Hlbewt Space Valued Traces M " 4 D Multiple Seratonovich Integrals . - with Statistical Applications

Multiple Stratonovich integrals (MSI) with respect to the Wiener process and the Brownian bridge are defined for a class of kernels having k-th order t-traces which are, in general different from the traces investigated in earlier work. Asymptotic distributions of V-statistics are derived and the limiting distribution expressed in terms of appropriate MSI. Another application yields an alternative proof of Filippova's theorem on the limiting distribution of von Mises statistical functions. 1. Introdnction. The study of Hilbert space valued traces and their connection with multiple Stratonovich integrals (MSI) originated, at least to our knowledge, in a paper of Hu and Meyer [43 in which a new approach to Feynman integrals was presented. In making this approach rigorous, Johnson and Kallianpur [7] introduced several different definitions of traces of which the limiting trace turned out to be the most appropriate one for the proof of the formulae in [4]. The MSI of [7] (the term "Stratonovich integral" was not used in the paper) were defined by using the ideas of lifting. While interesting from the point of view of furnishing formulae for certain types of Feynman integrals, these Stratonovich integrals do not meet the requirements of statistical applications since they are based essentially on Hilbert space techniques and do not take into account the values on the diagonals. In the present paper, we take a fresh look at the problem. The z-traces introduced in Section 2 are defined for a subclass (denoted by 9;) of the L2-space of p-th order symmetric kernels. Yi is made into a Hilbert space under a new inner product in such a manner that each of the k-th order z-traces (k = 0, 1, . . . , [ p / 2 ] ) is a continuous map from Yb to L2 [0, l]p-2k. * Research partially supported by the National Science Foundation and the Air Force Office of Scientific Research Grant No. 91-0030 and the Army Research Oflice Grant NO. DAAM3-92-G-008. 128 A. Budhiraja and G. Kall ianpur Results relating MSI to multiple Wiener integrals similar to those obtained in [77) are derived in Section 2. In Section 3, MSI are defined with respect to the Brownian bridge and a Hu-Meyer type formula is proved. The latter result is used in connecting the asymptotic distribution of a U-statistic with that of a V-statistic. Sections 4 and 5 are devoted to statistical applications of our results. In Theorem 4.3 the limiting distribution of a V-statistic is derived in terms of MSI. It is natural that Stratonovich integrals are involved since a V-statistic (in contrast to a U-statistic) allows repeated indices (see Definition 4.2). Hoeffding's pioneering 1948 result [3] is mentioned as a corollary to Theorem 4.3. An application to the asymptotic distribution of von Mises differentiable statistical functionals is made in Section 5. An alternative proof of Filippova's result is given in Theorem 5.3 vribere the limit is obtained as an MSI with respect to the Wiener process which is shown to be equivalent to the MSI with respect to the Brownian bridge obtained in [2], 2. Hilbert space valued traces and multiple stochastic integrals. In this section we will introduce the multiple Stratonovich integral. This integral, in general, is different from that considered by Johnson and Kallianpur [7], though the two integrals agree for step functions. The Stratonovich integral is closely tied to certain Hilbert space valued traces. In this section we will introduce these traces and discuss their connection with the "limiting traces" of Johnson and Kallianpur. DEFINITION 2.1 (the class 9, of step functions). A real valued symmetric function f, on COY 11, is in the class 9, of step functions iff there exists a partition {0 = ti < .. . < tm < t,,,+l = 1) of [O, 11 and constants {ai ,,..., ip; i , , . . ., i , = 0, 1, 2, .. . , rn) such that where Ail = ( t i j , t i ,+ l ] if 1 < ij < rn and A,, = (0) if i j = 0, j = 1 , 2 , . . . , p. Let (a, 9, P) be a probability space and (W,; 0 < t < 1) be a Wiener process on this space. We first define the multiple Stratonovich integral with respect to the Wiener process for integrands in Yp. This integral turns out to be the same as the multiple Stratonovich integral of Johnson and Kallianpur which will be denoted by 6; ( ) . DEFINITION 2.2 (multiple Stratonovich integral for functions in YP). Let f, E Yp be given by (2.1). Define the multiple Stratonovich integral (MSI) of f, as We will now briefly recall the limiting traces as introduced in M and then give a representation for Sp(fp) .in terms of those traces. Hilbert space valued traces , 129 DEFINITION 2.3 (limiting traces), Let f, be a symmetric function in L2 [O, 1]P. Fix k, 1 < k < b/2]. Suppose that for every complete orthonormal system (CONS) (#,I for L~ 10, 11 4 i I @ & i l n 4 i k @ 4 i k @ # i z k + l m $ z p > & i m + l . 4ip converges in L2 [0, 1IP-" to a limit which is independent of the choice of the CONS (#J. Then we say that the k-th limiting trace for f, exists, which, by definition is the limit of the series in (2.3) and is denoted by 3%. S0fp is defined to be the same as f,. The following proposition relates the MSI 6, with the multiple integral of Johnson and Kallianpur, and therefore with multiple Wiener integrals through the Hu-Meyer formula. PROP~S~TIUN 2.4. Let f, E YP. Then ??' fp exists for all k3 0 8 k Q [ p / 2 ] , and we h ~ v e Furthermore, (2.5) E C6, tfP)l2 G C { 1 f %.(ti , .. . , t,)dt, . . . dt, E0.13P