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MATHEMATICS
O. V. SARMANOV
PROPER CORRELATION FUNCTIONS AND THEIR APPLICATIONS IN THE THEORY OF STATIONARY MARKOV PROCESSES
(Presented by Academician A. N. Kolmogorov on 13 II 1960)
- Consider a stationary real process with continuous parameter \(\{x_t;\ -\infty < t < \infty\}\) and suppose that the random variables \(x_t\) have means equal to zero and variances equal to one. Denote \(x_{t_1}\) by \(x\), \(x_{t_1+t}\) by \(y\); by stationarity, \(x\) and \(y\) have identical a priori distributions.
Definition 1. \(r_1(t)\) is called the maximal correlation function of the process \(x_t\), if for \(t \ne 0\)
\[ |r_1(t)|=\sup_{f,g}|\mathbf{M}(f(x)g(y))| \tag{1} \]
in the class of functions \(f, g\) satisfying the conditions
\[ \mathbf{M}f(x)=\mathbf{M}g(y)=0;\qquad \mathbf{M}f^2(x)=\mathbf{M}g^2(y)=1, \tag{2} \]
where \(\mathbf{M}\) denotes mathematical expectation. (For \(t=0\), \(r_1(t)\) is completed by the condition \(r_1(0)=1\).)
Remark. If the functions \(f_1\) and \(g_1\) at which the upper bound (1) is attained are linear, then \(r_1(t)\) coincides with the ordinary correlation function \(r(t)\), where
\[ r(t)=\mathbf{M}(xy),\qquad t\ne 0. \tag{3} \]
The properties of \(r_1(t)\) are analogous to the properties of the maximal correlation coefficient \((^{1,2})\).
- We now considerably narrow the class of processes under consideration, assuming that the law of the joint distribution of \(x\) and \(y\) has density \(p(t,x,y)\), that this density is symmetric,
\[ p(t,x,y)=p(t,y,x) \tag{4} \]
in the region \(\Omega=[a\le x;\ y\le b]\) and for all \(t\ne 0\) satisfies the boundedness condition
\[ \iint_{(\Omega)} \frac{p^2(t,x,y)}{p(x)p(y)}\,dx\,dy<\infty, \tag{5} \]
where
\[ p(x)=\int_a^b p(t,x,y)\,dy,\qquad p(y)=\int_a^b p(t,x,y)\,dx. \tag{6} \]
By virtue of (4) and (5), the upper bound (1) is attained on identical functions, i.e. \(g_1(y)=f_1(y)\).
Definition 2. The sequence of eigenvalues \(r_k(t)\), \(k=1,2,\ldots\), of the kernel
\[ \frac{p(t,x,y)}{\sqrt{p(x)p(y)}} \tag{7} \]
is called the sequence of intrinsic correlation functions of the process \(x_{t_1}\), \(t \ne 0\).
Definition 3. If the eigenfunctions of the kernel (7) \(\{\varphi_k(x), \varphi_k(y)\}\) do not depend on \(t\), then the process is called maximally stationary.
Definition 4. The process \(x_{t_1}\) is called \(C\)-continuous if all \(r_k(t)\) are continuous; in particular,
\[ \lim_{t\to 0} r_k(t) = 1,\qquad k=1,2,\ldots \tag{8} \]
3. Suppose that \(p(t,x,y)\), in addition to the symmetry condition (4) and the constraint (5), satisfies the Markov equation
\[ p(t_1+t_2,x,y)=\int_a^b \frac{p(t_1,x,z)\,p(t_2,z,y)}{p(z)}\,dz; \tag{9} \]
then this density completely determines a stationary Markov process.
A consequence of the definitions introduced and equation (9) is
Theorem 1. In order that a symmetric two-dimensional density \(p(t,x,y)\), satisfying the constraint (5), be able to define a continuous Markov process, it is necessary and sufficient that the following conditions be fulfilled:
a) the eigenfunctions of the kernel (7) do not depend on \(t\), i.e. the process is maximally stationary;
b) the intrinsic correlation functions have the form (for \(k=1,2,\ldots\))
\[ r_k(t)=e^{-\lambda_k t},\qquad t\ge 0,\qquad 0<\lambda_1\le \lambda_2\le \cdots . \tag{10} \]
4. Suppose now that the limits exist
\[ \begin{aligned} A(x)&=\lim_{t\to 0}\frac{1}{t}\int_a^b (y-x)\frac{p(t,x,y)}{p(x)}\,dy,\\ B(x)&=\lim_{t\to 0}\frac{1}{t}\int_a^b (y-x)^2\frac{p(t,x,y)}{p(x)}\,dy \end{aligned} \tag{11} \]
and that the transition probability density
\[ f(t,x,y)=\frac{p(t,x,y)}{p(x)} \tag{12} \]
satisfies the well-known equations of A. N. Kolmogorov \((^3)\)
\[ \frac{\partial f}{\partial t} = A(x)\frac{\partial f}{\partial x} +\frac{1}{2}B(x)\frac{\partial^2 f}{\partial x^2}, \tag{I} \]
\[ \frac{\partial f}{\partial t} = -\frac{\partial}{\partial y}[A(y)f] +\frac{1}{2}\frac{\partial^2}{\partial y^2}[B(y)f]. \tag{II} \]
On the other hand, according to Theorem 1, for \(f(t,x,y)\) one obtains the bilinear expansion
\[ f(t,x,y)=p(y)\left[1+\sum_{k=1}^{\infty}\varphi_k(x)\varphi_k(y)e^{-\lambda_k t}\right]. \tag{13} \]
Substituting (13) into (I) and (II) and using the orthogonality of \(\varphi_k(x)\) with weight \(p(x)\), we obtain the ordinary differential equations
\[ \frac{1}{2}B(x)\varphi_k''(x)+A(x)\varphi_k'(x)+\lambda_k\varphi_k(x)=0,\qquad k=1,2,\ldots; \tag{14} \]
\[ B(x)p'(x)+[B'(x)-2A(x)]p(x)=0. \tag{15} \]
From (15) we find an explicit expression for \(p(x)\):
\[ p(x)=\frac{\gamma}{B(x)}\exp\left[\int \frac{2A(x)}{B(x)}\,dx\right], \tag{16} \]
where \(\gamma\) is a normalizing constant. (In particular, the process can be Gaussian only if \(A(x)\) is linear and \(B(x)=\mathrm{const}\).)
