Theory of Statistics - George Mason University

76 1 Probability Theory
Almost Sure Convergence
Definition 1.35 (almost sure (a.s.) convergence)
We say that {Xn} converges almost surely to X if
We write
lim
n→∞ Xn = X a.s. (1.153)
Xn
a.s.
→ X.
Writing this definition in the form of Definition 0.1.36 on page 718, with Xn
and X defined on the probability space (Ω, F, P), we have
P({ω : lim
n→∞ Xn(ω) = X(ω)}) = 1. (1.154)
This expression provides a very useful heuristic for distinguishing a.s. convergence
from other types of convergence.
Almost sure convergence is equivalent to
lim
n→∞ Pr(∪∞ m=nXm − X > ɛ) = 0, (1.155)
for every ɛ > 0 (exercise).
Almost sure convergence is also called “almost certain” convergence, and
a.c.
written as Xn → X.
The condition (1.153) can also be written as
Pr lim
n→∞ Xn
− X < ɛ = 1, (1.156)
for every ɛ > 0. For this reason, almost sure convergence is also called conver-
wp1
gence with probability 1, and may be indicated by writing Xn → X. Hence,
we may encounter three equivalent expressions:
a.s. a.c. wp1
→ ≡ → ≡ → .
Almost sure convergence of a sequence of random variables {Xn} to a
constant c implies lim sup n Xn = lim infn Xn = c, and implies {Xn = c i.o.};
by itself, however, {Xn = c i.o.} does not imply any kind of convergence of
{Xn}.
Convergence in r th Moment
Definition 1.36 (convergence in r th moment (convergence in Lr))
For fixed r > 0, we say that {Xn} converges in r th moment to X if
Theory of Statistics c○2000–2013 James E. Gentle
lim
n→∞ E(Xn − X r r) = 0. (1.157)