MATHEMATICS

V. P. LEONOV

THE USE OF THE CHARACTERISTIC FUNCTIONAL AND SEMI-INVARIANTS IN THE ERGODIC THEORY OF STATIONARY PROCESSES

(Presented by Academician A. N. Kolmogorov, 12 III 1960)

The present work is a continuation of paper (¹) and uses its basic definitions and notation: $\mathfrak S_1$, $\chi_\xi(g)$, $m_\xi^{(k)}(t_1,\ldots,t_k)$, $s_\xi^{(k)}(t_1,\ldots,t_k)$, $F_\xi^{(k)}(\Lambda)$, $S^\infty$, $\Phi^{(\infty)}$, $\Delta^{(\infty)}$ (see the introduction to (¹)). The results are formulated for processes with continuous time; however, all of them, with obvious changes, carry over also to the case of discrete time.

Let $\Omega$ be the space of all real-valued functions $\omega(t)$ of a real variable, $-\infty < t < \infty$; let $\mathfrak S$ be the $\sigma$-algebra of its subsets generated by cylinder sets with a finite-dimensional base; let $\xi(t)$ be a stationary, in the narrow sense, process with continuous time; let $L^p$ be the space of complex $\mathfrak S$-measurable functions $f(\omega)$ such that $M|f(\xi(t))|^p<\infty$, and let

\[ S^\tau \omega(t)=\omega(t+\tau),\qquad U^\tau f(\omega)=f(S^\tau\omega). \]

We shall say that the process $\xi(t)$ has one of the properties: ergodicity, mixing in the wide sense, mixing of degree $r$, if the corresponding translation group of one-to-one measure-preserving point transformations of the measurable space $(\Omega,\mathfrak S)$ has this property (see (²), Chap. IX, §1; for the definition of mixing in the wide sense see (³), Def. 11.2; for the definition of mixing of degree $r$ see (⁴), §1).

Definition 1. We shall call the process $\xi(t)$ a process with mixing of degree $r$ in the wide sense if

\[ \lim_{T\to\infty}\frac1T\int_0^T \left|M\{U^{\alpha_0\tau}f_0(\xi(t))\ldots U^{\alpha_r\tau}f_r(\xi(t))\} - M\{f_0(\xi(t))\}\ldots M\{f_r(\xi(t))\}\right|^2\,d\tau=0 \tag{1} \]

for all $f_0,\ldots,f_r\in L^{r+1}$ and for all real* $\alpha_0,\ldots,\alpha_r$; $\alpha_{i_1}\ne\alpha_{i_2}$ for $i_1\ne i_2$.

It is obvious that mixing of degree 1 coincides with mixing, and mixing of degree 1 in the wide sense coincides with mixing in the wide sense, as defined in (³).

Let $P_\xi(d\omega)$ be the probability measure on $(\Omega,\mathfrak S)$ corresponding to the random process $\xi(t)$.

Definition 2. We shall say that the process $\xi(t)$ belongs to

* In the case of discrete time, for all integer $\alpha_0,\ldots,\alpha_r$; $\alpha_{i_1}\ne\alpha_{i_2}$ for $i_1\ne i_2$.

class \(\overline{S^{(\infty)}}\) \(\left(\xi(t)\in \overline{S^{(\infty)}}\right)\), if:

1) \(\xi(t)\in S^{(\infty)}\);

2) for any random process \(\eta(t)\) such that
\[ m_\eta^{(k)}(t_1,\ldots,t_k)=m_\xi^{(k)}(t_1,\ldots,t_k) \]
for all \(k\geq 1,\ t_1,\ldots,t_k\), the equality \(P_\xi(A)=P_\eta(A)\) holds for all \(A\in\mathfrak S\). Obviously, if \(\xi(t)\in \overline{S^{(\infty)}}\), then \(\xi(t)\) is stationary in the narrow sense.

