Abstract
Full Text
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.
- 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.
- 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.
- 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
-
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