PDF of Lecture Notes - School of Mathematical Sciences

1. DISTRIBUTION THEORY
1.11 Limit Theorems
1.11.1 Convergence of random variables
Let X 1 , X 2 , X 3 , . . . , be an infinite sequence of RVs. We will consider 4 different types
of convergence.
1. Convergence in probability (weak convergence)
The sequence {X n } is said to converge to the constant α ∈ R, in probability if
for every ε > 0
lim
n→∞ P (|X n − α| > ε) = 0
2. Convergence in Quadratic Mean
lim
n→∞ E( (X n − α) 2) = 0
3. Almost sure convergence (Strong convergence)
The sequence {X n } is said to converge almost surely to α if for each ε > 0,
Remarks
(1),(2),(3) are related by:
|X n − α| > ε for only finite number of n ≥ 1.
Figure 15: Relationships of convergence
4. Convergence in distribution
The sequence of RVs {X n } with CDFs {F n } is said to converge in distribution
to the RV X with CDF F (x) if:
lim F n(x) = F (x) for all continuity points of F .
n→∞
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