On Smooth Behavior of Probability Distributions Under Polynomial Mappings

Let X be a random variable with probability distribution PX concentrated on [-1,1]$ and let Q(x) be a polynomial of degree $k\ge 2$. The characteristic function of a random variable Y=Q(X)is of order O(1/|t|1/k) as $|t|\to\infty$ if PX is sufficiently smooth. In addition, for every $\varepsilon \:1/...

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Bibliographic Details
Published in:Theory of probability and its applications 1998-01, Vol.42 (1), p.28-38
Main Authors: Gotze, F., Prokhorov, Yu. V., Ulyanov, V. V.
Format: Article
Language:English
Subjects:
Probability distribution
Random variables
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Summary:Let X be a random variable with probability distribution PX concentrated on [-1,1]$ and let Q(x) be a polynomial of degree $k\ge 2$. The characteristic function of a random variable Y=Q(X)is of order O(1/|t|1/k) as $|t|\to\infty$ if PX is sufficiently smooth. In addition, for every $\varepsilon \:1/k > \varepsilon > 0$ there exists a singular distribution PX such that every convolution $P^{n\star}_X$ is also singular while the characteristic function of Y is of order $O(1/|t|^{1/k-\varepsilon})$. While the characteristic function of X is small when "averaged," the characteristic function of the polynomial transformation Y of X is uniformly small.
ISSN:0040-585X
1095-7219
DOI:10.1137/S0040585X97975927