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The two-dimensional distribution of non-negative random dependent variables and defined by the density


are the Laguerre polynomials, orthonormalized on the positive semi-axis with weight , ;

and is an arbitrary distribution function on the segment . The correlation coefficient between and is . If , a symmetric gamma-correlation is obtained; in such a case , and the form of the corresponding characteristic function is

If is such that , then , , and is the correlation coefficient between and (). In this last case the density series can be summed using the formula (cf. [2]):

where is the Bessel function of an imaginary argument [2].


[1] I.O. Sarmanov, Trudy Gidrologichesk. Inst. , 162 (1969) pp. 37–61
[2] W. Myller-Lebedeff, "Die Theorie der Integralgleichungen in Anwendung auf einige Reihenentwicklungen" Math. Ann. , 64 (1907) pp. 388–416


This bivariate distribution is just one of the many possible multivariate generalizations of the (univariate) gamma-distribution. See [a1], Chapt. 40 for a survey as well as more details on this one.


[a1] N.L. Johnson, S. Kotz, "Distributions in statistics" , 2. Continuous multivariate distributions , Wiley (1972)
How to Cite This Entry:
Gamma-correlation. O.V. Sarmanov (originator), Encyclopedia of Mathematics. URL:
This text originally appeared in Encyclopedia of Mathematics - ISBN 1402006098