Sankhya: The Indian Journal of Statistics
1994, Volume 56, Series A, Pt. 3 , pp. 431-437
ON ENTIRE CHARACTERISTIC FUNCTIONS OF ORDER 2
ERIC S. KEY, University of Wisconsin
SUMMARY. Suppose that $\phi $ is the characteristic function of a random variable $X$ and that $\phi $ is an entire function satisfying $|\phi (z)| < C \; \exp (\eta |z|^2)$. Let $w \neq 0$ be a fixed complex number and suppose that $(1 + w)\phi (z) + (1-w)\phi (-z)$ has no zeros in the complex plane. We examine to what extent these conditions determine $\phi$, refining results due to Roberts (1971), and Funk and Rodine (1975). As an application we generalize a characterization of normality due to Funk and Rodine (1975) and Ramachandran (1975).
AMS (1980) subject classification. Primary 30D15; secondary 42D38, 60E10.
Key words and phrases. Characteristic function, Fourier transform, entire function, Hadamard's factorization theorem, normal distribution.
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