Topics in RMT by Terence Tao

Chapter 1: Preparatory material

A review of probability theory

• measure space (with total measure one)
• events or random variables
• sample space $\Omega \Rightarrow$ a probability space $(\Omega, \mathcal{B},P)$: set + $\sigma$-algebra (closed under complement & countable unions and intersections ) of related subset + probability measure
• extension of the sample space
• probability theory is only “allowed” to study concepts and perform operations which are preserved with respect to extension of the underlying sample space
• union bound: $P(\lor_i E_i) \le \sum_i P(E_i)$, also called zeroth moment method
• asymptotic notations: large and small O
• measuring the probability of an event:
• surely: equal to the event $\bar \emptyset$
• almost surely: occurs with proba 1
• with overwhelming probability: for evert fixed $A>0$, holds with proba $1- O_A(n^{-A})$
• with high proba: holds with proba $1-O(n^{-c})$ for some $c$ independent of $n$
• asymptotically almost surely: holds with proba $1- o(1)$, i.e., the pobab goes to 1 as $n \to \infty$.
• uniformly in some parameter if the constant of bound is independent of the parameter.
• random variables
• the affect of events of proba zero, as long as there are countable many of them
• joint random variable
• vector and matrix-valued r.v.
• point processes: e.g., the spectrum $\{\lambda_1, \ldots, \lambda_n\}$ of eigenvalues of a random (square) matrix.
• distribution $\mu_X$: probability measure/ law
• can we always create a r.v. with a specific distribution? extension may be necessary
• discrete distribution examples:
• Dirac distribution
• discrete uniform distribution
• (unsigned) Bernoulli distribution
• signed Bernoulli distribution
• Geometric distribution
• Possion distribution
• probability density function (PDF) $f$: Radon-Nikodym theorem tells us that there exists non-negative and absolutely integrable $f$ such that $\mu_X(S) = \int_S f dm$
• cumulative distribution function (CDF) $F_X$: indeed $\mu_X$ is the Lebesgue-Stieltjes measure of $F_X$
• examples of absolutely continuous scalar distribution:
• uniform distribution
• real normal distribution: standard normal distribution
• complex normal distribution
• expectation of mean (positive case) $\mathbf{E}X \equiv \int_0^\infty x d\mu_X(x)$ and by Fubini-Tonelli theorem $= \int_0^\infty P(X \ge \lambda ) d\lambda$, of course can be extended to R or C.
• expectation is linear and monotone if unsigned
• Markov inequality $P(|X| \ge \lambda) \le \frac1{\lambda} \mathbf{E}|X|$, also referred to as first moment method: but often gives suboptimal results since it does not explore any independence in the system
• Borel-Cantelli lemma: if $\sum_i P(E_i) < \infty$ then a.s. at most finitely many of the events $E_i$ occurs at once.
• expectation of function of a r.v.
• moments
• exponential moments
• Fourier moments/ characteristic function
• resolvent
• negative moment $|X|^{-k}$
• zeroth moment $\mathbf{E}|X|^0 = \mathbf{P}(X\neq 0)$