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UDC 159.21
MATHEMATICS
V. N. SOLEV
ON THE ASYMPTOTICS OF THE PREDICTION ERROR IN THE MULTIDIMENSIONAL CASE
(Presented by Academician Yu. V. Linnik on 8 VII 1968)
- Consider a stationary, in the broad sense, sequence
\(x(n)=(x^1(n),\ldots,x^m(n))\). With this sequence there is naturally associated a sequence of subspaces \(X(n)\), generated by the random variables \(x^1(n),\ldots,x^m(n)\). Denote by \(H_a^b\) the closed, in mean square, linear span of the subspaces \(X(s)\), \(a\leq s\leq b\). The linear space \(H_{-\infty}^{\infty}\) is a Hilbert space with scalar product \((x,y)=Ex\bar y\).
For two subspaces \(H_\varphi, H_\psi\) of the space \(H_{-\infty}^{\infty}\), put
\[ \tau(H_\varphi,H_\psi)=\sum_{k,s}|(\varphi_k,\psi_s)|^2, \]
where \(\{\varphi_k\}\), \(\{\psi_s\}\) are orthonormal bases in the subspaces \(H_\varphi, H_\psi\), respectively.
Let \(\bar H(n)\) be the orthogonal complement in the space \(H_{-\infty}^{\infty}\) to the subspace \(H_{-n}^{-1}\). Put \(\tau_n=\tau(X(0),\bar H(n))\). The quantity \(\tau_n\) can serve as a measure of the accuracy of the linear prediction of vectors \(\xi=(\xi^1,\ldots,\xi^m)\) with coordinates \(\xi^i\in X(0)\) from the past of length \(n\), i.e., from the random variables \(x^i(s)\), \(i=1,\ldots,m;\ s=-1,\ldots,-n\). In particular, if the random variables \(\xi^1,\ldots,\xi^m\) form a basis in \(X(0)\), then there are constants, independent of \(n\), \(0<m\leq M<\infty\) such that \(m\tau_n\leq \sigma_n^2(\xi)\leq M\tau_n\), where \(\sigma_n^2(\xi)=\sum_{i=1}^m\sigma_n^2(\xi^i)\), and \(\sigma_n^2(\xi^i)\) is the square of the error of the linear prediction of the random variable \(\xi^i\) from the past of length \(n\).
If the sequence \(x(n)\) is linearly regular (see (1)), then the quantity \(\tau_\infty>0\). It is clear that \(\delta_n=\tau_n-\tau_\infty\geq 0\) and \(\delta_n\to 0\) as \(n\to\infty\). We shall study the rate of decrease of the quantity \(\delta_n\) to zero depending on the properties of the spectral density (s.d.) \(f(\lambda)\) of a linearly regular sequence \(x(n)=(x^1(n),\ldots,x^m(n))\) of full rank. In the one-dimensional case an analogous problem was considered by I. A. Ibragimov ((2), see also (3,4)).
Theorem 1. In order that, as \(n\to\infty\),
\[ \delta_n=O(n^{-2p}),\qquad p>1/2,\quad p\ \text{nonintegral}, \tag{1} \]
it is necessary and sufficient that the s.d. \(f(\lambda)\) almost everywhere coincide with a continuous matrix-function having a strictly positive determinant, all elements of this matrix having absolutely continuous derivatives \(f_{ij}^{(r-1)}(\lambda)\), \(r=[p]\), the \(r\)-th derivatives \(f_{ij}^{(r)}(\lambda)\in L_2(-\pi,\pi)\) and satisfying there a Hölder condition of order \(\alpha=p-[p]\); the latter means that
\[ \sup_{t\leq h}\left(\int_{-\pi}^{\pi}|f_{ij}^{(r)}(\lambda)-f_{ij}^{(r)}(\lambda+t)|^2\,d\lambda\right)^{1/2} =O(h^\alpha). \tag{2} \]
- The proof of Theorem 1 is based on the following lemma.
Lemma 1. If condition (1) is satisfied, the collection \(x^i(s)\), \(i=1,\ldots,m;\ s=0,\pm1,\ldots\), forms a Riesz basis in the space \(H_{-\infty}\).
We first derive Theorem 1 from Lemma 1. Let condition (1) be satisfied. Denote by \(\Psi^i(s)\), \(i=1,\ldots,m;\ s=-1,\ldots,-k,\ldots\), the system conjugate to the system \(x^i(s)\), \(i=1,\ldots,m;\ s=-1,\ldots,-k,\ldots\), in the space \(H_{-\infty}^{-1}\). Let \(\widetilde H(n)\) be the orthogonal complement to the space \(H_{-n}^{-1}\) in the space \(H_{-\infty}^{-1}\). Obviously,
\[ \delta_n=\tau(X(0),\widetilde H(n)). \]
Since, by Lemma 1, the collection \(x^i(s)\), \(i=1,\ldots,m;\ s=-1,\ldots,-k,\ldots\), forms a Riesz basis in the space \(H_{-\infty}^{-1}\), the collection \(\Psi^i(s)\), \(i=1,\ldots,m;\ s=-1,\ldots,-k,\ldots\), forms a Riesz basis in the space \(H_{-\infty}^{-1}\). Consequently,
\[ \delta_n \asymp \sum_{-s=n+1}^{\infty}\sum_{j=1}^{m}\sum_{i=1}^{m} \left|(x^i(0),\Psi^j(s))\right|^2, \tag{3} \]
where the symbol \(a_n\asymp b_n\) means that
\[ 0<\varliminf_n \frac{a_n}{b_n}\leq \varlimsup_n \frac{a_n}{b_n}<\infty . \]
It also follows from Lemma 1 (see (5)) that there exists an everywhere defined, bounded, invertible, positive Hermitian operator \(B\) in \(H_{-\infty}^{-1}\) such that
\[ x^i(s)=B\Psi^i(s),\qquad i=1,\ldots,m;\ s=-1,-2,\ldots \tag{4} \]
From (3) and (4) it follows that
\[ \delta_n \asymp \sum_{-s=n+1}^{\infty}\sum_{j=1}^{m}\sum_{i=1}^{m} \left|(x^i(0),x^j(s))\right|^2 . \]
Hence, using the results of approximation theory (see (6), Ch. V), one can show that the matrix \(f(\lambda)\) coincides almost everywhere with a continuous matrix whose elements have derivatives of the order indicated in Theorem 1, and that condition (2) is satisfied. In particular, the trace of the mentioned continuous matrix is a bounded function and, consequently, its determinant is strictly positive (see Lemma 1).
