2 1. Preparatory material

1.1. A review of probability theory

Random matrix theory is the study of matrices whose entries are random

variables (or equivalently, the study of random variables which take values in

spaces of matrices). As such, probability theory is an obvious prerequisite

for this subject. As such, we will begin by quickly reviewing some basic

aspects of probability theory that we will need in the sequel.

We will certainly not attempt to cover all aspects of probability theory in

this review. Aside from the basic foundations, we will be focusing primarily

on those probabilistic concepts and operations that are useful for bounding

the distribution of random variables, and on ensuring convergence of such

variables as one sends a parameter n off to infinity.

We will assume familiarity with the foundations of measure theory, which

can be found in any text book (including my own text [Ta2011]). This is

also not intended to be a first introduction to probability theory, but is

instead a revisiting of these topics from a graduate-level perspective (and

in particular, after one has understood the foundations of measure theory).

Indeed, it will be almost impossible to follow this text without already having

a firm grasp of undergraduate probability theory.

1.1.1. Foundations. At a purely formal level, one could call probability

theory the study of measure spaces with total measure one, but that would

be like calling number theory the study of strings of digits which terminate.

At a practical level, the opposite is true: just as number theorists study

concepts (e.g., primality) that have the same meaning in every numeral sys-

tem that models the natural numbers, we shall see that probability theorists

study concepts (e.g., independence) that have the same meaning in every

measure space that models a family of events or random variables. And

indeed, just as the natural numbers can be defined abstractly without ref-

erence to any numeral system (e.g., by the Peano axioms), core concepts of

probability theory, such as random variables, can also be defined abstractly,

without explicit mention of a measure space; we will return to this point

when we discuss free probability in Section 2.5.

For now, though, we shall stick to the standard measure-theoretic ap-

proach to probability theory. In this approach, we assume the presence of

an ambient sample space Ω, which intuitively is supposed to describe all

the possible outcomes of all the sources of randomness that one is studying.

Mathematically, this sample space is a probability space Ω = (Ω, B, P)—a

set Ω, together with a σ-algebra B of subsets of Ω (the elements of which we

will identify with the probabilistic concept of an event), and a probability

measure P on the space of events, i.e., an assignment E → P(E) of a real

number in [0, 1] to every event E (known as the probability of that event),