Stat 501 – Probability Theory I Some Lecture Notes

These notes are based on the two lectures1 I gave in Stat 501, Probability Theory I, as a substitute for Professor Cheng Ouyang. There is a brief discussion of the so-called “basic grouping lemma” concerning independent σ-algebras, followed by a more detailed discussion of the Borel–Cantelli lemma, some applications, and some elaborations on independence. Independence, and the basic grouping lemma Let {Bt : t ∈ T} be a collection of independent σ-algebras, i.e., for any k ≥ 1, for any t1, . . . , tk, and any Bt1 , . . . , Btk in Bt1 , . . . ,Btk , the events Bt1 , . . . , Btk are independent. It makes sense that disjoint sub-collections of the σ-algebras are also independent. Here is the formal result. Lemma (Grouping Lemma). Let {Bt : t ∈ T} be an independent collection of σ-algebras. Let S be an index set with the property that, for s ∈ S, Ts ⊂ T and {Ts : s ∈ S} are pairwise disjoint. Define BTs = smallest σ-algebra containing all Bt, t ∈ Ts. Then {BTs : s ∈ S} is an independent collection of σ-algebras. Proof. Pretty easy, see pages 101–102 in Resnick. Despite the complicated σ-algebra terminology, the Grouping Lemma is quite intuitive. For example, let X1, . . . , Xn be a collection of independent random variables. Then the Grouping Lemma says that • σ({X1, . . . , Xk}) and σ({Xk+1, . . . , Xn}) are independent σ-algebras; • ∑k i=1Xi and ∑n i=k+1Xi are independent random variables; • and, more generally, f(X1, . . . , Xk) and g(Xk+1, . . . , Xn) are independent random variables for any suitable real-valued functions f and g. These lectures are based, in part, on Sidney Resnick’s A Probability Path.