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)

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