Vol. 58, No. 2, 1975

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A new characterization of characteristic functions of absolutely continuous distributions

Simeon M. Berman

Vol. 58 (1975), No. 2, 323–329
Abstract

It is well known that if g belongs to L2, then

∫
--g(∫x)g(x-+-y)dx
|g(x)|2dx

is the characteristic function of an absolutely continuous distribution function. Conversely, every such characteristic function has the representation given above. Rere we shown that if R(s,t) is a covariance function such that R(s,s) belongs to L1, then

∫
--R∫(s,s+-t)ds
R(s,s)ds

is the characteristic function of an absolutely continuous distribution. Conversely, every such characteristic function has the latter representation (put R(s,t) = g(s)g(t)). The use of this new result is that certain functions are directly seen to be of the second form but not the first; hence, they can be identifled as characteristic functions of absolutely continuous distributions.

Mathematical Subject Classification 2000
Primary: 60G10
Secondary: 60E05
Milestones
Received: 18 March 1974
Published: 1 June 1975
Authors
Simeon M. Berman