Abstract
Define ,\(S_{k,n} = \Sigma _{1 \leqslant i_1< \cdot \cdot \cdot< l_k \leqslant n} X_{i_1 } \cdot \cdot \cdot X_{i_k } ,n \geqslant k \geqslant {\text{1}}\) where {X, X n ,n≥1} are i.i.d. random variables withEX=0,EX 2=1 and letH k (·) denote the Hermite polynomial of degreek. By establishing an LIL for products of correlated sums of i.i.d. random variables, the a.s. decomposition
valid whenEX 4<∞, elicits an LIL forη k,n =k!S k,n −n k/2 H k (S 1n /n 1/2) under a reduced normalization. Moreover, whenE|X| p<∞ for somep in [2, 4], a Marcinkiewicz-Zygmund type strong law forη k,n is obtained, likewise under a reduced normalization.
Similar content being viewed by others
References
Avram, F. and Taqqu, M. S. (1986). Symmetric polynomials of random variables attracted to an infinitely divisible law.Prob. Th. Rel. Fields 71, 491–500.
Feller, W. (1946). A limit theorem for random variables with Infinite moments.Amer. J. Math. 18, 257–262.
Fernholtz, L. and Teicher, H. (1980). Stability of random variables and iterated logarithm laws for martingales and quadratic forms.Ann. Prob. 8, 765–774.
Finkelstein, H. (1971). The law of the iterated logarithm for empirical distributions.Ann. Math. Statist. 42, 607–615.
Giné, E. and Zinn, J. (1992). Markinkiewicz type laws of large numbers and convergence of moments forU-type statistics. In Dudley, R. M., Hahn, M. G., and Kuelbs, J., (eds.),Probability in Banach Spaces, Birkhäuser, Boston,8, 273–291.
Hoeffding, W. (1961). The Strong Law of Large Numbers for U-Statistics. Institute of Statistics Mimeo Series 302, University of North Carolina, Chapel Hill.
Rubin, H. and Vitale, R. A. (1981). Asymptotic distribution of symmetric statistics.Ann. Stat. 8, 165–170.
Teicher, H. (1992). Convergence of self-normalized generalizedU-statistics.J. Th. Prob. 5, 391–405.
Teicher, H. (1995). Moments of randomly stopped sums-revisited.J. Th. Prob. 8, 779–793.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Teicher, H., Zhang, CH. A decomposition for someU-type statistics. J Theor Probab 9, 161–170 (1996). https://doi.org/10.1007/BF02213738
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF02213738