1.1. A review of probability theory 33

that on compact sets, continuous functions are uniformly continu-

ous.)

(iv) Show that Xn converges in distribution to X if and only if μXn

converges to μX in the vague topology (i.e., f dμXn → f dμX

for all continuous functions f : R → R of compact support).

(v) Conversely, if Xn has a tight sequence of distributions, and μXn is

convergent in the vague topology, show that Xn is convergent in

distribution to another random variable (possibly after extending

the sample space). What happens if the tightness hypothesis is

dropped?

(vi) If X is deterministic, show that Xn converges in probability to X

if and only if Xn converges in distribution to X.

(vii) If Xn has a tight sequence of distributions, show that there is a

subsequence of the Xn which converges in distribution. (This is

known as Prokhorov’s theorem).

(viii) If Xn converges in probability to X, show that there is a subse-

quence of the Xn which converges almost surely to X.

(ix) Xn converges in distribution to X if and only if lim infn→∞ P(Xn ∈

U) ≥ P(X ∈ U) for every open subset U of R, or equivalently if

lim supn→∞ P(Xn ∈ K) ≤ P(X ∈ K) for every closed subset K of

R.

Exercise 1.1.26 (Skorokhod representation theorem). Let μn be a sequence

of probability measures on C that converge in the vague topology to an-

other probability measure μ. Show (after extending the probability space if

necessary) that there exist random variables Xn with distribution μn that

converge almost surely to a random variable X with distribution μ.

Remark 1.1.31. The relationship between almost sure convergence and

convergence in probability may be clarified by the following observation. If

En is a sequence of events, then the indicators I(En) converge in probability

to zero iff P(En) → 0 as n → ∞, but converge almost surely to zero iff

P(

n≥N

En) → 0 as N → ∞.

Example 1.1.32. Let Y be a random variable drawn uniformly from [0, 1].

For each n ≥ 1, let En be the event that the decimal expansion of Y begins

with the decimal expansion of n, e.g., every real number in [0.25, 0.26) lies

in E25. (Let us ignore the annoying 0.999 . . . = 1.000 . . . ambiguity in the

decimal expansion here, as it will almost surely not be an issue.) Then the

indicators I(En) converge in probability and in distribution to zero, but do

not converge almost surely.

If yn is the

nth

digit of Y , then the yn converge in distribution (to the

uniform distribution on {0, 1,..., 9}), but do not converge in probability or