Summary
Letξ i be a strictly stationary, absolutely regular process defined on a probability space (Ω,A,P), i.e.,ξ i's satisfy the condition
whereξ ba (a≦b) denote theσ-algebra of events generated byσ a,...σb.(It is known thatσ i is absolutely regular if {σi} isφ-mixing, i.e.
For some class of one-sample rank-order statistics, generated by absolutely regular processes, we shall prove theorems concerning the following problems under the assumption thatσ 1 has a continuous (not necessarily symmetric) distribution function.
-
(a)
weak convergence to a processU(t): O≦t≦1 defined byU(t) \(U(t) = \int\limits_0^t {h(s)dW(s)} ,\)
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(b)
functional laws of the iterated logarithm, and
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(c)
almost sure invariance principles and integral tests.
Some of them are extensions of Sen's results [(Ann. Statist.2, 49–62. (1974; Zbl. 273, 60005) ibid.2, 1358 (1974; Zbl.292, 60012)] and Stigler's ones [ibid.2, 676–693 (1974; Zbl.286, 62028)].
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Yoshihara, Ki. Limiting behavior of one-sample rank-order statistics for absolutely regular processes. Z. Wahrscheinlichkeitstheorie verw Gebiete 43, 101–127 (1978). https://doi.org/10.1007/BF00668453
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DOI: https://doi.org/10.1007/BF00668453