Spectral density estimation for p-adic stationary processes

In this paper, we propose two asymptotically unbiased and consistent estimators for the spectral density of a stationary p-adic process X = . The first estimator is constructed from observations X = Un being the p-adic ball with center 0 and radius and the second, from observations where (Tk)kez is a sequence of random variables taking their values in Qp, associated to a Poisson counting process N.


Introduction
In mathematical physics, we use real and complex numbers, since space time coordinates are well described by means of real numbers.
Recently, to answer many questions in physics, an increasing interest has been given to p-adic numbers: they are used in superstrings theory (using very small distances, of the order of Planck length) where there are no grounds for believing the usual ideas to be valid. P-adic numbers are going to be used, not only in mathematical physics, but also in other scientific grounds, where are met fractals and hierarchical structures (turbulence theory, dynamical physics, biology... )(cf. [13] and [1]). Brillinger, in (2~, was the first to introduce spectral estimation for stationary p-adic processes, , and he constructed the peridogram analogously to the real case. This paper has two focuses developing a consistent estimates of the spectrum of a p-adic stationary process: observed on a p-adic ball and observed at the points process like in [8] and [9]. . First we give some preliminaries about p-adic numbers. where E Z.
.~p is a norm on Q and is called p-adic norm. The completion of Q for that norm is denoted Qp, which called the field of p-adic numbers.
where Fx is the spectral measure of X, and is uniquely determined from c2.
As c2 ~ L 1 ( Q p ) , the spectral density f X is defined by First, we study the periodogram. The studied process is observed for all t belonging to U,~.
Proof. From proposition 3.1, and with the change of variates v = Mn03BB -Mnu, we get:

QP Mn
Since c2 E L1(Qp), fX is continuous; and as Qp W(u)du =1, then by dominated convergence theorem, we get the result.
In the sequel, for a~ E Qp, we denote by b~ the p-adic Dirac delta function, given for all a E Qp by: 3B4x(03BB) = 0 otherwise. and shortly, 03B4(03BB) = 03B40(03BB). Proposition 4.2 Let X = {X(t)}t~Qp be a real variate p-adic stationary second order process, with mean zero, contanuous covariance function c2 element of L1 (Qp), such that hypothesis ~l~) and ~l2) are satisfied, then for all al, ~2 in Qp Therefore X,n (a) is a consistent estimator for fX (03BB).
If 03BB1~03BB2 : Since  This implies mean quadratic convergence. 5 Spectral density estimation of a p-adic stationary process from random sampling 5 We suppose that, for every A element of BQp, the random variable N(A) has a Poisson distribution P (A(A)) (such a process exists by [3,7]), where A(A) = 03B2 (A) and p, is the Haar measure on Qp. In the sequel, we suppose also that also the mean intensity 03B2 = IE{U0} is known. This definition may be written, too: Z(t + dt) = X (t)N(t + dt).
Since X and N are independent, we get The Haar measure being a-finite on Qp, then the measures 03B8N and 03B8Z are also a-finite.
The spectral density f Z associated to the process Z is defined, for A E Qp by: In order to estimate fX, we write from the formula (15): So we have to estimate c2(o) and fz.
This implies mean quadratic convergence.

Discussion and extensions
This paper has been concerned with the case of real-valued process. Extensions to the complex-valued and r-vector valued cases are immediate. We think that, it will be very important to treat the case of p-adic valued process, and afterward, observe the almost sure convergence and give the asymptotic distributions of the estimates.
As the convergence rate of the estimators depends on the sequence (Mn)n~N, we think that the choice of this sequence is crucial, and methods like Cross-Validation procedure's (cf. [11] and [5]), will solve this problem.

Aknowledgements :
We would like to thank professor M. Bertrand, the editor and the anonymous referee for their careful reading of an earlier version. Their comments and criticisms substantially improved the paper.