and C ( j ) s +1 are disjoint events, and the fact that D ( j ) 1 = C ( j ) 1 we obtain: R$_{1}$ ( p$_{1}$ ) ∑ r =1 Pr ( C ( j ) r | P$_{2}$$_{j}$ ≤ rq$_{2}$ R$_{1}$ ( p$_{1}$ ) , P$_{1}$ = p$_{1}$ ) = Pr ( D ( j ) 1 | P$_{2}$$_{j}$ ≤ q$_{2}$ R$_{1}$ ( p$_{1}$ ) , P$_{1}$ = p$_{1}$ ) + R$_{1}$ p$_{1}$ ) ∑ r =2 [ Pr ( D ( j ) r | P$_{2}$$_{j}$ ≤ rq$_{2}$ R$_{1}$ ( p$_{1}$ ) , P$_{1}$ = p$_{1}$ ) − Pr ( D ( j ) r − 1 | P$_{2}$$_{j}$ ≤ rq$_{2}$ R$_{1}$ ( p$_{1}$ ) , P$_{1}$ = p$_{1}$ )] = R$_{1}$ ( p$_{1}$ ) r =1 Pr D ( j ) r | P$_{2}$$_{j}$ ≤ rq$_{2}$ R$_{1}$ ( p$_{1}$ ) , P$_{1}$ = p$_{1}$ − R$_{1}$ ( p$_{1}$ ) − 1 ∑ r =1 Pr ( D ( j ) r | P$_{2}$$_{j}$ ≤ ( r + 1) q$_{2}$ R$_{1}$ ( p$_{1}$ ) , P$_{1}$ = p$_{1}$ ) ≤ R$_{1}$ ( p$_{1}$ ) ∑ r =1 Pr ( D ( j ) r | P$_{2}$$_{j}$ ≤ rq$_{2}$ R$_{1}$ ( p$_{1}$ ) , P$_{1}$ = p$_{1}$ ) − R$_{1}$ ( p$_{1}$ ) − 1 r =1 Pr D ( j ) r | P$_{2}$$_{j}$ ≤ rq$_{2}$ R$_{1}$ ( p$_{1}$ ) , P$_{1}$ = p$_{1}$ ) (F.20) = Pr ( D ( j ) R$_{1}$ ( p$_{1}$ ) | P$_{2}$$_{j}$ ≤ q$_{2}$, P$_{1}$ = p$_{1}$ ) = 1 , where the inequality in (F.20) follows from (F.19). Proof of item 2. Let p$_{1}$ be arbitrary fixed. Then, E ( ∑ j ∈ I$_{10}$ R$_{j}$/ max( R, 1) | P$_{1}$ = p$_{1}$ ) = j ∈ I$_{10}$ ∩R$_{1}$ ( p$_{1}$ ) R$_{1}$ ( p$_{1}$ ) ∑ r =1 1 r I [ p$_{1}$$_{j}$ ≤ rq$_{1}$ m ] Pr ( P$_{2}$$_{j}$ ≤ rq$_{2}$ R$_{1}$ ( p$_{1}$ ) , C ( j ) r | P$_{1}$ = p$_{1}$ ) ≤ ∑ j ∈ I$_{10}$ ∩R$_{1}$ ( p$_{1}$ ) R$_{1}$ ( p$_{1}$ ) ∑ r =1 1 r Pr ( P$_{2}$$_{j}$ ≤ rq$_{2}$ R$_{1}$ ( p$_{1}$ ) , C ( j ) r | P$_{1}$ = p$_{1}$ ) (F.21) = ∑ j ∈ I$_{10}$ ∩R$_{1}$ ( p$_{1}$ ) R$_{1}$ ( p$_{1}$ ) ∑ r =1 1 r Pr ( P$_{2}$$_{j}$ ≤ rq$_{2}$ R$_{1}$ ( p$_{1}$ ) | P$_{1}$ = p$_{1}$ ) Pr ( C ( j ) r | P$_{2}$$_{j}$ ≤ rq$_{2}$ R$_{1}$ ( p$_{1}$ ) , P$_{1}$ = p$_{1}$ ) ≤ q$_{2}$ R$_{1}$ ( p$_{1}$ ) ∑ j ∈ I$_{10}$ ∩R$_{1}$ ( p$_{1}$ ) R$_{1}$ ( p$_{1}$ ) ∑ r =1 Pr ( C ( j ) r | P$_{2}$$_{j}$ ≤ rq$_{2}$ R$_{1}$ ( p$_{1}$ ) , P$_{1}$ = p$_{1}$ ) ≤ q$_{2}$ R$_{1}$ ( p$_{1}$ ) | I$_{10}$ ∩ R $_{1}$( p$_{1}$ ) | . (F.22) 46