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L-statistic; GL-statistic; strong law of large numbers; stationary ergodic process
It is pointed out that a strong law of large numbers for L-statistics established by van Zwet (1980) for i.i.d. sequences, remains valid for stationary ergodic data. When the underlying process is weakly Bernoulli, the result extends even to generalized L-statistics considered in Helmers et al. (1988).
[1] Aaronson J.: An ergodic theorem with large normalising constants. Israel J. Math. 38 (1981), 182-188. MR 0605376 | Zbl 0468.60036
[2] Aaronson J., Burton R., Dehling H., Gilat D., Hill T., Weiss B.: Strong laws for L- and U-statistics. Trans. Am. Math. Society 348 (1996), 2845-2866. MR 1363941 | Zbl 0863.60032
[3] Helmers R., Janssen P., Serfling R.: Glivenko-Cantelli properties of some generalized empirical df's and strong convergence of generalized L-statistics. Probab. Th. Rel. Fields 79 (1988), 75-93. MR 0952995 | Zbl 0631.60032
[4] van Zwet W.R.: A strong law for linear functions of order statistics. Ann. Probab. 8 (1980), 986-990. MR 0586781 | Zbl 0448.60025
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