## A central limit theorem for generalized quadratic forms.(English)Zbl 0596.60022

Random variables of the form $$W(n)=\sum_{1\leq i\leq n}\sum_{1\leq j\leq n}w_{ijn}(X_ i,X_ j)$$ are considered with $$X_ i$$ independent (not necessarily identically distributed), and $$w_{ijn}(.,.)$$ Borel functions, such that $$w_{ijn}(X_ i,X_ j)$$ is square integrable and has vanishing conditional expectations: $E(w_{ijn}(X_ i,X_ j)| X_ i)=E(w_{ijn}(X_ i,X_ j)| X_ j)=0,\quad a.s.$ A central limit theorem is proved under the condition that the normed fourth moment tends to 3. Under some restrictions the condition is also necessary. Finally conditions on the individual tails of $$w_{ijn}(X_ i,X_ j)$$ and an eigenvalue condition are given that ensure asymptotic normality of W(n).

### MSC:

 60F05 Central limit and other weak theorems 62E20 Asymptotic distribution theory in statistics
Full Text:

### References:

 [1] Bloemena, A.R.: Sampling from a graph. Mathematical Centre Tract 2, Amsterdam (1964) · Zbl 0142.15002 [2] Barbour, A. D.; Eagleson, G. K., Multiple comparisons and sums of dissociated random variables, Adv. Appl. Prob., 17, 147-162 (1985) · Zbl 0559.60027 [3] Beran, R. J., Rank spectral processes and tests for serial dependence, Ann. Math. Statist., 43, 1749-1766 (1972) · Zbl 0257.62028 [4] Brown, B. M.; Kildea, D. G., Reduced U-statistics and the Hodge-Lehmann estimator, Ann. Statist., 6, 828-835 (1978) · Zbl 0408.62024 [5] Chung, K. L., A course in probability theory (1974), New York: Academic Press, New York · Zbl 0345.60003 [6] Feller, W., An introduction to probability theory and its applications II (1971), New York: Wiley, New York · Zbl 0219.60003 [7] Heyde, C. C.; Brown, B. M., On the departure from normality of a certain class of martingales, Ann. Math. Statist., 41, 2165-2165 (1970) · Zbl 0225.60026 [8] Hall, P., Central limit theorem for integrated square error of multivariate nonparametric density estimators, J. Multivar. Anal., 14, 1-16 (1984) · Zbl 0528.62028 [9] 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 [10] Karlin, S.; Rinott, Y., Applications of ANOVA type decompositions of conditional variance statistics including Jackknife estimates, Ann. Statist., 10, 485-501 (1982) · Zbl 0491.62036 [11] Kester, A., Asymptotic normality of the number of small distances between random points in a cube, Stochastic Process. Appl., 3, 45-54 (1975) · Zbl 0307.60021 [12] McGinley, W. G.; Sibson, R., Dissociated random variables, Math. Proc. Cambridge. Phil. Soc., 77, 185-188 (1975) · Zbl 0353.60018 [13] Noether, G. E., A central limit theorem with non-parametric applications, Ann. Math. Statist., 41, 1753-1755 (1970) · Zbl 0216.22102 [14] Robinson, J.: Limit theorems for standardized partial sums of exchangeable and weakly exchangeable arrays. (Preprint) (1985) · Zbl 0707.62036 [15] Rotar’, V. I., Some limit theorems for polynomials of second degree, Theor. Probab. Appl., 18, 499-507 (1973) · Zbl 0304.60037 [16] Sevast’yanov, B. A., A class of limit distributions for quadratic forms of normal stochastic variables, Theor. Probab. Appl., 6, 337-340 (1961) · Zbl 0286.60028 [17] Shapiro, C. P.; Hubert, L., Asymptotic normality of permutation statistics derived from weighted sums of bivariate functions, Ann. Statist., 1, 788-794 (1979) · Zbl 0423.62020 [18] Weber, N. C., Central limit theorems for a class of symmetric statistics, Math. Proc. Cambridge Philos. Soc., 94, 307-313 (1983) · Zbl 0563.60025 [19] Whittle, P., On the convergence to normality of quadratic forms in independent variables, Theor. Probab. Appl., 9, 113-118 (1964) · Zbl 0146.40905
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.