Limit distributions of studentized means

Chistyakov G, Götze F (2004)
ANNALS OF PROBABILITY 32(1A): 28-77.

Zeitschriftenaufsatz | Veröffentlicht | Englisch
 
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Autor/in
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Abstract / Bemerkung
Let X, X-j, j is an element of N, be independent, identically distributed random variables with probability distribution F. It is shown that Student's statistic of the sample {X-j}(n)(j=1), has a limit distribution G such that G({-1, 1}) not equal 1, if and only if: (1) X is in the domain of attraction of a stable law with some exponent 0 < alpha less than or equal to 2; (2) EX = 0 if 1 < alpha < 2; (3) if alpha = 1, then X is in the domain of attraction of Cauchy's law and Feller's condition holds: lim(n-->infinity) nEsin(X/a(n)) exists and is finite, where a(n) is the infimum of all x > 0 such that nx(-2)(l + integral((-x,x))y(2)F{dy}) less than or equal to 1. If G({-1, 1}) = 1, then Student's statistic of the sample (X-j)(n)(j=1) has a limit distribution if and only if P(\X\ > x), x > 0, is a slowly varying function at +infinity.
Stichworte
the Cauchy type integral; regular functions; domains of attraction; stable distributions; self-normalized sums; student t-statistic
Erscheinungsjahr
2004
Zeitschriftentitel
ANNALS OF PROBABILITY
Band
32
Ausgabe
1A
Seite(n)
28-77
ISSN
0091-1798
Page URI
https://pub.uni-bielefeld.de/record/1608262

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Chistyakov G, Götze F. Limit distributions of studentized means. ANNALS OF PROBABILITY. 2004;32(1A):28-77.
Chistyakov, G., & Götze, F. (2004). Limit distributions of studentized means. ANNALS OF PROBABILITY, 32(1A), 28-77.
Chistyakov, G., and Götze, F. (2004). Limit distributions of studentized means. ANNALS OF PROBABILITY 32, 28-77.
Chistyakov, G., & Götze, F., 2004. Limit distributions of studentized means. ANNALS OF PROBABILITY, 32(1A), p 28-77.
G. Chistyakov and F. Götze, “Limit distributions of studentized means”, ANNALS OF PROBABILITY, vol. 32, 2004, pp. 28-77.
Chistyakov, G., Götze, F.: Limit distributions of studentized means. ANNALS OF PROBABILITY. 32, 28-77 (2004).
Chistyakov, Gennadiy, and Götze, Friedrich. “Limit distributions of studentized means”. ANNALS OF PROBABILITY 32.1A (2004): 28-77.