^{1}

^{*}

^{2}

In the paper, the deviation of the spline estimator for the unknown probability density is approximated with the Gauss process. It is also found zeros for the infimum of variance of the derivation from the approximating process.

The present work is a continuation of the work [

Let

where

Remind that

where

of Wiener processes.

Denote by

and by

where

In the second section of the work, Theorem 2 and 3 are proven:

and

And it is also stated (Theorem 5) that

It holds the following

Theorem 1. Let

The proof of this statement is easy, therefore we omit it.

Theorems 2 and Theorem 3 will be proved by the mthods given in [

Theorem 2. Let

Then under our assumption a) and b) concerning

Proof. By the main Theorem from [

and for any

Set

Theorem 2 follows now from Theorem 1, relations (2) from [

random variables

Theorem 3. If conditions of Theorem 2 hold and

where

Proof. From the interpolation condition

we have

One can easily note that

in the points of interpolation

where

The relation (5) implies that for arbitrary

It remains to choose

Relations

Theorem 4. First order mean square derivations of the Gauss process

Let now

Theorem 5. 1) The variance of mean square derivations of the Gauss process

2) If the variance vanishes also in intervals

Proof. At the beginning of the proof of the theorem, we proceed as in [

we get for

Substituting into (6)

and taking into account that

or

We find analogously

and also

Generalizing the obtained results, we have

Denote

implies

On the other hand,

where

Obviously,

i.e. at

The first part of Theorem 5 is proved.

Let pass to the proof of the second part. Both in the case of

is valid for

The explicit form of

Note, in this case

One can easily see that

The first part of Theorem 5 is proved.

At last, Theorems 2 and 3 imply that limit distributions of the random variables

coincide. However, the Gauss process

polation points for the spline, and

to investigate the distribution of the maximum of