Independence

Independent events and random variables

• finite collection of events: independent if P(nnn B_(i_j)) = Pi P(B_(i_j))`
• collection of events: independent if every finite subcollection is independent
• `A` nonempty, `AA alpha in A, G_alpha sub F` a collection of events, family `{G_alpha: alpha in A}`: independent if `AA B_alpha in G_alpha` for `alpha in A`, `{B_alpha: alpha in A}` is independent
• collection of random variables:
  • independent if family of `sigma`-algebras is indepedent (if family of `pi`-systems is indepedent, then `sigma`-algebra generated from then will also be indepedent)
  • indepedent if joint cdf is always the product of the marginals (if ac, equivalent can replace cdf with pdf)
  • indepedent if `E[Pi h_i(X_i)] = Pi E[h_i(X_i)]` for all bounded Borel measurable functions `h_i: RR -> RR`

• If `X_1 and X_2` are independent and `E|X_1|, E|X_2| < oo` then `E|X_1 X_2| < oo` and `E[X_1 X_2] = E[X_1] * E[X_2]`

Borel-Cantelli lemmas, tail `sigma`-algebras and Kolmogorov’s zero-one law

Let `(Omega, F)` be a measurable space, and `{A_n}_(n>=1)` be a sequence of sets, then:

• `barlim A_n = nnn_(k=1)^(oo)(uuu_(n>=k) A_k ) = {omega: omega in A_n "infinitely often"}`
• `ul lim A_n = uuu_(k=1)^(oo)(nnn_(n>=k) A_k ) = {omega: omega in A_n "for all but finitely many" n}`

• If `sum P(A_n) < oo` then `P(barlim A_n) = 0`.
• If `sum P(A_n) = oo` and `{A_n}` are pairwises independent, then `P(barlim A_n) = 1`.

Called the zero-one law.

• If `sum P(|X_n| > epsi) < oo AA epsi > 0` then `P(lim_(n->oo) X_n = 0) = 1`
• If `{X_n}` are pairwise independent, and `P(lim_(n->oo) X_n = 0) = 1` then `sum P(|X_n| > epsi) < oo AA epsi > 0`

The tail `sigma`-algebra of a sequence of random variables `{X_n}` on a ps, is `tau = nnn_(n=1)^(oo) sigma(: {X_j: j >= n} :)`, any `A in tau` is called a tail event, and any `tau`-measurable rv is called a tail random variable. Tail events are determined by behaviour for large n and remain unchanged if any finite subcollection is dropped or changed.

Kolmogorov’s 0-1 law: Let `{X_n}_(n>=1)` be a sequence of independent rv’s on a probabily space `(Omega, F, P)`, and `tau` the tail `sigma`-algebra on `{X_n}`, then `P(A) = 0 or 1 AA A in tau`.

Let `tau` be the tail `sigma`-algebra of `{X_n}_(n>=1)`, and let `X` be a tail random variable `X: Omega -> barRR` then `EE c in barRR "st" P(X = c) = 1`.