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Estimators of g-monotone dependence functions

Applicationes Mathematicae

The notion of g-monotone dependence function introduced in [4] generalizes the notions of the monotone dependence function and the quantile monotone dependence function defined in [2], [3] and [6]. In this paper we study the asymptotic behaviour of sample g-monotone dependence functions and their strong properties.

On monotone dependence functions of the quantile type

Applicationes Mathematicae

We introduce the concept of monotone dependence function of bivariate distributions without moment conditions. Our concept gives, among other things, a characterization of independent and positively (negatively) quadrant dependent random variables.

Necessary and sufficient conditions for weak convergence of random sums of independent random variables

Commentationes Mathematicae Universitatis Carolinae

Let $\left\{{X}_{n},\phantom{\rule{0.166667em}{0ex}}n\ge 1\right\}$ be a sequence of independent random variables such that $E{X}_{n}={a}_{n}$, $E{\left({X}_{n}-{a}_{n}\right)}^{2}={\sigma }_{n}^{2}$, $n\ge 1$. Let $\left\{{N}_{n},\phantom{\rule{0.166667em}{0ex}}n\ge 1\right\}$ be a sequence od positive integer-valued random variables. Let us put ${S}_{{N}_{n}}={\sum }_{k=1}^{{N}_{n}}{X}_{k}$, ${L}_{n}={\sum }_{k=1}^{n}{a}_{k}$, ${s}_{n}^{2}={\sum }_{k=1}^{n}{\sigma }_{k}^{2}$, $n\ge 1$. In this paper we present necessary and sufficient conditions for weak convergence of the sequence $\left\{\left({S}_{{N}_{n}}-{L}_{n}\right)/{s}_{n},\phantom{\rule{0.166667em}{0ex}}n\ge 1\right\}$, as $n\to \infty$. The obtained theorems extend the main result of M. Finkelstein and H.G. Tucker (1989).

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