 On normal characterizations by the distribution of linear forms, assuming finite varianceBarry C. Arnold and Dean L. Isaacson Stochastic Processes and their Applications, 1978, vol. 7, issue 2, 227-230 Abstract: If X1 and X2 are independent and identically distributed (i. i. d.) with finite variance, then (X1+X2)/[radical sign]2 has the same distribution as X1 if and only if X1 is normal with mean zero (Pólya [9]). The idea of using linear combinations of i. i. d. random variables to characterize the normal has since been extended to the case where [sigma][infinity]i=1aiXi has the same distribution as X1. In particular if at least two of the ai's are non-zero and X1 has finite variance, then Laha and Lukacs [8] showed that X1 is normal. They also [7] established the same result without the assumption of finite variance. The purpose of this note is to present a different and easier proof of the characterization under the assumption of finite variance. The idea of the proof follows closely the approach used by Pólya in [9]. The same technique is also used to give a characterization of the exponential distribution. Date: 1978 Downloads: (external link) Related works: Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text Persistent link: https://EconPapers.repec.org/RePEc:eee:spapps:v:7:y:1978:i:2:p:227-230 Ordering information: This journal article can be ordered from Access Statistics for this article Stochastic Processes and their Applications is currently edited by T. Mikosch More articles in Stochastic Processes and their Applications from Elsevier |
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