For \(g\in\mathfrak G_1\), denote
\[ V^\tau g(dt)=g[d(t-\tau)]. \]

Theorem 1. For the ergodicity of \(\xi(t)\) it is necessary and sufficient that the relation
\[ \lim_{T\to\infty}\frac1T\int_0^T \chi_\xi(g_0+V^\tau g_1)\,d\tau =\chi_\xi(g_0)-\chi_\xi(g_1) \quad \text{for all } g_0,\ g_1\in\mathfrak G_1 \tag{2} \]
be fulfilled.

In order that \(\xi(t)\) be a mixing process of degree \(r\) in the broad sense, it is necessary and sufficient that the relation
\[ \lim_{T\to\infty}\frac1T\int_0^T \left|\chi_\xi\left(V^{\alpha_0\tau}g_0+\cdots+V^{\alpha_r\tau}g_r\right) -\chi_\xi(g_0)\cdots\chi_\xi(g_r)\right|^2\,d\tau=0 \tag{3} \]
hold for all \(g_0,\ldots,g_r\in\mathfrak G_1\) and for all real \(\alpha_0,\ldots,\alpha_r;\ \alpha_{i_1}\ne\alpha_{i_2}\) when \(i_1\ne i_2\).*

In order that \(\xi(t)\) be a mixing process of degree \(r\), it is necessary and sufficient that the relation
\[ \lim_{\min_{0\leq i<j\leq r}|\tau_i-\tau_j|\to\infty} \chi_\xi\left(V^{\tau_0}g_0+\cdots+V^{\tau_r}g_r\right) =\chi_\xi(g_0)\cdots\chi_\xi(g_r). \]
\[ \text{for all } g_0,\ldots,g_r\in\mathfrak G_1. \tag{4} \]

Remark 1. The theorem remains valid if, in its formulation, the space \(\mathfrak G_1\) is replaced by the space of generalized measures concentrated at a finite number of points; in this case, in (2), (3), (4) the characteristic functionals become finite-dimensional characteristic functions.

It is well known \({}^{(5-7)}\) that if \(\xi(t)\) is a Gaussian process with covariance function \(B_\xi(\tau)=s_\xi^{(2)}(t,t+\tau)\) and spectral function
\[ F_\xi(\lambda)=F_\xi^{(2)}((-\infty,\lambda)X(-\infty,\lambda)), \]
then, in order that \(\xi(t)\) be a mixing process in the broad sense, it is necessary and sufficient that
\[ \lim_{T\to\infty}\frac1T\int_0^T |B_\xi(\tau)|^2\,d\tau=0, \]
or, equivalently, that \(F_\xi(\lambda)\) be continuous; and, in order that \(\xi(t)\) be a mixing process, it is necessary and sufficient that
\[ \lim_{T\to\infty} B_\xi(\tau)=0. \]

A direct generalization of these facts is given by the following theorems **.

Theorem 2. In order that a process \(\xi(t)\in\overline{S^{(\infty)}}\) be a mixing process of degree \(r\) in the broad sense, it is necessary and sufficient that the relation ***
\[ \lim_{T\to\infty}\frac1T\int_0^T \left|s_\xi^{(k)}(t_{01}+\alpha_0\tau,\ldots,t_{0l_0}+\alpha_0\tau,\ t_{11}+\alpha_1\tau,\ldots,t_{1l_1}+\alpha_1\tau,\ldots\right. \]
\[ \left.\ldots,\ t_{r1}+\alpha_r\tau,\ldots,t_{rl_r}+\alpha_r\tau)\right|^2\,d\tau=0 \tag{5} \]

\[ \text{* See the footnote to Definition 1.} \]

\[ \text{** The statements given above concerning Gaussian processes are contained} \]
in Theorems 2, 3, 5, since in this case \(\xi(t)\in\overline{S^{(\infty)}}\cap\Phi^{(\infty)}\).

