Khatri, C. G. ; Rao, C. Radhakrishna
(1972)
*Functional equations and characterization of probability laws through linear functions of random variables*
Journal of Multivariate Analysis, 2
(2).
p. 162.
ISSN 0047-259X

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Official URL: http://www.sciencedirect.com/science/article/pii/0...

Related URL: http://dx.doi.org/10.1016/0047-259X(72)90024-3

## Abstract

General functional equations of the type ∑φ_{i}(A_{i}'t+B_{i}'u)=C_{a}(u|t)+D_{b}(t|u)+P_{k}(t,u) and ∑φ_{i}(Ci||t) = P_{k}(t) have been solved, where P_{k} represents a polynomial of degree k in all the arguments, C_{n}(u short parallel t), a polynomial of degree a in u given t, and D_{b}(t short parallel u), a polynomial of degrce b in t given u. The results are applied in characterizing the multivariate normal variable by nonuniqueness of linear structure, independence of sets of linear functions, and constancy of regression of one set on another set of linear functions. The problem of characterization of probability distributions of individual random variables (which may be vectors), given the joint distribution of a relatively few linear functions of all the variables, has been studied. The results provide a generalization of all previous work on the characterization of probability laws of vector random variables through linear functions.

Item Type: | Article |
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Source: | Copyright of this article belongs to Elsevier Science. |

Keywords: | Functional Equations; Characterization Of Probability Distributions; Multivariate Normal; Regression; Linear Structure |

ID Code: | 58143 |

Deposited On: | 31 Aug 2011 12:29 |

Last Modified: | 31 Aug 2011 12:29 |

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