The proof of sufficiency is carried out analogously.
- In the proof of Lemma 1 some results from the theory of orthogonal matrix polynomials are used (see also (7)).
Consider the space \(L_2^m(f)\) of matrices \(S(\lambda)\) of size \(m\times n\) such that
\[ \|S\|_f^2=\operatorname{sp}\frac{1}{2\pi}\int_{-\pi}^{\pi} S(\lambda)f(\lambda)S^*(\lambda)\,d\lambda<\infty . \]
Put
\[ (S,P)_f=\frac{1}{2\pi}\int_{-\pi}^{\pi}SfP^*\,d\lambda . \]
The condition
\[ (\varphi_n,\varphi_m)_f= \begin{cases} 0, & m\ne n,\\ E, & m=n, \end{cases} \tag{5} \]
uniquely determines a sequence of polynomials orthogonal in the sense of (5),
\(\varphi_n(z)=k_n z^n+\cdots\), where \(k_n\) is a positive Hermitian matrix.
Lemma 2. All zeros of the function \(\operatorname{Det}\varphi_n(z)\) lie inside the unit circle.
Put
\[ s_n(x,z)=\sum_0^n \varphi_n^*(x)\varphi_n(z). \]
Lemma 3. Let \(g(z)\) be an arbitrary polynomial of degree \(n\). Then
\[ \bigl(g(z),s_n(x,z)\bigr)_f=g(x),\qquad z=e^{i\lambda}. \]
Lemma 4. The identity
\[ s_n(0,z)=\sum_0^n \varphi_n^*(0)\varphi_n(z)=k_n z^n \varphi_n^*(z) \]
holds.
In particular,
\[ s_n(0,0)=\sum_0^n \varphi_n^*(0)\varphi_n(0)=k_n^2. \]
Lemma 5. The recurrence formula
\[ k_n\varphi_{n+1}(z)=k_{n+1}z\varphi_n(z)+\varphi_{n+1}(0)z^n\varphi_n^*(z) \]
holds.
Lemma 6. Let \(g(z)\) be a polynomial of degree \(n\) with coefficient \(E\) at the leading term. Then
\[ \operatorname{Det}(g,g)_f \geqslant \operatorname{Det} k_n^{-2}. \]
The polynomial \(k_n^{-1}\varphi_n(z)\) minimizes the quantity \(\operatorname{Det}(g,g)_f\) among polynomials of degree \(n\) with coefficient \(E\) at the leading term.
The following Lemmas 7 and 8 are proved almost in the same way as the corresponding assertions in the one-dimensional case \(\left({}^{2}\right)\).
Lemma 7. Under condition (1),
\[ \sum |\varphi_k(0)|^2<\infty . \]
Lemma 8. Under condition (1), there exist constants independent of \(n\), \(0<m\leq M<\infty\), such that
\[ mE\leqslant s_n(0,z)s_n^*(0,z)\leqslant ME,\qquad z=e^{i\lambda}. \]
Denote by \(G(z)\) the maximal analytic matrix inside the unit disk (see \(\left({}^{1}\right)\)) such that
\[ f(\lambda)=G(e^{i\lambda})G^*(e^{i\lambda}). \]
It is easy to show, using Lemmas 3 and 7, that under condition (1) the sequence \(s_n(0,z)\) converges on the unit circle to the matrix \([G^{-1}(0)]^*G^{-1}(z)\) in the uniform metric. Hence, and from Lemma 8, follows the existence of constants \(0<m\leq M<\infty\) such that the relation \(mE\leq f(\lambda)\leq ME\) is satisfied for almost all \(\lambda\in[-\pi,\pi]\). The obtained relation is equivalent to the assertion of Lemma 1.
- Consider the case when \(\delta_n=O(e^{-cn})\), \(c>0\). The following two theorems are proved by the same method as Theorem 1 was proved (cf. Theorems 4 and 5 of \(\left({}^{2}\right)\)).
Theorem 2. The relation
\[ \delta_n=O(e^{-cn}),\qquad c>0, \]
holds if and only if the s.p. \(f(\lambda)\) almost everywhere coincides with a matrix admitting analytic continuation into the strip of values of the complex argument \(z=\lambda+i\mu\) of width \(c\) and having a strictly positive determinant.
Theorem 3. The relation
\[ \delta_n = O(e^{-cn}) \]
holds for all \(c > 0\) if and only if the analytic continuation of the matrix mentioned in Theorem 2 is an entire function and its determinant is strictly positive.
Leningrad Branch
of the V. A. Steklov Mathematical Institute
Academy of Sciences of the USSR
Received
18 III 1968
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