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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: 

<img src='content_image/313556.jpg'>

Note that for each p$_{1}$, j ∈ R $_{1}$( p$_{1}$ ) ∩ I$_{10}$ and l ∈ { 1 , . . . , R$_{1}$ ( p$_{1}$ ) } : 

<img src='content_image/313558.jpg'>

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$_{1}$ ) } , 

since for j ∈ I$_{10}$, Pr( P$_{2}$$_{j}$ ≤ x ) ≤ x for all x ≥ 0 , in particular Pr( P$_{2}$$_{j}$ = 0) = 0 . 

<img src='content_image/313560.jpg'>

