Convergence of Random Variable

Pointwise or sure convergence

A sequence of random variables {Xn}n∈N is said to converge point-wise or surely to X if
Xn(ω) → X(ω),  ∀ ω∈Ω

Almost sure convergence

A sequence of random variables {Xn}n∈N is said to converge almost surely or with probability 1 to X if
P ({ω|Xn(ω) → X(ω)}) = 1

This convergence is also known as convergence with probability 1. It is denoted by a.s. or w.p.1. To get the intuition, sure convergence says that the set of ω’s where the random variables converge has a probability one.

Convergence in probability

A sequence of random variables {Xn}n∈N is said to converge in probability to X if
limn→∞ P(|Xn −X|>ε)=0, ∀ ε>0.

This is denoted by i.p. or p.

Convergence in rth mean

A sequence of random variables {Xn}n∈N is said to converge in rth mean to X if
limn→∞E[|Xn − X|r]=0

When r=2 this is called convergence in mean squared/quadratic/L2 mean. This is the most common form of convergence used. This is denoted by q.m.

Convergence in distribution

A sequence of random variables {Xn}n∈N is said to converge in distribution to X if
limn→∞FXn (x) = FX (x),  ∀ x ∈ R where FX (·) is continuous.

This convergence is also known as weak convergence.

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