A functional equation from probability theory

Author:
John A. Baker

Journal:
Proc. Amer. Math. Soc. **121** (1994), 767-773

MSC:
Primary 39B22; Secondary 62E10

DOI:
https://doi.org/10.1090/S0002-9939-1994-1186127-5

MathSciNet review:
1186127

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Abstract | References | Similar Articles | Additional Information

Abstract: The functional equation

() |

has been used by Laha and Lukacs (Aequationes Math.

**16**(1977), 259-274) to characterize normal distributions. The aim of the present paper is to study (1) under somewhat different assumptions than those assumed by Laha and Lukacs by using techniques which, in the author's opinion, are simpler than those employed by the afore-mentioned authors. We will prove, for example, that if and for , where

*k*is a natural number, , (1) holds for and exists then either or there exists a real constant

*c*such that for all .

**[1]**John A. Baker,*Functional equations, tempered distributions and Fourier transforms*, Trans. Amer. Math. Soc.**315**(1989), 57-68. MR**979965 (90k:39006)****[2]**W. Jarczyk,*A recurrent method of solving iterative functional equations*, Uniwersytet Slaski, Katowice, 1991. MR**1135795 (92k:39005)****[3]**M. Kuczma, B. Choczewski, and R. Ger,*Iterative functional equations*, Cambridge Univ. Press, London and New York, 1990. MR**1067720 (92f:39002)****[4]**M. Laczkovich,*Non-negative measurable solutions of a difference equation*, J. London Math. Soc. (2)**34**(1986), 139-147. MR**859155 (87m:39003)****[5]**R. G. Laha and E. Lukacs,*On a functional equation which occurs in a characterization problem*, Aequationes Math.**16**(1977), 259-274. MR**0471023 (57:10765)****[6]**E. Vincze,*Bemerkung zur Charakterisierung der Gauss'schen Fehlergesetzes*, Magyar Tud. Acad. Mat. Kutató Int. Kösl.**7**(1962), 357-61. MR**0152773 (27:2748)****[7]**E. Zeidler,*Nonlinear functional analysis and its applications*. I, Springer-Verlag, New York, Berlin, Heidelberg, and Tokyo, 1986. MR**816732 (87f:47083)**

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9939-1994-1186127-5

Keywords:
Functional equation,
probability

Article copyright:
© Copyright 1994
American Mathematical Society