- Suppose, for example, that \(\varphi_k(x)\) is a polynomial of degree \(k\), \(k=1,2,\ldots\), and that the domain \(\Omega\) is infinite. In this case, using (13) and the orthogonality of the eigenfunctions, we immediately obtain
\[ \begin{aligned} A(x)&=-\lambda_1 x,\\ B(x)&=(2\lambda_1-\lambda_2)x^2+c(\lambda_2-\lambda_1)x+\lambda_2, \end{aligned} \tag{17} \]
where
\[ c=Mx^3 \tag{18} \]
is the coefficient of asymmetry of the a priori distribution of \(x_{t_1}\) (since \(Mx_{t_1}=0\), and \(Mx_{t_1}^2=1\)).
Since \(B(x)\geq 0\) on an infinite domain, the condition
\[ \lambda_2\leq 2\lambda_1 \tag{19} \]
is necessary.
On the other hand, substituting into (14), instead of \(\varphi_k(x)\), a polynomial of degree \(k\) with undetermined coefficients, and instead of \(A(x)\) and \(B(x)\) their expressions (17), we find
\[ \lambda_k=k\left[\frac{k-1}{2}\lambda_2-(k-2)\lambda_1\right]. \tag{20} \]
Expression (20), when \(\lambda_2<2\lambda_1\), becomes negative for sufficiently large \(k\); consequently, in inequality (19) only the equality sign is possible, and then in general
\[ \lambda_k=k\lambda_1,\qquad k=1,2,\ldots, \tag{21} \]
and equation (14) is reduced to the form
\[ \left(\frac{c}{2}x+1\right)\varphi_k''(x)-x\varphi_k'(x)+k\varphi_k(x)=0,\qquad k=1,2,\ldots, \tag{22} \]
where
\[ p(x)=\frac{\gamma_1}{cx+2}\exp\left[-\int \frac{2x}{cx+2}\,dx\right], \qquad cx+2\geq 0,\quad c\geq 0,\quad \gamma_1=\mathrm{const}. \tag{23} \]
- Let \(c=0\); then \(p(x)=\frac{1}{\sqrt{2\pi}}e^{-x^2/2}\), and \(\varphi_k(x)\) are Hermite polynomials, and the process is Gaussian, while for \(t>0\)
\[ p(t,x,y)=\frac{1}{2\pi\sqrt{1-e^{-2\lambda_1 t}}} \exp\left[-\frac{x^2+y^2-2e^{-\lambda_1 t}xy}{2(1-e^{-2\lambda_1 t})}\right]. \tag{24} \]
is the density of a normal correlation with positive correlation coefficient \(R=e^{-\lambda_1 t}\).
- For \(c>0\), equation (22) defines generalized Laguerre polynomials, orthogonal on the interval \(-\dfrac{2}{c}\leqslant x<\infty\) with weight
\[ p(x)=\gamma_2 e^{-\frac{2}{c}x}\left(x+\frac{2}{c}\right)^{\frac{4}{c^2}-1},\qquad \gamma_2=\mathrm{const}. \tag{25} \]
The corresponding density of the joint probability distribution has been studied in detail in paper \((^4)\); we note here only that the characteristic function of this distribution has the form
\[ \varphi_c(\tau_1,\tau_2)= \frac{\exp\left[-\dfrac{2i}{c}(\tau_1+\tau_2)\right]} {\left[1-\dfrac{c}{2}i(\tau_1+\tau_2)-\dfrac{c^2}{4}\left(1-e^{-\lambda_1 t}\right)\tau_1\tau_2\right]^{\frac{4}{c^2}}}, \tag{26} \]
and as \(c\to 0\) it converges to the characteristic function of the normal distribution (24).
Remark. If \(c<0\), then the corresponding correlation dependence between \(x\) and \(y\) is specified in the quadrant \(-\infty<x;\ y\leqslant \dfrac{2}{c}\), and a simultaneous change of signs of \(x\) and \(y\) reduces this case to the one already considered.
- In items 5–7 the following assertion has been proved.
Theorem 2. The normal correlation (24) and the correlation with characteristic function (26), constructed from generalized Laguerre polynomials, exhaust the class of densities determining a continuous stationary Markov process and such that the eigenfunctions of the kernel (7) constitute a complete system of polynomials orthogonal on an infinite interval.
Steklov Mathematical Institute
Academy of Sciences of the USSR
Received
10 II 1960
REFERENCES
\(^1\) O. V. Sarmanov, DAN, 120, No. 4 (1958).
\(^2\) O. V. Sarmanov, DAN, 121, No. 1 (1958).
\(^3\) A. N. Kolmogorov, UMN, issue 5 (1938).
\(^4\) O. V. Sarmanov, DAN, 132, No. 2 (1960).