UDC 519.272.129

MATHEMATICS

I. O. SARMANOV

THE GAMMA-CORRELATION PROCESS AND ITS PROPERTIES

(Presented by Academician Yu. V. Linnik on 17 VII 1969)

1. Theorem. The function \(p(t_1,x;t_2,y)\), defined for \(0<t_1<t_2\), \(x,y\ge 0\) by the bilinear expansion

\[ p(t_1,x;t_2,y)=p(t_1,x)p(t_2,y)\sum_{k=0}^{\infty} a_k(t_1,t_2) L_k^{\alpha(t_1)}\!\left(\frac{x}{\beta(t_1)}\right) L_k^{\alpha(t_2)}\!\left(\frac{y}{\beta(t_2)}\right), \tag{1} \]

where

\[ p(t,x)=x^{\alpha(t)}e^{-x/\beta(t)}\big/[\beta(t)]^{\alpha(t)+1}\Gamma(\alpha(t)+1); \tag{2} \]

\[ a_k(t_1,t_2)= \]

\[ =\left[\omega(t_1)/\omega(t_2)\right]^k \left[ \frac{\Gamma(\alpha(t_2)+1)\Gamma(\alpha(t_1)+k+1)} {\Gamma(\alpha(t_1)+1)\Gamma(\alpha(t_2)+k+1)} \right]^{1/2}; \tag{3} \]

\(\beta(t)>0\); \(\alpha(t)>-1\) is a monotonically increasing continuous function, \(\omega(t)\) is a continuous strictly increasing function, and \(L_k^\alpha(x/\beta)\) are Laguerre polynomials, is a two-dimensional density defining a Markov process \(\xi(t)\ge 0\) for \(t>0\).

Proof. By virtue of the orthogonality and normality of the Laguerre polynomials

\[ L_k^{\alpha(t)}\!\left(\frac{x}{\beta(t)}\right) = \left[ \frac{\Gamma(\alpha(t)+1)\Gamma(\alpha(t)+k+1)}{k!} \right]^{1/2} \sum_{r=0}^{k} \frac{(-1)^r\binom{k}{r}}{\Gamma(\alpha(t)+r+1)} \left(\frac{x}{\beta(t)}\right)^r \tag{4} \]

with weight (2) on the half-axis \(0\le x<\infty\), the function (1) satisfies the following Markov equation for the two-dimensional density:

\[ p(t_1,x;t_2,y)= \int_{0}^{\infty} \frac{p(t_1,x;t,z)\,p(t,z;t_2,y)}{p(t,z)}\,dz, \tag{5} \]

where

\[ p(t,z)=\int_{0}^{\infty}p(t_1,x;t,z)\,dx \]

is the marginal density (2) of the gamma distribution, \(0<t_1<t<t_2\).

The series (1) for \(t_1<t_2\) converges in the mean, since the squares of the coefficients of the series

\[ a_k^2(t_1,t_2)=[\omega(t_1)/\omega(t_2)]^{2k} \]

are less than the terms of a geometric progression with ratio \(\omega^2(t_1)/\omega^2(t_2)<1\).

Consequently, to prove the validity of the assertion of the theorem, it remains to show the nonnegativity of the sum of the series (1) for \(t_1<t_2\) and \(x,y\ge 0\).

Noting that the Fourier transform of polynomial (4) with weight (2) has the form1

\[ \left[\Gamma(\alpha+k+1)/\Gamma(\alpha+1)k!\right]^{1/2} \frac{(-i\tau\beta)^k}{(1-i\tau\beta)^{\alpha+k+1}}, \]

we find the two-dimensional Fourier transform of function (1)

\[ \varphi(\tau_1,\tau_2)= \frac{\left[1-i\tau_2\beta(t_2)\right]^{\alpha(t_1)-\alpha(t_2)}} {\left[1-i\tau_1\beta(t_1)-i\tau_2\beta(t_2)-\left(1-\omega(t_1)/\omega(t_2)\right)\tau_1\tau_2\beta(t_1)\beta(t_2)\right]^{\alpha(t_1)+1}}. \tag{6} \]

Expression (6) is a two-dimensional characteristic function, since the numerator is a one-dimensional characteristic function of the gamma distribution with nonnegative parameter \(\alpha(t_2)-\alpha(t_1)\), and the second factor is the two-dimensional characteristic function of the symmetrized gamma correlation studied in [2]. Consequently, (1) is a two-dimensional density. The theorem is proved.

