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
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
(i) Xn converges almost surely to X if, for almost every ω ∈ Ω, Xn(ω)
converges to X(ω), or equivalently
d(Xn,X) ≤ ε) = 1
for every ε 0.
(ii) Xn converges in probability to X if, for every ε 0, one has
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
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
metric space is σ-compact if it is the countable union of compact sets.