VI. Probability spaces and independence

Probability spaces and random variables

• Kolmogorov’s probability model. See pages 189-190 of A&L.

• Random variable

• CDF

• Probability distribution definition.

• Random vector of finite dimension.

• Joint probability distribution.

• Marginal probability distribution.

• Expected value.

• Change of variable formula.

• Moments.

• Moment generating function.

• MGF finite in neighborhood of zero implies: i) all moments finite, ii) power series expansion, iii) infinitely differentiable, \(r\)th derivative formula. Follow Proposition 6.2.3 of A&L.

• Inequalities:

  • Markov’s
  • Chebychev’s
  • Hölder’s
  • Cauchy-Schwarz
  • Minkowski’s

• Mention joint MGF.

• Kolmogorov’s consistency theorem? Should I mention this or save for later? Definition of a stochastic process?

Independence

• Independence of a finite number of events.

• Independence of a collection of events, arbitrarily indexed.

• Independence of collections of arbitrarily indexed collections of events.

• Independence of a collection of random variables, arbitrarily indexed.

• Independence of \(\pi\)-classes implies independence of \(\sigma\)-algebras generated by the \(\pi\)-classes. Show proof for just two \(\pi\)-classes. Page 64 of my handwritten notes. Present Proposition 7.1.1.

• A collection of random variables is independent iff the joint CDFs of finite subcollections are factorizable. Follow Corollary 7.1.2 of A&L. Prove by showing \(\{X^{-1}((-\infty,x]), x\in \mathbb{R}\}\) is a \(\pi\)-class and invoking the previous result.

• Set of random variables independent if expected value of product of functions is equal to the product of expectations—for all Borel measurable functions. Expectation of a product is the product of expectations. Follow Prop 7.1.3 of A&L.

Borel-Cantelli, Kolmogorov zero-one laws

• Define \(\liminf_{n \to \infty}A_n\) as a union of intersections and \(\limsup_{n \to \infty}A_n\) as an intersection of unions.

• Give interpretations that \(\omega\) occurs “in all but finitely many” (\(\liminf\)) or “infinitely often” (\(\limsup\)).

• First and second Borel-Cantelli Lemmas. Follow Theorem 7.2.2 in A&L.

• Condition for a sequence of random variables to converge to zero with probability one. A converse. Follow Proposition 7.2.3 of A&L.

• Definition of tail \(\sigma\)-algebra.

• Kolmogorov’s 0-1 law.

• Definition of extended real-valued random variable.

• Random variable measurable with respect to a tail \(\sigma\)-algebra converges almost surely to a (possibly infinite) constant. Corollary 7.2.5 of A&L.

• Example 7.2.2 of A&L perhaps.