1. In [3] it is noted that the correlation function of a continuous Markov process for \(t_1<t_2\) must have the form \(R(t_1,t_2)=\psi(t_1)/\psi(t_2)\), where \(\psi(t)\) is a continuous strictly increasing function.

In our case \(\psi(t)=\omega(t)[\alpha(t)+1]^{1/2}\), and

\[ R(t_1,t_2)=a_1(t_1,t_2)= \omega(t_1)[\alpha(t_1)+1]^{1/2}/\omega(t_2)[\alpha(t_2)+1]^{1/2}, \tag{7} \]

since, in general, \(a_k(t_1,t_2)\) is the correlation coefficient between
\(L_k^{\alpha(t_1)}(\xi(t_1)/\beta(t_1))\) and
\(L_k^{\alpha(t_2)}(\xi(t_2)/\beta(t_2))\); hence \(a_1(t_1,t_2)\) is the correlation coefficient between \(\xi(t_1)\) and \(\xi(t_2)\).

Definition. The Markov process \(\xi(t)\) specified by density (1) will be called a gamma-correlation process or a gamma process.

1. If \(\alpha(t)=\alpha\), \(\beta(t)=\beta\) do not depend on \(t\), and \(\omega(t)=e^{\lambda t}\), where \(\lambda\) is a positive constant, then \(a_k(t_1,t_2)=\exp\{-\lambda(t_2-t_1)k\}\), and (1) defines the stationary Markov process studied in [4] (for \(\beta=1\)).

If \(\alpha=-1/2\) (does not depend on \(t\)), \(\beta(t)=\omega(t)=t\), then \(a_k(t_1,t_2)=(t_1/t_2)^k\), and we obtain a gamma process \(\xi(t)\) with density

\[ \frac{(x/t_1)^{-1/2}e^{-x/t_1}}{\Gamma(1/2)t_1} \frac{(y/t_2)^{-1/2}e^{-y/t_2}}{\Gamma(1/2)t_2} \sum_{k=0}^{\infty}\left(\frac{t_1}{t_2}\right)^k L_k^{-1/2}\left(\frac{x}{t_1}\right) L_k^{-1/2}\left(\frac{y}{t_2}\right) = \]

\[ = \frac{\operatorname{ch}\left[2(xy)^{1/2}/(t_2-t_1)\right]} {\pi\left[t_1(t_2-t_1)xy\right]^{1/2}} \exp\left\{-\frac{x/t_1+y/t_2}{1-t_1/t_2}\right\}. \tag{8} \]

It is easy to verify that, if one considers the Wiener process \(\eta(t)\) with parameters

\[ \mathbf{M}\eta(t)=0,\qquad \mathbf{D}\eta(t)=t,\qquad R(t_1,t_2)=(t_1/t_2)^{1/2}, \]

then (8) is the two-dimensional density of the process \(\xi(t)=\eta^2(t)/2\).

Definition. If, for the process \(\xi(t)\) with density (1), the correlation function \(R(t_1,t_2)=t_1/t_2\) is equal to the square of the correlation function of the Wiener process, then the process \(\xi(t)\) will be called a gamma process of Wiener type.

