32 1. Preparatory material

strength) will be almost sure convergence, convergence in probability, con-

vergence in distribution, and tightness of distribution.

Definition 1.1.29 (Modes of convergence). Let R = (R, d) be a

σ-compact7

metric space (with the Borel σ-algebra), and let Xn be a sequence of random

variables taking values in R. Let X be another random variable taking values

in R.

(i) Xn converges almost surely to X if, for almost every ω ∈ Ω, Xn(ω)

converges to X(ω), or equivalently

P(lim sup

n→∞

d(Xn,X) ≤ ε) = 1

for every ε 0.

(ii) Xn converges in probability to X if, for every ε 0, one has

lim inf

n→∞

P(d(Xn,X) ≤ ε) = 1,

or equivalently if d(Xn,X) ≤ ε holds asymptotically almost surely

for every ε 0.

(iii) Xn converges in distribution to X if, for every bounded continuous

function F : R → R, one has

lim

n→∞

EF (Xn) = EF (X).

(iv) Xn has a tight sequence of distributions if, for every ε 0, there

exists a compact subset K of R such that P(Xn ∈ K) ≥ 1 − ε for

all suﬃciently large n.

Remark 1.1.30. One can relax the requirement that R be a σ-compact

metric space in the definitions, but then some of the nice equivalences and

other properties of these modes of convergence begin to break down. In our

applications, though, we will only need to consider the σ-compact metric

space case. Note that all of these notions are probabilistic (i.e., they are

preserved under extensions of the sample space).

Exercise 1.1.25 (Implications and equivalences). Let Xn,X be random

variables taking values in a σ-compact metric space R.

(i) Show that if Xn converges almost surely to X, then Xn converges

in probability to X. (Hint: Use Fatou’s lemma.)

(ii) Show that if Xn converges in distribution to X, then Xn has a tight

sequence of distributions.

(iii) Show that if Xn converges in probability to X, then Xn converges

in distribution to X. (Hint: First show tightness, then use the fact

7A

metric space is σ-compact if it is the countable union of compact sets.