\[ \text{*** By virtue of the symmetry of } s_\xi^{(k)}(t_1,\ldots,t_k) \text{, from the fulfillment of relation (5) follows} \]
the fulfillment of the corresponding relations under arbitrary permutations of the arguments of \(s_\xi^{(k)}\).

for all \(k \geqslant 2,\ 0<l_0<k,\ l_i \geqslant 0,\ l_0+l_1+\cdots+l_r=k,\ t_{ij}\), \(0 \leqslant i \leqslant r,\ 1 \leqslant j \leqslant l_i\), real* \(\alpha_0,\ldots,\alpha_r;\ \alpha_{i_1}\ne \alpha_{i_2}\) for \(i_1\ne i_2\).

Denote by \(L_k^l(\nu)\) the hyperplane of the space \(R^k\) of the variables \(\lambda_1,\ldots,\lambda_k\), defined by the relation \(\lambda_1+\cdots+\lambda_l=\nu\).

Theorem 3. In order that the process \(\xi(t)\in \overline{S}^{(\infty)}\cap \Phi^{(\infty)}\) be a process with mixing in the broad sense, it is necessary and sufficient that the relation***
\[ \left|F_\xi^k\right|\bigl(L_k^l(\nu)\bigr)=0 \tag{6} \]
hold for all \(k\geqslant 2,\ 0<l<k,\ -\infty<\nu<\infty\), where \(\left|F_\xi^{(k)}\right|(\Lambda)\) is the total variation of the complex measure \(F_\xi^{(k)}(\Lambda)\).

Theorem 4. In order that the process \(\xi(t)\in \overline{S}^{(\infty)}\cap \Phi^{(\infty)}\) be a process with mixing of all degrees in the broad sense, it is necessary and sufficient that the relation
\[ \left|F_\xi^{(k)}\right|(L)=0 \tag{7} \]
hold for all \(k\geqslant 2\) and for all hyperplanes**** \(L\) of the space \(R^k\), distinct from \(L_k^k(0)\).

Remark 2. Since for \(\xi(t)\in \overline{S}^{(\infty)}\cap \Phi^{(\infty)}\) the measure \(F_\xi^{(k)}(\Lambda)\) is concentrated on \(L_k^k(0)\) (see (1)), Theorem 4 may be formulated as follows:

In order that the process \(\xi(t)\in \overline{S}^{(\infty)}\cap \Phi^{(\infty)}\) be a process with mixing of all degrees in the broad sense, it is necessary and sufficient that relation (7) hold for all \(k\geqslant 2\) and for all hyperplanes \(L\) of the subspace \(L_k^k(0)\).

Theorem 5. In order that the process \(\xi(t)\in \overline{S}^{(\infty)}\) be a process with mixing of degree \(r\), it is necessary and sufficient that the relation*
\[ \lim_{\min_{0\leqslant i<j\leqslant r}|\tau_i-\tau_j|\to\infty} s_\xi^{(k)}(t_{01}+\tau_0,\ldots,t_{0l_0}+\tau_0,t_{11}+\tau_1,\ldots,t_{1l_1}+\tau_1,\ldots \]
\[ \ldots,t_{r1}+\tau_r,\ldots,t_{rl_r}+\tau_r)=0 \tag{8} \]
hold for all \(k\geqslant 2,\ 0<l_0<k,\ l_i\geqslant 0,\ l_0+l_1+\cdots+l_r=k,\ t_{ij}\)*, \(0\leqslant i\leqslant r,\ 1\leqslant j\leqslant l_i\).

Theorem 6. In order that a stochastically continuous process \(\xi(t)\in \overline{S}^{(\infty)}\) be a process with mixing of all degrees, it is necessary and sufficient that the relation
\[ \lim_{\max_{i,j}|t_i-t_j|\to\infty} s_\xi^{(k)}(t_1,\ldots,t_k)=0 \quad \text{for all } k\geqslant 2. \tag{9} \]

From the theorems stated there follow:

Corollary 1. If \(\xi(t)\in \overline{S}^{(\infty)}\cap \Delta^{(\infty)}\), then \(\xi(t)\) is a process with mixing of all degrees.

