3. METRICS PROBABILITY DISTRIBUTIONS F AND P 9 Theorem 3.1 A.dpr(P,Q)2 dBL-(P,Q)<2dpr(P,Q). B.H2(P,Q)≤dv(P,Q)≤H(P,Q){2-H2(P,Q)}/2 C.dpr(P,Q)<drv(P;Q). D.For distributions P,Q on the real line,dL<dk drv. Proof.We proved B in chapter 2.For A,see Dudley (1989)section 11.3,problem 5,and section 11.6,corollary 11.6.5.Also see Huber (1981),corollary 2.4.3,page 33.Another useful reference is Whitt(1974).▣ Theorem 3.2 (Strassen).The following are equivalent: (a)dprr(P,Q)≤e. (b)There exist X~P,Y~Q defined on a common probability space (F,Pr)such that Pr(d(X,Y)≤e)≥1-e. Proof.(b)implies (a)is easy:for any Borel set B, [X∈B]=[X∈B,d(X,Y)≤eU[X∈B,d(X,Y)>d c[X∈B]U[d(X,Y)>, so that P(B)≤Q(B)+e. For the proof of (a)implies (b)see Strassen (1965),Dudley (1968),or Schay (1974).A nice treatment of Strassen's theorem is given by Dudley (1989).3. METRICS PROBABILITY DISTRIBUTIONS F AND P 9 Theorem 3.1 A. dP r(P, Q)2 ≤ dBL∗ (P, Q) ≤ 2dP r(P, Q). B. H2(P, Q) ≤ dT V (P, Q) ≤ H(P, Q){2 − H2(P, Q)}1/2. C. dP r(P, Q) ≤ dT V (P, Q). D. For distributions P, Q on the real line, dL ≤ dK ≤ dT V . Proof. We proved B in chapter 2. For A, see Dudley (1989) section 11.3, problem 5, and section 11.6, corollary 11.6.5. Also see Huber (1981), corollary 2.4.3, page 33. Another useful reference is Whitt (1974). ✷ Theorem 3.2 (Strassen). The following are equivalent: (a) dP r(P, Q) ≤ '. (b) There exist X ∼ P, Y ∼ Q defined on a common probability space (Ω, F, P r) such that P r(d(X, Y ) ≤ ') ≥ 1 − '. Proof. (b) implies (a) is easy: for any Borel set B, [X ∈ B] = [X ∈ B, d(X, Y ) ≤ '] ∪ [X ∈ B, d(X, Y ) > '] ⊂ [X ∈ B$ ] ∪ [d(X, Y ) > '], so that P(B) ≤ Q(B$ ) + '. For the proof of (a) implies (b) see Strassen (1965), Dudley (1968), or Schay (1974). A nice treatment of Strassen’s theorem is given by Dudley (1989). ✷