zbMATH — the first resource for mathematics

A central limit theorem for generalized multilinear forms. (English) Zbl 0709.60019
Let \(X_ i\), \(i\geq 1\), be independent random variables, and, for each \(I\subset \{1,2,...,n\}\), let \(W_ I\) be a function of \(\{X_ i\), \(i\in I\}\). Suppose further that the \(W_ I\) have mean zero and finite fourth moments, and that \({\mathbb{E}} W_ IW_ J=0\) if \(I\neq J\). Such random variables arise naturally in the decomposition of W. Hoeffding [Ann. Math. Stat. 19, 293-325 (1948; Zbl 0032.04101)].
The author is interested in conditions under which the d-homogeneous sum \(W=\sum_{| I| =d}W_ I\), standardized to have variance 1, converges to \(N(0,1)\) as \(n\to \infty\). This is not necessarily to be expected: if \(d\geq 2\), a weighted mixture of centred chi-squared distributions is usual, but the setting is general enough to allow the possibility that all the weights become small as n increases. The main result is that under an asymptotic negligibility condition, it is sufficient that \({\mathbb{E}} W^ 4\to 3\).
Reviewer: A.D.Barbour

60F05 Central limit and other weak theorems
Full Text: DOI
[1] Barbour, A.D.; Eagleson, G.K., Multiple comparisons and sums of dissociated random variables, Adv. appl. probab., 17, 147-162, (1985) · Zbl 0559.60027
[2] Chung, K.L., ()
[3] De Jong, P., A central limit theorem for generalized quadratic forms, Probab. theory related fields, 75, 261-277, (1987) · Zbl 0596.60022
[4] De Jong, P., (), Tract 61
[5] Hall, P., Central limit theorem for integrated square error of multivariate nonparametric density estimators, J. multivariate anal., 14, 1-16, (1984) · Zbl 0528.62028
[6] Heyde, C.C.; Brown, B.M., On the departure from normality of a certain class of martingales, Ann. math. statist., 41, 2161-2165, (1970) · Zbl 0225.60026
[7] Hoeffding, W., A class of statistics with asymptotically normal distribution, Ann. math. statist., 19, 293-325, (1948) · Zbl 0032.04101
[8] Jammalamadaka, R.S.; Janson, S., Limit theorems for a triangular scheme of U-statistics with applications to interpoint distances, Ann. probab., 14, 1347-1358, (1986) · Zbl 0604.60023
[9] McGinley, W.G.; Sibson, R., Dissociated random variables, (), 185-188 · Zbl 0353.60018
[10] Noether, G.E., A central limit theorem with non-parametric applications, Ann. math. statist., 41, 1753-1755, (1970) · Zbl 0216.22102
[11] Rotar’, V.I., Some limit theorems for polynomials of second degree, Theory probab. appl., 18, 499-507, (1973) · Zbl 0304.60037
[12] Rotar’, V.I., Limit theorems for polylinear forms, J. multivariate anal., 9, 511-530, (1979) · Zbl 0426.62013
[13] Sevast’yanov, B.A., A class of limit distributions for quadratic forms of normal stochastic variables, Theory probab. appl., 6, 337-340, (1961) · Zbl 0286.60028
[14] Van Zwet, W.R., A Berry-Esseen bound for symmetric statistics, Z. wahrsch. verw. gebiete, 66, 425-440, (1984) · Zbl 0525.62023
[15] Weber, N.C., Central limit theorems for a class of symmetric statistics, (), 307-313 · Zbl 0563.60025
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.