Prelim Study Guide

Resnick Definitions

  • ℬ(ℝ)\mathcal{B}(\mathbb{R})
  • countable/co-countable σ\sigma-field
  • field
  • σ\sigma-field generated by 𝒞\mathcal{C}
  • lim inf⁡\liminf and lim sup⁡\limsup and lim⁡\lim
  • non-decreasing, non-increasing, and monotone sequences
  • σ\sigma-field
  • distribution function
  • general construction of probability model
  • λ\lambda-system
  • π\pi-system
  • probability space
  • semialgebra
  • continuous function
  • measure induced by random element
  • σ\sigma-field generated by a map
  • measurability w.r.t. a σ\sigma-field
  • measurable map
  • measurable space
  • metric
  • preimage
  • random element
  • σ\sigma-field generated by a collection of maps
  • ⋁i∈Iℬi\bigvee_{i \in I} \mathcal{B}_i
  • almost trivial σ\sigma-field
  • independent classes
  • independent events
  • independent random variables
  • tails

    • Resnick Theorems

  • minimal postulates for a field 𝒜\mathcal{A}
  • properties of indicators
  • intuitive interpretation of lim inf⁡ \liminf and lim sup⁡ \limsup
  • properties of lim inf⁡ \liminf and lim sup⁡ \limsup
  • properties of monotone sequences
  • minimal postulates for a σ\sigma-field 𝒜\mathcal{A}
  • key properties of σ\sigma-fields
  • restriction of σ\sigma-fields
  • combo extension
  • Dynkin
  • characterization of the field generated by a semialgebra 𝒮\mathcal{S}
  • first measure extension
  • the eight properties of probability measures
  • second measure extension
  • measurability of composition
  • measurability and continuity
  • equivalent definition of σ(X)\sigma(X) for a random variable XX
  • examples of r.v.s produced by measurable maps
  • image of random vector under measurable map
  • measurability of limits
  • necessary sufficient conditions for measurability
  • properties of preimages
  • preimages of σ\sigma-fields
  • random sequences are sequences of random variables
  • random vectors are tuples of random variables
  • almost trivial σ\sigma-fields
  • Borel-Cantelli
  • Borel zero-one law
  • dyadic expansion of uniformly distributed r.v.
  • factorization criterion for independence
  • grouping lemma
  • basic criteria for independence
  • Kolmogorov zero-one law
  • corollaries of Kolmogorov zero-one law
  • Renyi

    • Bickel & Doksum Definitions

  • identifiability
  • statistical model
  • parameter
  • parametrization

    • Bickel & Doksum Theorems

  • identification of parameter