Functional generalizations of Hoeffding’s covariance lemma and a formula for Kendall’s tau

Central Limit Theorems for Coupled Particle Filters

“…) is upper-bounded by a term which converges in probability to a finite constant. This can be verified in an almost identical manner to the approach for (23) in the proof of Lemma C.2; hence we have verified that (31) converges in probability to zero.…”

“…) converges in probability to a finite constant and our proof is concluded by the argument associated to (23).…”

“…any coupling of (η l n , η l−1 n ). To see this, one can construct a functional covariance equality for µ( φ(v 4ξ ⊗ v 8ξ )) (µ is any coupling of (η l n , η l−1 n ) such that µ( φ(v 4ξ ⊗ v 8ξ )) < +∞) as in [23,Theorem 3.1] (that is, in terms of the CDFs of µ, η l n and η l−1 n as in Hoeffding's Lemma) and then when centering with (η l n ⊗ η l−1 n )( φ(v 4ξ ⊗ v 8ξ )) and applying Hoeffding-Fréchet bounds, one observes that ηW n is not optimal. As a result, one cannot transfer between ηW n and ηC n as is done in the proofs of Lemmata F.7-F.8, nor can one use Kantorovich duality.…”

Central limit theorems for coupled particle filters

“…One of the issues of the proof is that the Wasserstein coupling is not the coupling which minimizes the expectation ofφ(v 4ξ ⊗ v 8ξ ) with respect to any coupling of (η l n , η l−1 n ). To see this, one can construct a functional covariance equality for μ(φ(v 4ξ ⊗ v 8ξ )) (μ is any coupling of (η l n , η l−1 n ) such that μ(φ(v 4ξ ⊗ v 8ξ )) < +∞) as in [24,Theorem 3.1] (that is, in terms of the CDFs of μ, η l n , and η l−1 n , as in Hoeffding's lemma), and then when centering with (η l n ⊗ η l−1 n )(φ(v 4ξ ⊗ v 8ξ )) and applying Hoeffding-Fréchet bounds, one observes thatη W n is not optimal. As a result, one cannot transfer betweenη W n andη C n as is done in the proofs of Lemmata F.7-F.8, nor can one use Kantorovich duality.…”

A general treatment of alternative expectation formulae

“…More recently, we noticed attempts to generalize these formulae. Besides a specific moment, Lo (2017) studied alternative formulae for the covariance between transformations of two random variables. Song and Wang (2019b) extended these formulae to more general moment generating functions for both continuous and discrete, as well as univariate and multivariate, cases.…”