Given these n eigenvalue estimates ˆ λ$_{1}$, ..., ˆ λ$_{n}$ , one could then simulate from ∑ n k =1 ˆ λ$_{k}$ ( Z 2 k − 1) to approximate ∑ ∞ k =1 λ$_{k}$ ( Z 2 k − 1). While this seems to be a plausible approach, a formal study establishing regularity conditions under which this procedure is valid is beyond the scope of this paper. We note that it also does not fall within the scope of results in Gretton et al. (2009) since their kernel does not depend on estimated nuisance parameters. We refer the reader to Franz (2006) for possible sufficient conditions under which this approach may be valid. In practice, it suffices to give a data-dependent asymptotic upper bound on q$_{1}$$_{−}$$_{α}$ . We will refer to ˆ q ub 1 − $_{α}$, which depends on P$_{n}$ , as an asymptotic upper bound of q$_{1}$$_{−}$$_{α}$ if If q$_{1}$$_{−}$$_{α}$ is consistently estimated, one possible choice of ˆ q ub 1 − α is this estimate of q$_{1}$$_{−}$$_{α}$ – the inequality above would also become an equality provided the conclusion of Theorem 4 holds. It is easy to derive a data-dependent upper bound with this property using Chebyshev’s inequality. To do so, we first note that where we have interchanged the variance operation and the limit using the L 2 martingale convergence theorem and the last equality holds because λ$_{k}$ , k = 1 , 2 , . . . , are the eigenvalues of the Hilbert-Schmidt integral operator with kernel ˜ Γ$_{0}$. Under mild regularity conditions, P 2 $_{0}$Γ 2 0 can be consistently estimated using$_{U}$$_{n}$Γ 2 $_{n}$. Provided P 2 $_{0}$Γ 2 0 > 0, we find that where the limit variate has mean zero and unit variance. The following theorem gives a valid choice of ˆ q ub 0 . $_{95}$. 18 $^{(}$2$_{U}$$_{n}$ Γ 2 n ) − 1 / 2 nψ$_{n}$ ⇝ $^{(}$2 P 2 $_{0}$Γ 2 0 ) − 1 / 2 ∞ ∑ k =1 λ$_{k}$ $^{(}$Z 2 k − 1 ) , (7) V ar$_{P}$$_{0}$ [ ∞ ∑ k =1 λ$_{k}$ $^{(}$Z 2 k − 1 ) ] = ∞ ∑ k =1 λ 2 $_{k}$V ar$_{P}$$_{0}$ $^{(}$Z 2 k ) = 2 ∞ ∑ k =1 λ 2 k = 2 P 2 $_{0}$Γ 2 0 , lim sup n →∞ P n 0 $^{(}$nψ$_{n}$ > ˆ q ub 1 − α ) ≤ 1 − α . (6)