and therefore Since ∑ l ( o$_{C}$$_{l}$ − e$_{C}$$_{l}$ ) 2 e$_{C}$$_{l}$ or ( o$_{C}$ − e$_{C}$ ) 2 e$_{C}$ is part of the sum that defines T $^{I}$, it follows from equations (A.5) and (A.4) that T I N − m +1 converges uniformly almost surely to a positive constant greater than c $^{′}$. Let | Γ | denote the cardinality of Γ { ( x$_{1}$, y$_{1}$ ) , . . . , ( x$_{N}$, y$_{N}$ ) } . Since I ∈ Γ { ( x$_{1}$, y$_{1}$ ) , . . . , ( x$_{N}$, y$_{N}$ ) } was arbitrary fixed, and since the convergence for fixed I of t$_{C}$/ [ N − ( m − 1)] to a limit bounded from below by a positive constant was uniform, it follows that 1 | Γ | ∑ I∈ Γ T I N − m +1 converges almost surely to a positive constant at least as large as c $^{′}$. To see this, note that from the uniform convergence in equation (A.4), it follows that for an arbitrary fixed ϵ > 0, there exists N ( ϵ ) (which does not depend on C ) such that for all N > N ( ϵ ), 1 N − ( m − 1) ( o$_{C}$ − e$_{C}$ ) 2 e$_{C}$ ≥ c ′ − ϵ for every C , and therefore that 1 | Γ | ∑ I∈ Γ T I N − m +1 ≥ c ′ − ϵ for all N > N ( ϵ ). Since S$_{m}$$_{×}$$_{m}$ ≥ ∑ I∈ $_{Γ}$T $^{I}$, it follows that almost surely We shall show that lim$_{N}$$_{→∞}$ | Γ | / ( N m − 1 ) is bounded below by a positive constant. First, we shall consider the case where m is finite. Then, a subset of Γ { ( x$_{1}$, y$_{1}$ ) , . . . , ( x$_{N}$, y$_{N}$ ) } is the set of all partitions with m − 2 sample points in A$_{1}$ , and one sample point in A$_{2}$ . Therefore, | Γ | ≥ ( N$_{1}$ m − 2 $^{)}$N$_{2}$ . Simple algebraic manipulations lead to the following expression for ( N 1 m − $_{2}$) N$_{2}$ ( N m − $_{1}$) : 45 ( m − 1) N$_{2}$ N N$_{1}$ N − 1 · · · N$_{1}$ − m + 3 N − m + 2 . lim N →∞ S$_{m}$$_{×}$$_{m}$ | Γ | ( N − m + 1) > c $^{′}$. (A.6) ( o$_{C}$ − e$_{C}$ ) 2 e$_{C}$ ≤ ∑ l ( o$_{C}$$_{l}$ − e$_{C}$$_{l}$ ) 2 e$_{C}$$_{l}$ . (A.5)