In a continuous time nonlinear regression model the residual correlogram is considered as an estimator of the stationary Gaussian random noise covariance function. For this estimator the functional central limit theorem is proved in the space of continuous functions. The result obtained shows that the limiting sample continuous Gaussian random process coincides with the limiting process in the central limit theorem for standard correlogram of the random noise in the specified regression model.

Estimation of the signal parameters in the “signal+noise” observation model is a classic problem of statistics of stochastic processes. If the signal (regression function) nonlinearly depends on parameters, then this is a problem of nonlinear time-series regression analysis. Another problems arise when there is a need to estimate the functional characteristics of the correlated random noise in the given functional regression model. For the stationary noise it can be estimation of the noise spectral density or covariance function. Asymptotic properties of the Whittle and Ibragimov estimators of spectral density parameters in the continuous time nonlinear regression model were considered in Ivanov and Prykhod’ko [

However, unlike the residual sum of squares and usual correlogram, the results on the residual correlogram are not sufficiently represented in statistical literature except for a few theorems for discrete time linear regression with stationary correlated observation errors (see Anderson [

In this paper we prove the functional central limit theorem (CLT) in the space of continuous functions for the normed residual correlogram as an estimator of the stationary Gaussian random noise covariance function in continuous time nonlinear regression model. The first result of such a kind has been obtained in Ivanov and Moskvychova [

Suppose the observations are of the form

The assumption about domain

Obviously, if

LSE of unknown parameter

The existence of at least one such a vector follows from the Pfanzagl results [

As an estimator of

From the condition

Consider the normalized residual correlogram

We will consider the processes

Let

A family

Since

We assume that the process

Introduce the function (see section 6.4 of the chapter 6 in Buldygin and Kozachenko [

If

Below we are going to formulate a theorem obtained in Buldygin and Kozachenko [

In particular, for any

As it is shown in the Remark 6.4.1 in [

Taking into account the Theorem

Thus, to obtain a functional theorem in

To prove (

Assume that for any

Introduce now the normalized LSE

Instead of the words “for all sufficiently large

We have

For basic observation model (

Condition

The measure

Taking into account (

Sufficient conditions of the assumption

Consider the diagonal elements (measures)

A function

Consider some sufficient conditions on

For

By Lebesgue monotonic convergence theorem from

Under conditions

On the other hand,

Put

In this section, we formulate and prove the CLT for the normalized residual correlogram

In view of the Theorem

Obviously, by conditions

We will use the notation

Apply the Taylor formula to the integral

Subject

Thus all finite-dimensional distributions of the stationary Gaussian processes

Since by condition

According to the Kolmogorov theorem (see, for example, Gikhman and Skorokhod [

Let

Note that

From the inequalities (

We write

Consider sample continuous Gaussian processes

For

Besides, under conditions

We have proved that

Note further that similarly to (

Theorem

In the proofs of the sections 3 and 4 the condition

In this section, we consider the example of trigonometric regression function

To apply the results obtained in the paper to the function (

The LSE in the Walker sense of unknown parameter (

The relations (

Due to the smoothness of function (

To check the fulfillment of the condition

In the conditions

Check condition

Using formula (

Passing to condition

Since

As to the condition

To obtain such a result, it was first proved in [

It remains to check the last condition

First of all for

Using formulas (