An Exponential Inequality for U-Statistics Under Mixing Conditions

Abstract

The family of U-statistics plays a fundamental role in statistics. This paper proves a novel exponential inequality for U-statistics under the time series setting. Explicit mixing conditions are given for guaranteeing fast convergence, the bound proves to be analogous to the one under independence, and extension to non-stationary time series is straightforward. The proof relies on a novel decomposition of U-statistics via exploiting the temporal correlatedness structure. Such results are of interest in many fields where high dimensional time series data are present. In particular, applications to high dimensional time series inference are discussed.

Probability inequalities for high dimensional time series under a triangular array framework

“…case have been extensively studied [Hoeffding, 1963;Arcones and Gine, 1993;Giné et al, 2000;Adamczak, 2006]. Assumed {X t,n } t∈Z to be geometrically φ-mixing, Han [2018] established the following theorem that gives an exponential inequality for dependent U-statistics.…”

“…Theorem 2.12. [Theorem 2.1, Han [2018]] Let {X t,n } t∈Z satisfies φ({X t,n } t∈Z ; m) ≤ c n exp(−C n m) for m = 1, 2, . .…”

Exponential inequalities for dependent V-statistics via random Fourier features

“…However, there are few results on non-asymptotic concentration bounds for Vand U-statistics. Exceptions include Borisov and Volodko [2015] and Han [2018], who proved Hoeffding-type inequalities for U-and V-statistics under φ-mixing conditions. There, the results either rely on assumptions difficult to verify, or are limited to nondegenerate ones.…”

Tail behavior of dependent V-statistics and its applications

“…There are few papers in this area. Exceptions include Han [2018] and Borisov and Volodko [2015], who proved Hoeffding-type inequalities for U-and Vstatistics under φ-mixing conditions. There, the results are either limited to nondegenerate ones, or relying on assumptions difficult to verify.…”

“…With the aid of our newly developed exponential inequalities, we show that the penalized Honoré-Powell estimator proposed in Han et al [2017b] is able to recover the optimal s log p/n consistency rate under some explicitly stated mixing conditions on the data-generating scheme. It would not have been possible to establish this result using the Hoeffding-type results in Borisov and Volodko [2015] or Han [2018]. Second, we consider a simultaneous independence test of bivariate time series, where, thanks to the developed Cramér-type moderate deviation, we allow the number of pairs to grow at an arbitrarily fast polynomial rate while still maintaining an asymptotically valid size.…”