Il Prof. Andrei Volodin (Universita' di Regina-Canada) sara' ospite del Dipartimento dall'11 Dicembre al 22 Dicembre prossimi. Terra' due seminari, sui seguent argomenti: 1. Legge dei Grandi Numeri e Convergenza Completa 2. Teoremi Limite per il "bootstrap" dipendente della media
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13-12-2004 (15:00) - aula P del Polo Fibonacci Andrei Volodin (Universita' di Regina-Canada) : Legge dei Grandi Numeri e Convergenza Completa
The concept of complete convergence was introduced by Hsu and Robbins (1947) as follows. A sequence of random variables ${U_n, n \geq 1}$ is said to converge completely to a constant $C$ if $\sum^\infty_{n=1} P{|U_n - C | > \epsilon} < \infty$ for all $\epsilon > 0$. In view of the Borel-Cantelli lemma, this implies that $U_n \rightarrow C$ almost surely. Hsu and Robbins proved that the sequence of arithmetic means of independent and identically distributed random variables converges completely to the expected value if the variance of the summands is finite. We extend and generalize some recent results on complete convergence for arrays of rowwise independent real and Banach space valued random variables. In the main result, no assumptions are made concerning the existence of expected values or absolute moments of the random variables and no assumptions are made concerning the geometry of the underlying Banach space. Some well-known results from the literature are obtained easily as corollaries. The corresponding convergence rates are also established.
******************************************************************* Giorno Aula e orario da comunicare per il II seminario:
Teoremi Limite per il "bootstrap" dipendente della media
Let ${X_n, n\ge 1}$ be a sequence of random variables defined on a probability space $(\Omega, {\cal F}, P)$. Let ${m(n), n\ge 1}$ and ${k(n), n\ge 1}$ be two sequences of positive integers such that for all $n\ge 1: m(n)\le n k(n). $ For $\omega \in \Omega$ and $n \geq 1$, the dependent bootstrap is defined as the sample of size $m(n)$, denoted ${\hat{X}^{(\omega)}_{n, j}, 1 \leq j \leq m(n)}$, drawn without replacement from the collection of $n k(n)$ items made up of $k(n)$ copies each of the sample observations $X_1(\omega), \cdots, X_n(\omega)$. Let $\overline{X}_n(\omega) = \frac{1}{n} \sum_{j=1}^n X_j(\omega)$ denote the sample mean of ${X_j(\omega), 1 \leq j \leq n}$. This dependent bootstrap procedure is proposed as a procedure to reduce variation of estimators and to obtain better confidence intervals.
Giulia Curciarello Segreteria Didattica tel: 050-2213219 e-mail curciare@dm.unipi.it _______________________________________________ Settimanale mailing list Settimanale@mail.dm.unipi.it https://mail.dm.unipi.it/mailman/listinfo/settimanale