Rita Giuliano Antonini

On the rate of complete convergence for weighted sums of arrays of Banach space valued random elements with application to moving average processes

We obtain complete convergence results for arrays of rowwise independent Banach space valued random elements. In the main result no assumptions are made concerning the geometry of the underlying Banach space. As corollaries we obtain a result on complete convergence in stable type p Banach spaces and on the complete convergence of moving average processes.

Topography of Gng2- and NetrinG2-Expression Suggests an Insular Origin of the Human Claustrum

The claustrum has been described in the forebrain of all mammals studied so far. It has been suggested that the claustrum plays a role in the integration of multisensory information: however, its detailed structure and function remain enigmatic. The human claustrum is a thin, irregular, sheet of grey matter located between the inner surface of the insular cortex and the outer surface of the putamen. Recently, the G-protein gamma2 subunit (Gng2) was proposed as a specific claustrum marker in the rat, and used to better delineate its anatomical boundaries and connections. Additional claustral markers proposed in mammals include Netrin-G2 in the monkey and latexin in the cat. Here we report the expression and distribution of Gng2 and Netrin-G2 in human post-mortem samples of the claustrum and adjacent structures. Gng2 immunoreactivity was detected in the neuropil of the claustrum and of the insular cortex but not in the putamen. A faint labelling was present also in the external and extreme capsules. Double-labelling experiments indicate that Gng2 is also expressed in glial cells. Netrin-G2 labelling was seen in neuronal cell bodies throughout the claustrum and the insular cortex but not in the medially adjacent putamen. No latexin immunoreactive element was detected in the claustrum or adjacent structures. Our results confirm that both the Gng2 and the Netrin-G2 proteins show an affinity to the claustrum and related formations also in the human brain. The presence of Gng2 and Netrin-G2 immunoreactive elements in the insular cortex, but not in the putamen, suggests a possible common ontogeny of the claustrum and insula.

On the Strong Rates of Convergence for Arrays of Rowwise Negatively Dependent Random Variables

The strong convergence rate and complete convergence results for arrays of rowwise negatively dependent random variables are established. The results presented generalize the results of Chen et al. [1] and Sung et al. [2]. As applications, some well-known results on independent random variables can be easily extended to the case of negatively dependent random variables.

ON THE COMPLETE CONVERGENCE FOR ARRAYS OF ROWWISE EXTENDED NEGATIVELY DEPENDENT RANDOM VARIABLES

Abstract. A general result for the complete convergence of arrays ofrowwise extended negatively dependent random variables is derived. Asits applications eight corollaries for complete convergence of weightedsums for arrays of rowwise extended negatively dependent random vari-ables are given, which extend the corresponding known results for inde-pendent case. 1. IntroductionThe concept of complete convergence of a sequence of random variables wasintroduced by Hsu and Robbins ([5]) as follows. A sequence {U n ,n≥ 1} ofrandom variables converges completely to the constant θifX ∞n=1 P{|U n −θ| >ǫ} 0.Moreover, they proved that the sequence of arithmetic means of independentidentically distribution (i.i.d.) random variables converges completely to theexpected value if the variance of the summands is finite. This result has beengeneralized and extended in several directions, see Gut ([3], [4]), Hu et al. ([7],[8]), Chen et al. ([2]), Sung ([14], [15], [17]), Zarei and Jabbari ([20]), Baek etal. ([1]). In particular, Sung ([14]) obtained the following two Theorems A andB.Theorem A. Let {X

The Intersective ASCLT

Abstract In a recent work of the second named author on the Almost Sure Central Limit Theorem (ASCLT), we showed the usefulness of the concept of quasi-orthogonal system of random variables introduced by Bellman and later developed by Kac, Salem and Zygmund. In this paper, we propose an optimal formulation of the ASCLT again by using this idea and new correlation inequalities for sums of independent random variables. We also introduce and develop the notion of “intersective ASCLT” by proving some new results generalizing and improving substancially the classical formulation of the ASCLT. Essential tools for this approach are correlation inequalities recently developed by the first named author and some extensions of these ones obtained in the present paper.

ON THE STRONG CONVERGENCE OF WEIGHTED SUMS

Let {X n ,n≥1} be a sequence of independent and identically distributed random variables and {a ni ,1≤i≤n,n≥1} an array of constants. Some strong convergence results for the weighted sums ∑ i=1 n a ni X i are obtained.

Limiting behaviour of moving average processes under ρ-mixing assumption

Let be a doubly infinite sequence of identically distributed -mixing random variables, an absolutely summable sequence of real numbers. In this paper, we prove the complete convergence and Marcinkiewicz-Zygmund strong law of large numbers for the partial sums of the moving average processes .

A strong law of large numbers for generalized almost sure central limit theorems

We prove a Strong Law of Large Numbers in which the variables are assumed to be asymptotically negligible and a generalized Almost Sure Central Limit Theorem is given. As an application we obtain a result about the so-called intersective ASCLT.