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
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