The exact law of large numbers via Fubini extension and characterization of insurable risks

2006 ◽  
Vol 126 (1) ◽  
pp. 31-69 ◽  
Author(s):  
Yeneng Sun
Keyword(s):  
Law Of Large Numbers ◽  
Fubini Extension ◽  
Large Numbers

scholarly journals A characterization of a new type of strong law of large numbers

2015 ◽  
Vol 368 (1) ◽  
pp. 539-561 ◽  
Author(s):  
Deli Li ◽  
Yongcheng Qi ◽  
Andrew Rosalsky
Keyword(s):  
Law Of Large Numbers ◽  
Strong Law ◽  
Large Numbers ◽  
New Type

Hyperfinite Law of Large Numbers

1996 ◽  
Vol 2 (2) ◽  
pp. 189-198 ◽  
Author(s):  
Yeneng Sun
Keyword(s):  
Law Of Large Numbers ◽  
Nonstandard Analysis ◽  
Random Variables ◽  
Triangular Array ◽  
Large Numbers ◽  
Loeb Space ◽  
Space Construction ◽  
Pairwise Independence ◽  
Necessary And Sufficient

AbstractThe Loeb space construction in nonstandard analysis is applied to the theory of processes to reveal basic phenomena which cannot be treated using classical methods. An asymptotic interpretation of results established here shows that for a triangular array (or a sequence) of random variables, asymptotic uncorrelatedness or asymptotic pairwise independence is necessary and sufficient for the validity of appropriate versions of the law of large numbers. Our intrinsic characterization of almost sure pairwise independence leads to the equivalence of various multiplicative properties of random variables.

Maximal Moment Inequalities for Weakly Negatively Dependent Variables

2019 ◽  
pp. 251-276
Author(s):  
Florence Merlevède ◽  
Magda Peligrad ◽  
Sergey Utev
Keyword(s):  
Law Of Large Numbers ◽  
Lipschitz Functions ◽  
Quadratic Variation ◽  
Partial Sums ◽  
Moment Inequalities ◽  
Large Numbers ◽  
Dependent Variables ◽  
Alternative Characterization ◽  
Maximal Moment

The aim of this chapter is to prove Khintchine–Marcinkiewicz–Zygmund or Rosenthal-type moment inequalities for the partial sums or for the maximum of partial sums, in terms of weak dependence coefficients. We start with general probability and moment inequalities for Lipschitz functions of weakly negatively dependent variables. Besides being of interest in themselves, they will be key tools for proving moment inequalities for the partial sums associated with weakly negatively dependent variables. Some parts of this chapter are devoted to the weak law of large numbers, useful to get the convergence of quadratic characteristics such as the quadratic variation and also to obtain an alternative characterization of the notion of weakly negatively dependent vectors.

Introduction

2017 ◽  
Author(s):  
Jochen Rau
Keyword(s):  
Probability Theory ◽  
Phase Transitions ◽  
Statistical Mechanics ◽  
Conservation Laws ◽  
Law Of Large Numbers ◽  
Macroscopic Scale ◽  
Classical Probability ◽  
Large Numbers ◽  
Microscopic World ◽  
New Phenomena

Statistical mechanics concerns the transition from the microscopic to the macroscopic realm. On a macroscopic scale new phenomena arise that have no counterpart in the microscopic world. For example, macroscopic systems have a temperature; they might undergo phase transitions; and their dynamics may involve dissipation. How can such phenomena be explained? This chapter discusses the characteristic differences between the microscopic and macroscopic realms and lays out the basic challenge of statistical mechanics. It suggests how, in principle, this challenge can be tackled with the help of conservation laws and statistics. The chapter reviews some basic notions of classical probability theory. In particular, it discusses the law of large numbers and illustrates how, despite the indeterminacy of individual events, statistics can make highly accurate predictions about totals and averages.

scholarly journals Limit theorems for multi-type general branching processes with population dependence

2020 ◽  
Vol 52 (4) ◽  
pp. 1127-1163
Author(s):  
Jie Yen Fan ◽  
Kais Hamza ◽  
Peter Jagers ◽  
Fima C. Klebaner
Keyword(s):  
Carrying Capacity ◽  
Population Model ◽  
General Framework ◽  
Limit Theorems ◽  
Mating Systems ◽  
Law Of Large Numbers ◽  
Branching Processes ◽  
Special Section ◽  
Large Numbers ◽  
Type Population

AbstractA general multi-type population model is considered, where individuals live and reproduce according to their age and type, but also under the influence of the size and composition of the entire population. We describe the dynamics of the population as a measure-valued process and obtain its asymptotics as the population grows with the environmental carrying capacity. Thus, a deterministic approximation is given, in the form of a law of large numbers, as well as a central limit theorem. This general framework is then adapted to model sexual reproduction, with a special section on serial monogamic mating systems.

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