1.1. A review of probability theory 15

when X is real-valued and

(1.16) EF (X) =

C

F (z) dμX(z)

when X is complex-valued. The same formula applies to signed or complex

F if it is known that |F (X)| is absolutely integrable. Important examples

of expressions such as EF (X) are moments

(1.17)

E|X|k

for various k ≥ 1 (particularly k = 1, 2, 4), exponential moments

(1.18)

EetX

for real t, X, and Fourier moments (or the characteristic function)

(1.19)

EeitX

for real t, X, or

(1.20)

Eeit·X

for complex or vector-valued t, X, where · denotes a real inner product. We

shall also occasionally encounter the resolvents

(1.21) E

1

X − z

for complex z, though one has to be careful now with the absolute conver-

gence of this random variable. Similarly, we shall also occasionally encounter

negative moments

E|X|−k

of X, particularly for k = 2. We also sometimes

use the zeroth moment

E|X|0

= P(X = 0), where we take the somewhat

unusual convention that x0 := limk→0+ xk for non-negative x, thus x0 := 1

for x 0 and

00

:= 0. Thus, for instance, the union bound (1.1) can be

rewritten (for finitely many i, at least) as

(1.22) E|

i

ciXi|0

≤

i

|ci|0E|Xi|0

for any scalar random variables Xi and scalars ci (compare with (1.12)).

It will be important to know if a scalar random variable X is “usually

bounded”. We have several ways of quantifying this, in decreasing order of

strength:

(i) X is surely bounded if there exists an M 0 such that |X| ≤ M

surely.

(ii) X is almost surely bounded if there exists an M 0 such that

|X| ≤ M almost surely.

(iii) X is sub-Gaussian if there exist C, c 0 such that P(|X| ≥ λ) ≤

C

exp(−cλ2)

for all λ 0.