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  • Author or Editor: A. Földes x
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We present a joint functional iterated logarithm law for the Wiener process and the principal value of its local times.

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In reliability and survival-time studies one frequently encounters the followingrandom censorship model:X 1,Y 1,X 2,Y 2,… is an independent sequence of nonnegative rv's, theX n' s having common distributionF and theY n' s having common distributionG, Z n =min{X n ,Y n },T n =I[X n <-Y n ]; ifX n represents the (potential) time to death of then-th individual in the sample andY n is his (potential) censoring time thenZ n represents the actual observation time andT n represents the type of observation (T n =O is a censoring,T n =1 is a death). One way to estimateF from the observationsZ 1.T 1,Z 2,T 2, … (and without recourse to theX n' s) is by means of theproduct limit estimator (Kaplan andMeier [6]). It is shown that a.s., uniformly on [0,T] ifH(T )<1 wherel−H=(l−F) (l−G), uniformly onR if whereT F =sup {x:F(x)<1}; rates of convergence are also established. These results are used in Part II of this study to establish strong consistency of some density and failure rate estimators based on .

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Sample path properties of the Cauchy principal values of Brownian and random walk local times are studied. We establish LIL type results (without exact constants). Large and small increments are discussed. A strong approximation result between the above two processes is also proved.

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