Abstract
Let X i , i=1, …, n be independent P i-dimensional random vectors. Consider two linear functions
where A i , B i are (mxP i ) matrices. In the paper the condition of identically distribution of and L1 and L2 is studied. The results are applied to a characterization of the normal law and to the problem of characterization of probability distributions of random vectors given the joint distribution of linear functions of them.
The results provide a generalization of those concerning characterization of probability laws throught properties of linear functions.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
Cramér H. (1936): Uber eine Eigenschaft der normalen Verteilungsfunktion. Math. Zs. 41 (1936), 405–414.
Darmois G. (1953): Analyse générale des liaisons stochastiques. Rev. Inst. Intern. Statist. 21 (1953), 2–8.
Ghurye S. G.,Olkin I. (1962): A characterization of the multivariate normal distribution. Ann. Math. Statist. 33 (1962), No. 2, 533–541.
Heyde C.C. (1970): Characterization of the normal law by the symmetry of a certain conditional distribution. Sankhyä A 32 (1970), No. l, 115–118.
Kagan A. M., Linnik Yu.V., Rao C. R. (1972): Characterization problems of mathematical statistics. Russian edition, Moscow 1972. English edition, John Wiley, New York 1973.
Khatri C. G.,Rao C.R. (1968): Solutions to some functional equations and their applications to characterization of probability distributions. Sankhyä A 30 (1968), 167–180.
Khatri C.G. (1971): On characterization of gamma and normal distributions by solving some functional equations in vector variables, J. Multivariate Analysis 1 (1971), 70–89.
Khatri C. G.,Rao C.R. (1971): Functional equations and characterization of probability lavs through linear functions of random variables. Indian Stat. Inst., Discussion Paper (1971), No.68, 1–18.
Klebanov L.B. (1975): On the condition of the identical distribution of linear forms in a speciel case. Teoriya Veroyatn. i yeye priminen. 20 (1975), No. 3, 684–685. (in Russian).
Kotlarski I. (1971): On a characterisation of probability distributions by the joint distribution of their linear functions. Sankhyä a 33 (1971), 73–80.
Marcinkiewicz I. (1938): Sur une propriété de la loi de Gansa. Math. 2s. 44 (1938), No. 4–5, 622–638.
Rao C.R. (1971) Characterization of probability laws through linear functions. Sankhyä A 33 (1971), 255–259.
Skitovič V.P. (1953): On a property of normal distribution. Doklady Acad. Mauk SSSR 18 (1953), No.2, 217–219.
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 1978 ACADEMIA, Publishing House of the Czechoslovak Academy of Sciences, Prague
About this chapter
Cite this chapter
Klebanov, L. (1978). When are Two Special Linear Forms of Independent Random Vectors Identically Distributed?. In: Transactions of the Eighth Prague Conference. Czechoslovak Academy of Sciences, vol 8A. Springer, Dordrecht. https://doi.org/10.1007/978-94-009-9857-5_34
Download citation
DOI: https://doi.org/10.1007/978-94-009-9857-5_34
Publisher Name: Springer, Dordrecht
Print ISBN: 978-94-009-9859-9
Online ISBN: 978-94-009-9857-5
eBook Packages: Springer Book Archive