Moderate deviation principle for $ m $-dependent random variables under the sub-linear expectation

<abstract><p>Let $ \{X_n, n\geq1\} $ be a sequence of $ m $-dependent strictly stationary random variables in a sub-linear expectation $ (\Omega, \mathcal{H}, \mathbb{E}) $. In this article, we give the definition of $ m $-dependent sequence of random variables under sub-linear expectation spaces taking values in $ \mathbb{R} $. Then we establish moderate deviation principle for this kind of sequence which is strictly stationary. The results in this paper generalize the result that in the case of independent identically distributed samples. It provides a basis to discuss the moderate deviation principle for other types of dependent sequences.</p></abstract>

The Buffon needle problem for Lévy distributed spacings and renewal theory

What is the probability that a needle dropped at random on a set of points scattered on a line segment does not fall on any of them? We compute the exact scaling expression of this hole probability when the spacings between the points are independent identically distributed random variables with a power-law distribution of index less than unity, implying that the average spacing diverges. The theoretical framework for such a setting is renewal theory, to which the present study brings a new contribution. The question posed here is also related to the study of some correlation functions of simple models of statistical physics.

Some strong limit theorems for weighted sums of measurable operators

The aim of this study is to provide some strong limit theorems for weighted sums of measurable operators. The almost uniform convergence and the bilateral almost uniform convergence are considered. As a result, we derive the strong law of large numbers for sequences of successively independent identically distributed measurable operators without using the noncommutative version of Kolmogorov’s inequality.

Characterization of the Geometric Distribution Via Linear Combinations of Observations and of Records

In a sequence of independent identically distributed geometric random variables, the sum of the first two record values is distributed as a simple linear combination of geometric variables. It is verified that this distributional property characterizes the geometric distribution. A related characterization conjecture is also discussed. Related discussion in the context of weak records is also provided.

Spherical Distributions Used in Evolutionary Algorithms

Performance of evolutionary algorithms in real space is evaluated by local measures such as success probability and expected progress. In high-dimensional landscapes, most algorithms rely on the normal multi-variate, easy to assemble from independent, identically distributed components. This paper analyzes a different distribution, also spherical, yet with dependent components and compact support: uniform in the sphere. Under a simple setting of the parameters, two algorithms are compared on a quadratic fitness function. The success probability and the expected progress of the algorithm with uniform distribution are proved to dominate their normal mutation counterparts by order n!!.

On Some Laws of Large Numbers for Uncertain Random Variables

Baoding Liu created uncertainty theory to describe the information represented by human language. In turn, Yuhan Liu founded chance theory for modelling phenomena where both uncertainty and randomness are present. The first theory involves an uncertain measure and variable, whereas the second one introduces the notions of a chance measure and an uncertain random variable. Laws of large numbers (LLNs) are important theorems within both theories. In this paper, we prove a law of large numbers (LLN) for uncertain random variables being continuous functions of pairwise independent, identically distributed random variables and regular, independent, identically distributed uncertain variables, which is a generalisation of a previously proved version of LLN, where the independence of random variables was assumed. Moreover, we prove the Marcinkiewicz–Zygmund type LLN in the case of uncertain random variables. The proved version of the Marcinkiewicz–Zygmund type theorem reflects the difference between probability and chance theory. Furthermore, we obtain the Chow type LLN for delayed sums of uncertain random variables and formulate counterparts of the last two theorems for uncertain variables. Finally, we provide illustrative examples of applications of the proved theorems. All the proved theorems can be applied for uncertain random variables being functions of symmetrically or asymmetrically distributed random variables, and symmetrical or asymmetrical uncertain variables. Furthermore, in some special cases, under the assumption of symmetry of the random and uncertain variables, the limits in the first and the third theorem have forms of symmetrical uncertain variables.

Equivalent Conditions of Complete p th Moment Convergence for Weighted Sums of I. I. D. Random Variables under Sublinear Expectations

We investigate the complete p th moment convergence for weighted sums of independent, identically distributed random variables under sublinear expectations space. Using moment inequality and truncation methods, we prove the equivalent conditions of complete p th moment convergence of weighted sums of independent, identically distributed random variables under sublinear expectations space, which complement the corresponding results obtained in Guo and Shan (2020).

Exponential Growth of Products of Non-Stationary Markov-Dependent Matrices

Let $(\xi _j)_{j\ge 1} $ be a non-stationary Markov chain with phase space $X$ and let $\mathfrak {g}_j:\,X\mapsto \textrm {SL}(m,{\mathbb {R}})$ be a sequence of functions on $X$ with values in the unimodular group. Set $g_j=\mathfrak {g}_j(\xi _j)$ and denote by $S_n=g_n\ldots g_1$, the product of the matrices $g_j$. We provide sufficient conditions for exponential growth of the norm $\|S_n\|$ when the Markov chain is not supposed to be stationary. This generalizes the classical theorem of Furstenberg on the exponential growth of products of independent identically distributed matrices as well as its extension by Virtser to products of stationary Markov-dependent matrices.

Random Power Series in Q p , 0 Spaces

Aulaskari et al. proved if 0 < p < 1 and ε n is sequence of independent, identically distributed Rademacher random variables on a probability space, then the condition Σ n = 0 ∞ n 1 − p a n 2 < ∞ implies that the random power series R f z = ∑ n = 0 ∞ a n ε n z n ∈ Q p almost surely. In this paper, we improve this result showing that the condition Σ n = 0 ∞ n 1 − p a n 2 < ∞ actually implies R f ∈ Q p , 0 almost surely.

Convergence for sums of i.i.d. random variables under sublinear expectations

In this paper, we obtain equivalent conditions of complete moment convergence of the maximum for partial weighted sums of independent identically distributed random variables under sublinear expectations space. The results obtained in the paper are extensions of the equivalent conditions of complete moment convergence of the maximum under classical linear expectation space.