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.
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