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The purpose of this note is to show by constructing counterexamples that two conjectures of Móri and Székely for the Borel-Cantelli lemma are false.

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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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In a string ofn independent coin tosses we consider the difference between the lengths of the longest blocks of consecutive heads resp. tails. A complete characterization of the a.s. limit properties of this quantity is proved.

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We characterize the lower classes of the integrated fractional Brownian motion by an integral test.

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Let { B H;K ( t ), t ≧ 0} be a bifractional Brownian motion with indexes 0 < H < 1 and 0 < K ≦ 1 and define the statistic

\documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$V_T = \mathop {\sup }\limits_{0 \leqq s \leqq T - a_T } \beta _T \left| {B_{H,K} (s + a_T ) - B_{H,K} (s)} \right|$$ \end{document}
where β T and α T are suitably chosen functions of T ≧ 0. We establish some laws of the iterated logarithm for V T .

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LetS n be the partial sums of ?-mixing stationary random variables and letf(x) be a real function. In this note we give sufficient conditions under which the logarithmic average off(S n/sn) converges almost surely to ?-8 8 f(x)dF(x). We also obtain strong approximation forH(n)=?k=1 n k -1 f(S k/sk)=logn ?-8 8 f(x)dF(x) which will imply the asymptotic normality ofH(n)/log1/2 n. But for partial sums of i.i.d. random variables our results will be proved under weaker moment condition than assumed for ?-mixing random variables.

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We obtain new lower and upper bounds for probabilities of unions of events. These bounds are sharp. They are stronger than earlier ones. General bounds may be applied in arbitrary measurable spaces. We have improved the method that has been introduced in previous papers. We derive new generalizations of the first and second parts of the Borel-Cantelli lemma.

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Let {B H,K(t); t ≧ 0} be a bifractional Brownian motion with indices 0 < H < 1 and 0 < K ≦ 1. We characterize the upper classes of some increments of B H,K by an integral test.

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