ON THE DENSITY OF SOME WIENER FUNCTIONALS : AN APPLICATION OF MALLIAVIN CALCULUS

ON THE DENSITY OF SOME WIENER FUNCTIONALS: AN APPLICATION OF MALLIAVIN CALCULUS


Introduction
Let F be a probability distribution function on R, with characteristic function c(t) := E(exp{itX}), X being a real random variable with distribution function F.
Let X1 , X2, . . .be a sequence of independent copies of X defined on some probability space (S2, .P, P), and F n the empirical distribution function of the first n variables .
The empirical characteristic function c,(t) is the (random) characteristic function of F, and has been used in several statistical applications since at least Cramer's famous book.
In the late 70's a systematic study of its properties and statistical applications was initiated by Feuerverger and Mureika .In particular they proved the first limit theorems for the empirical characteristic process Y,~(t) := ~,l,-n,(c (t)-c(t)), under strong morrient conditions on F, see [2] .
In 1981 M. Marcus, [3], found necessary and sufficient conditions for the processes {X,,(t)},°,°__i, t E to weakly converge to a limit process of the following form where B is Brownian bridge on [0,1], as C[T1,T2] valued random elements.
From this strong approximations S. Csórgó derived rates .,of convergence for several functiónals of the paths of the empirical characteristic procese, under the further assumption of existente and boundedness of a density for the limit functional.
Here we want to study the following one which is useful in testing for symmetry of F., namely where 0-V is a given distribution function with support in [T1 ,T2], and -oo < T I < T2 < +oo .
The main result in the present work is that with great generality, T has a density which in fact satisfies much stronger boundedness conditions than those needed for Csórgó's rates of convergente to hold true.It is the following Theorem .Assume that the random variable T, as defined abone, is non degenerate (this is a condition on both, F and Hl) .Then T has a smooth density, which is either a tempered function or an exponentially decaying one (at infinity) .
The main ingrediente of the proof are Malliavin calculus, Cauchy's formula and Fourier transform .In the next paragraph we recall, very briefly, the principal definitions and results we are going to use, and give some referentes where complete and detailed expositions can be found.Finally, in paragraph, 3 we give the proof of the theorem.

The tools
Let L2 [0, 1] be the Hilbert space of square integrable functions with respect to the Lebesgue measure on the Borel u-field, C3, ir1 the unit interval [0,1] .
Around 1950 K. Ito showed that each square integrable functional F E P) can be developed in the form where I,,(fj are multiple Ito-Wiener integrals of the (deterministic) functions f, E L2([0, 1]m, 13'n , A'n ) .In particular II (fI) = W(f1) .This expansion is sometimes known as the Wiener chaos decomposition of F. Stroock's formula identifies the kernels, f, in terms of iterated Malliavin derivatives of F: For F E L2 (52, .F, P) and h E L2 [0,1], the Malliavin derivative of F in the h direction, DjLF, can be defined as if the series converges in L2 (S2, .F, P), where (, ) is inner product .All this (and much more) can be found in [4] and [5] .
To apply this theory to our functional T, we write it down in terms of as follows : Now we recall some more definitions and results that will also be needed in the next paragraph .
A function f on R is called a tempered function if it is a smooth function and for each N E Nl (the set of natural numbers) sup sup(1+x2)N jf (p)(x)1 < +oo p<N xER where f (p) is the p-th derivative of f .
Let S denote the space of tempered functions on R .The Fourier transform of a function f E S is defined as f(t) :_ (27r)-z ~ exp(-itx)f(x)dx 00 It turns out that this transformation defines a continuous one-to-one mapping of S onto S, whose inverse is also continuous .Moreover, it has period 4, due to the inversion formula which is fundamental A very good referente for this is [6] .

Proof of the theorem
The idea of the proof is quite simple : to show that the characteristic function of T is a tempered function in the properly infinite case; and direct calculation in the finite case (distinetion of the two cases will be clear in a moment) .
Let us go back to Stroock's representation of T; and recall the explicit form of the kernel in the' double Ito-Wiener integral there .
It is known that such a kernel can be developed in a L2 -convergent series .
It is also well known that for h E L2 [0, 1], with 1111112 = 1 and combination of this facts leads us to the representation we had in mind, namely with F-00, '\i < +00 .
We remark here that from this representation it is clear that T is non-degenerate iff Stroock's kernel, D2 T, is non-degenerate .
To proceed we distinguish two cases : the one where only a finite nurriber of the Ai are non zero, and the one where there are infinitely many non zero.We treat only the second case, as the first one is very elementary.So, to finish the proof of the theorem it is enough to prove next lemma, Lemma.Let {%i} .°°,be independent identically distributed random variables with a common chi-square distribution, ¡.e .a distribution with densityx -z -exp(-2)110,-1(r,) .where i := V/---1.Observe that for z in this strip Z < Re(1 -2¡,\z) < 2 , so that we can use the branch of y/'z -which coincides with the usual square root function on the positive real numbers ., and we can apply Theorem (15 .6)from [7] ; (here we use the hypothesis -°°1 ~i < + oo) .For t > 0 apply Cauchy's formula to the rectangle F defined by the points (t/2) T-i(4A1) -1 , 2t :F i(4A1) -' ; to get j=1 rj where {Fj}~_1 are the four sides of the rectangle F.
To bound this integrals we use the fact that for each natural number MEN For instante the integral on the left side, let us say FI , of the rectangle F, can be bounded as follows where we used (1) ; and the same type of bound is obtained for the integral on the right side, 1'3, of F. In a similar way the bound in ( 2) is used to treat the integrals on the horitzontal sides of the rectangle, F2 and I'4.And this proves the lemma in the case t > 0. For t < 0, with small changos, all works the same way, and for t = 0 there is no problem at all .Thus the lemma is proved .
Remark .In the above proof we could have used the asymptotic representation obtained in [8], instead of our lemma.It is clear from [8] that the density f(x) is exporientially decaying at infinity, even in the properly infinito case.However, it is not clear how can we get the temperate character of f from Zolotarev's representation, without further work.In that point our proof seems to be shorter and clearer .