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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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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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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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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 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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The sub-bifractional Brownian motion, which is a quasi-helix in the sense of Kahane, is presented. The upper classes of some of its increments are characterized by an integral test.

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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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