[CvGmt News] [Notiziario] [Settimanale] AVVISI SEMINARI

[Giulia Curciarello]

 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 de Grandi Numeri e Convergenza Completa
2. Teoremi Limite per  il "bootstrap" dipendente della media

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Legge de 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.

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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.





Aula e orario da comunicare





Giulia Curciarello
Segreteria Didattica
tel: 050-2213219
e-mail curciare at dm.unipi.it

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