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Further results on laws of large numbers for uncertain random variables

Feng Hu, Xiaoting Fu, Ziyi Qu, Zhaojun Zong (2023)

Kybernetika

The uncertainty theory was founded by Baoding Liu to characterize uncertainty information represented by humans. Basing on uncertainty theory, Yuhan Liu created chance theory to describe the complex phenomenon, in which human uncertainty and random phenomenon coexist. In this paper, our aim is to derive some laws of large numbers (LLNs) for uncertain random variables. The first theorem proved the Etemadi type LLN for uncertain random variables being functions of pairwise independent and identically...

Further study on complete convergence for weighted sums of arrays of rowwise asymptotically almost negatively associated random variables

Haiwu Huang, Hanjun Zhang, Qingxia Zhang, Jiangyan Peng (2015)

Kybernetika

In this paper, the authors further studied the complete convergence for weighted sums of arrays of rowwise asymptotically almost negatively associated (AANA) random variables with non-identical distribution under some mild moment conditions. As an application, the Marcinkiewicz-Zygmund type strong law of large numbers for weighted sums of AANA random variables is obtained. The results not only generalize the corresponding ones of Wang et al. [19], but also partially improve the corresponding ones...

Gebelein's inequality and its consequences

M. Beśka, Z. Ciesielski (2006)

Banach Center Publications

Let ( X i , i = 1 , 2 , . . . ) be the normalized gaussian system such that X i ∈ N ( 0 , 1 ) , i = 1,2,... and let the correlation matrix ρ i j = E ( X i X j ) satisfy the following hypothesis: C = s u p i ≥ 1 ∑ j = 1 ∞ | ρ i , j | < ∞ . We present Gebelein’s inequality and some of its consequences: Borel-Cantelli type lemma, iterated log law, Levy’s norm for the gaussian sequence etc. The main result is that (f(X₁) + ⋯ + f(Xₙ))/n → 0 a.s. for f ∈ L¹(ν) with (f,1)ν = 0.

Growth-optimal portfolios under transaction costs

Jan Palczewski, Łukasz Stettner (2008)

Applicationes Mathematicae

This paper studies a portfolio optimization problem in a discrete-time Markovian model of a financial market, in which asset price dynamics depends on an external process of economic factors. There are transaction costs with a structure that covers, in particular, the case of fixed plus proportional costs. We prove that there exists a self-financing trading strategy maximizing the average growth rate of the portfolio wealth. We show that this strategy has a Markovian form. Our result is obtained...

Inequalities and limit theorems for random allocations

István Fazekas, Alexey Chuprunov, József Túri (2011)

Annales UMCS, Mathematica

Random allocations of balls into boxes are considered. Properties of the number of boxes containing a fixed number of balls are studied. A moment inequality is obtained. A merge theorem with Poissonian accompanying laws is proved. It implies an almost sure limit theorem with a mixture of Poissonian laws as limiting distribution. Almost sure versions of the central limit theorem are obtained when the parameters are in the central domain.

Józef Marcinkiewicz (1910-1940) - on the centenary of his birth

Lech Maligranda (2011)

Banach Center Publications

Józef Marcinkiewicz’s (1910-1940) name is not known by many people, except maybe a small group of mathematicians, although his influence on the analysis and probability theory of the twentieth century was enormous. This survey of his life and work is in honour of the 100 t h anniversary of his birth and 70 t h anniversary of his death. The discussion is divided into two periods of Marcinkiewicz’s life. First, 1910-1933, that is, from his birth to his graduation from the University of Stefan Batory in Vilnius,...

Józef Marcinkiewicz: analysis and probability

N. H. Bingham (2011)

Banach Center Publications

We briefly review Marcinkiewicz's work, on analysis, on probability, and on the interplay between the two. Our emphasis is on the continuing vitality of Marcinkiewicz's work, as evidenced by its influence on the standard works. What is striking is how many of the themes that Marcinkiewicz studied (alone, or with Zygmund) are very much alive today. What this demonstrates is that Marcinkiewicz and Zygmund, as well as having extraordinary mathematical ability, also had excellent mathematical taste.

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