Corollary 2. For Gaussian processes, mixing of all degrees coincides with mixing, and mixing of all degrees in the broad sense coincides with mixing in the broad sense.

* Including also such sets \(t_{ij}\) in which not all \(t_{ij}\) are distinct.

** See the footnote to Definition 1.

*** By virtue of the symmetry of the measure \(F_\xi^{(k)}(\Lambda)\) (see the introduction to (1)), it follows from relation (6) that the corresponding relations are satisfied for all hyperplanes of the form \(\lambda_{i_1}+\cdots+\lambda_{i_l}=\nu\).

**** In the discrete-time case, for all hyperplanes of the form \(\alpha_1\lambda_1+\cdots+\alpha_k\lambda_k=\nu\) with integer \(\alpha_i\), distinct from \(L_k^k(0)\).

***** See the footnote to Theorem 2.

Let us note that the condition \(\xi(t)\in \overline{S^{(\infty)}}\) contained in the formulations of Theorems 2–6 is necessary if we wish to decide the question of the ergodic properties of the process \(\xi(t)\) on the basis only of its moments (or semi-invariants), namely:

If \(\xi(t)\in S^{(\infty)}\setminus \overline{S^{(\infty)}}\) and is stationary in the narrow sense, then there exists a continuum of nonergodic processes, all of whose moments coincide with the moments of the process \(\xi(t)\).

A proper strengthening of condition (9) leads to the asymptotic normality of the integral

\[ \zeta_T=\int_0^T \xi(t)\,dt, \]

i.e., if we denote

\[ b_T=D\left(\int_0^T \xi(t)\,dt\right) =\int_0^T\int_0^T B_\xi(t_1-t_2)\,dt_1\,dt_2, \]

then the following holds:

Theorem 7. If \(\xi(t)\in S^{(\infty)}\) and

\[ \int_0^T\cdots\int_0^T S_\xi^{(k)}(t_1,\ldots,t_k)\,dt_1\cdots dt_k =o\left(b_T^{k/2}\right) \quad \text{for all } k\ge 3, \tag{10} \]

then

\[ \lim_{T\to\infty} P_\xi\left\{ \frac{\zeta_T-m_\xi^{(1)}T}{\sqrt{b_T}}<y \right\} = \frac{1}{\sqrt{2\pi}}\int_{-\infty}^{y} e^{-z^2/2}\,dz . \tag{11} \]

In essence, Theorem 7 is of interest only in the case when \(b_T\to\infty\) as \(T\to\infty\). In this connection we make the following remarks.

Remark 3. If

\[ \lim_{\lambda\to 0} \frac{F_\xi(\lambda)-F_\xi(-\lambda)}{(2\lambda)^\gamma} =f_\gamma,\qquad 0\le \gamma<2, \]

\[ \lim_{T\to\infty}\frac{b_T}{T^{2-\gamma}} \ge \frac{4}{\pi^2}(2\pi)^\gamma f_\gamma . \]

Remark 4. If

\[ F_\xi(\lambda)=\int_{-\infty}^{\lambda} f_\xi(\nu)\,d\nu,\qquad f_\xi(\lambda)=\lambda^\beta f_1(\lambda),\qquad -1<\beta<1, \]

and \(f_1(\lambda)\) is continuous at the point \(\lambda=0\), then

\[ b_T=d_\beta f_1(0)T^{1-\beta}+o\left(T^{1-\beta}\right), \qquad \text{where }\quad d_\beta=2\int_{-\infty}^{\infty}\frac{1-\cos\nu}{\nu^2-\beta}\,d\nu . \]

In conclusion the author expresses his gratitude to A. N. Kolmogorov, under whose supervision the present work was carried out.

Mathematical Institute named after V. A. Steklov
Academy of Sciences of the USSR

Received
9 III 1960

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