Limit theorems for the canonical von Mises statistics with dependent data

Abstract

We study the limit behavior of the canonical (i.e., degenerate) von Mises statistics based on samples from a sequence of weakly dependent stationary observations satisfying the ψ-mixing condition. The corresponding limit distributions are defined by the multiple stochastic integrals of nonrandom functions with respect to the nonorthogonal Hilbert noises generated by Gaussian processes with nonorthogonal increments.Keywords: limit theorems, stochastic integral, multiple stochastic integral, elementary stochastic measure, Gaussian processes, stationary sequences of random variables, mixing, U -and V -statistics Statement of the Main ResultsIn the paper we study the limit behavior of the canonical von Mises statistics based on samples from a sequence of weakly dependent stationary observations satisfying the ψ-mixing condition. The approach uses the representation of such statistics as multiple stochastic integrals with respect to the corresponding normalized empirical product-measure as well as the results of [1]. These results allow us to interpret the corresponding limit random element as a multiple stochastic integral of the kernel of the statistic under consideration with respect to a Gaussian noise which plays a role of the weak limit for the above-mentioned normalized empirical measures.Let {X i ; i ∈ Z} be a stationary sequence of random variables with values in an arbitrary measurable space {X, A } and marginal distribution P . Consider a measurable function f (t 1 , . . . , t d ) : X d → R. Define the von Mises statistic (or V -statistic) by the formula V n := n −d/2 1≤i 1 ,...,i d ≤n f (X i 1 , . . . , X i d ), n = 1, 2, . . . ,where d ≥ 2 and the subscripts i k range independently over the integers from 1 to n, and the function f satisfies the degeneracy condition Ef (t 1 , . . . , t k−1 , X k , t k+1 , . . . , t d ) = 0 (2) for all t 1 , . . . , t d ∈ X and k = 1, . . . , d. The function f is called a kernel of a von Mises statistic, and the statistics with degenerate kernel are called canonical.In the case of independent observations {X i } such statistics were studied in the second half of the last century (see the references and examples of such statistics in [2]). Some limit theorems for these statistics were obtained for the first time by von Mises [3] and Hoeffding [4]. Moreover, in these articles there were introduced the so-called U -statistics: U n := n −d/2or U 0 n := n −d/21≤i 1 <···<i d ≤n f 0 (X i 1 , . . . , X i d ),

Degenerate $$U$$ - and $$V$$ -statistics under ergodicity: asymptotics, bootstrap and applications in statistics

“…Then one can employ empirical process technology to derive the limit distribution that is in this case described as a multiple stochastic integral of the kernel under consideration, with respect to increments of a centered Gaussian process; see e.g. Babbel (1989) and Borisov and Bystrov (2006). Similar techniques have been invoked by Dehling and Taqqu (1991) to obtain the asymptotics of U -statistics under long-range dependence.…”

A note on exponential inequalities for the distribution tails of canonical von Mises’ statistics of dependent observations

“…which is an analogue of Höffding's inequality (4) for the mth power of the normed sum of independent identically distributed centered and bounded random variables, i. e., for an elementary example of canonical V -statistics of order m. In Arcones and Gine, 1993, some inequality close to (6) was proved without condition (5), and relation (7) is given as a consequence. In Gine et al, 2000, some refinement of (7) is obtained for m = 2, and in Adamczak, 2006, the later result was extended to canonical U-statistics of an arbitrary order.…”

Limit distributions of V- and U-statistics in terms of multiple stochastic Wiener-type integrals

“…The construction of this stochastic integral follows the lines of Filippova [8] who considers the special case S = [0, 1] endowed with Q the uniform distribution and G Q may be identified with the traditional standard Brownian bridge. Borisov and Bystrov [3] extend Filippova's [8] stochastic integral approach to Ψ -mixing strictly stationary sequences and completely degenerated kernels. Another kind of extension is given in Denker et al [6], who derive functional limit theorems for k-sample V-and U-statistics.…”

“…Especially, this implies convergence in distribution. 3. Next, let f be an arbitrary function in  F .…”