Now it follows from the arguments that lead from (F.3) to (F.8) that the first term of the sum in (A.1) is bounded by | I$_{0}$ | q$_{1}$/m. Let us now find an upper bound for the second term of the sum in (A.1). For each p$_{1}$, j ∈ R $_{1}$( p$_{1}$ ) ∩ I$_{10}$, r ∈ { 1 , . . . , R$_{1}$ ( p$_{1}$ ) } and l ∈ { 1 , . . . , R$_{1}$ ( p$_{1}$ ) } , let us define: p$_{jrl}$ ( p$_{1}$ ) = Pr ( P$_{2}$$_{j}$ ∈ ( ( l − 1) q$_{2}$ R$_{1}$ ( p$_{1}$ ) ∑ R$_{1}$ ( p$_{1}$ ) s =1 1 /s , lq$_{2}$ R$_{1}$ ( p$_{1}$ ) ∑ R$_{1}$ ( p$_{1}$ ) s =1 1 /s ] , C ( j ) r ∣ P$_{1}$ = p$_{1}$ ) . (G.1) Note that for each p$_{1}$, j ∈ R $_{1}$( p$_{1}$ ) ∩ I$_{10}$ and l ∈ { 1 , . . . , R$_{1}$ ( p$_{1}$ ) } : R$_{1}$ ( p$_{1}$ ) ∑ r =1 p$_{jrl}$ ( p$_{1}$ ) = Pr ( ∪ R$_{1}$ ( p$_{1}$ ) r =1 C ( j ) r , P$_{2}$$_{j}$ ∈ ( ( l − 1) q$_{2}$ R$_{1}$ ( p$_{1}$ ) ∑ R$_{1}$ ( p$_{1}$ ) s =1 1 /s , lq$_{2}$ R$_{1}$ ( p$_{1}$ ) ∑ R$_{1}$ ( p$_{1}$ ) s =1 1 /s ] ∣ P$_{1}$ = p$_{1}$ ) = Pr P$_{2}$$_{j}$ ∈ [ ( l − 1) q$_{2}$ R$_{1}$ ( p$_{1}$ ) ∑ R$_{1}$ ( p$_{1}$ ) s =1 1 /s , lq$_{2}$ R$_{1}$ ( p$_{1}$ ) ∑ R$_{1}$ ( p$_{1}$ ) s =1 1 /s ] ∣ P$_{1}$ = p$_{1}$ ) . (G.2) The equalities follow from the fact that given P$_{1}$ = p$_{1}$, ∪ R$_{1}$ ( p$_{1}$ ) r =1 C ( j ) r is the whole sample space for P ( j ) 2 , represented as a union of disjoint events. In addition, note that for each p$_{1}$, j ∈ R $_{1}$( p$_{1}$ ) ∩ I$_{10}$ and r ∈ { 1 , . . . , R$_{1}$ ( p$_{1}$ ) } , Pr ( P$_{2}$$_{j}$ ≤ rq$_{2}$ R$_{1}$ ( p$_{1}$ ) ∑ R$_{1}$ ( p$_{1}$ ) s =1 1 /s , C ( j ) r | P$_{1}$ = p$_{1}$ ) = r ∑ l =1 p$_{jrl}$ ( p$_{1}$ ) , (G.3) since for j ∈ I$_{10}$, Pr( P$_{2}$$_{j}$ ≤ x ) ≤ x for all x ≥ 0 , in particular Pr( P$_{2}$$_{j}$ = 0) = 0 . 49