℙ(F) = 1. Then a sequence converges almost surely if and only if Xn(ω) approaches X(ω) for all ω contained in F.
This is the strongest mode of convergence listed here. We will see that it is useful in probability theory, because it allows us to neglect sets of measure zero.
Convergence in probability.
In this mode of convergence, random variables become arbitrarily
close to one another, such that the probability of them being
a certain non-zero distance from one another is zero. This
can be written as ℙ(|Xn - X| greater than ε)
= 0 for all ε greater than 0.
Convergence in Lr.
This is reminciant of convergence in probability, but instead of
using the probability measure we use the expectation of the
rth moment to determine when random variables draw
near eachother. The expectation is denoted 𝔼, and this mode
of convergence is written: 𝔼(|Xn - X|r) = 0
for some fixed r.
Convergence in distribution (or weakly).
This is when the distribution functions, or cumulative distribution
functions, are equal to eachother in the limit. This is written
as limnFn(x) = F(x) for all x
where F is continuous. Note the condition in the last part
— it is easy to forget.
This is the weakest of all the modes of convergence, hence the alternative name convergence weakly.