1. Let us consider a more general case of a gamma process of Wiener type. Put \(\alpha(t)+1=t\), \(\omega(t)=t^{1/2}\), \(\beta(t)=1\); then the process will have the basic parameters

\[ \mathbf{M}\xi(t)=t,\qquad \mathbf{D}\xi(t)=t,\qquad R(t_1,t_2)=t_1/t_2. \tag{9} \]

The marginal density has the form

\[ p(t,x)=x^{t-1}e^{-x}/\Gamma(t),\qquad t>0,\qquad x\ge 0. \]

The two-dimensional density is written in the form of a series

\[ p(t_1,x;t_2,y)= \frac{x^{t_1-1}e^{-x}}{\Gamma(t_1)} \frac{y^{t_2-1}e^{-y}}{\Gamma(t_2)} \sum_{k=0}^{\infty} \left(\frac{t_1}{t_2}\right)^{k/2} \left[ \frac{\Gamma(t_2)\Gamma(t_1+k)} {\Gamma(t_1)\Gamma(t_2+k)} \right]^{1/2} L_k^{t_1-1}(x)L_k^{t_2-1}(y), \tag{10} \]

where \(0<t_1<t_2\).

Let us note that the trajectories of the process defined by density (10) may, as in the Wiener case, be regarded as issuing from the origin; they all lie in the first quadrant of the plane \(XOT\), where time \(t\) is plotted along the abscissa axis. The mean value \(M\xi(t)=t\) grows without bound as \(t\) increases, whereas in the Wiener case \(M\eta(t)=0\).

In conclusion, we write down some other characteristics of the process (10). The characteristic function of the two-dimensional distribution has the form

\[ \varphi(\tau_1,\tau_2)= \frac{1}{(1-i\tau_2)^{t_2-t_1}}\, \frac{1}{ \left[1-i\tau_1-i\tau_2-\left(1-(t_1/t_2)^{1/2}\right)\tau_1\tau_2\right]^{t_1} }, \quad 0<t_1<t_2 . \]

The initial moments of the marginal distribution are expressed by the formula

\[ M\xi^k(t)=t(t+1)\cdots(t+k-1),\qquad k=1,2,\ldots \]

The conditional moments \(m_k(t_1,t_2,x)\) of the quantity \(\xi(t_2)\) under the condition \(\xi(t_1)=x\) are expressed as follows:

\[ m_k(t_1,t_2,x)= \Gamma(t_2+k) \sum_{\mu=0}^{k} \frac{\binom{k}{\mu}\left[x(t_1/t_2)^{1/2}\right]^\mu} {\Gamma(t_2+\mu)} \sum_{\nu=0}^{k-\mu} \binom{k-\mu}{\nu} \left[(t_1/t_2)^{1/2}\right]^\nu \times \]

\[ \times \frac{\Gamma(t_2+\mu)\Gamma(t_1+\mu+\nu)} {\Gamma(t_1+\mu)\Gamma(t_2+\mu+\nu)} . \tag{11} \]

With the aid of (11) one can find the conditional variance

\[ D\bigl(\xi(t_2)\mid \xi(t_1)=x\bigr) = t_2-t_1+t_1\left[1-(t_1/t_2)^{1/2}\right]^2+ 2(t_1/t_2)^{1/2}\left[1-(t_1/t_2)^{1/2}\right]x . \tag{12} \]

Remark. We have put \(\beta=1\) only to shorten the notation; replacing \(x\) and \(y\) respectively by \(x/\beta(t_1)\) and \(y/\beta(t_2)\) will not change the correlation function and will make it possible, without difficulty, to pass to a more general process of Wiener type.

Institute of Water Problems
Academy of Sciences of the USSR
Moscow

Received
2 VII 1969

REFERENCES

1. A. Kratzer, W. Franz, Transcendental Functions, IL, 1963.
2. O. V. Sarmanov, DAN, 132, No. 2, 299 (1960).
3. S. N. Bernstein, Collected Works, 4, Moscow, 1964.
4. O. V. Sarmanov, Proceedings of the V. A. Steklov Mathematical Institute, Academy of Sciences of the USSR, 60, 238 (1961).

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