On smooth behavior of probability distributions under polynomial mappings

Götze F, Prokhorov YV, Ulyanov VV (1998)
THEORY OF PROBABILITY AND ITS APPLICATIONS 42(1): 28-38.

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Zeitschriftenaufsatz | Veröffentlicht | Englisch
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Abstract / Bemerkung
Let X be a random variable with probability distribution P-X concentrated on [-1,1] and let Q(x) be a polynomial of degree k greater than or equal to 2. The characteristic function of a random variable Y = Q(X) is of order O(1/\t\(1/k)) as \t\ --> infinity if P-X is sufficiently smooth. In addition, for every epsilon: 1/k > epsilon > 0 there exists a singular distribution P-X such that every convolution P-X(n star) is also singular while the characteristic function of Y is of order O(1/\t\(1/k-epsilon)). While the characteristic function of X is small when "averaged," the characteristic function of the polynomial transformation Y of X is uniformly small.
Erscheinungsjahr
Zeitschriftentitel
THEORY OF PROBABILITY AND ITS APPLICATIONS
Band
42
Ausgabe
1
Seite(n)
28-38
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Götze F, Prokhorov YV, Ulyanov VV. On smooth behavior of probability distributions under polynomial mappings. THEORY OF PROBABILITY AND ITS APPLICATIONS. 1998;42(1):28-38.
Götze, F., Prokhorov, Y. V., & Ulyanov, V. V. (1998). On smooth behavior of probability distributions under polynomial mappings. THEORY OF PROBABILITY AND ITS APPLICATIONS, 42(1), 28-38.
Götze, F., Prokhorov, Y. V., and Ulyanov, V. V. (1998). On smooth behavior of probability distributions under polynomial mappings. THEORY OF PROBABILITY AND ITS APPLICATIONS 42, 28-38.
Götze, F., Prokhorov, Y.V., & Ulyanov, V.V., 1998. On smooth behavior of probability distributions under polynomial mappings. THEORY OF PROBABILITY AND ITS APPLICATIONS, 42(1), p 28-38.
F. Götze, Y.V. Prokhorov, and V.V. Ulyanov, “On smooth behavior of probability distributions under polynomial mappings”, THEORY OF PROBABILITY AND ITS APPLICATIONS, vol. 42, 1998, pp. 28-38.
Götze, F., Prokhorov, Y.V., Ulyanov, V.V.: On smooth behavior of probability distributions under polynomial mappings. THEORY OF PROBABILITY AND ITS APPLICATIONS. 42, 28-38 (1998).
Götze, Friedrich, Prokhorov, YV, and Ulyanov, VV. “On smooth behavior of probability distributions under polynomial mappings”. THEORY OF PROBABILITY AND ITS APPLICATIONS 42.1 (1998): 28-38.