Details

Autor(en) / Beteiligte

• Gotze, F.
• Prokhorov, Yu. V.
• Ulyanov, V. V.

Titel

On Smooth Behavior of Probability Distributions Under Polynomial Mappings

Ist Teil von

• Theory of probability and its applications, 1998-01, Vol.42 (1), p.28-38

Ort / Verlag

Philadelphia: Society for Industrial and Applied Mathematics

Erscheinungsjahr

1998

Link zum Volltext

Quelle

Alma/SFX Local Collection

Beschreibungen/Notizen

• 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.