A Cut-Invariant Law of Large Numbers for Random Heaps

We consider the framework of Bernoulli measures for heap monoids. We introduce in this framework the notion of asynchronous stopping time, which generalizes the notion of stopping time for classical probabilistic processes. A strong Bernoulli property is proved. A notion of cut-invariance is formulated for convergent ergodic means. Then, a version of the strong law of large numbers is proved for heap monoids with Bernoulli measures. We study a sub-additive version of the law of large numbers in this framework.     

Laws of large numbers for cooperative St. Petersburg gamblers

Summary General linear combinations of independent winnings in generalized \St~Petersburg games are interpreted as individual gains that result from pooling strategies of different cooperative players. A weak law of large numbers is proved for all such combinations, along with some almost sure results for the smallest and largest accumulation points, and a considerable body of earlier literature is fitted into this cooperative framework. Corresponding weak laws are also established, both conditionally and unconditionally, for random pooling strategies.     

Random Matching of Several Infinite Populations

Random matching models with infinite populations present formal difficulties due to the lack of a law of large numbers for a continuum of random variables. In a previous paper (Alós-Ferrer, 1999), random matching processes for a continuum of agents were studied, proving their existence. The present work extends the analysis to the case of several populations, which includes general equilibrium buyers–sellers models, and evolutionary models with underlying asymmetric games.     

Strong laws of large numbers for sub-linear expectations

We investigate three kinds of strong laws of large numbers for capacities with a new notion of independently and identically distributed(IID) random variables for sub-linear expectations initiated by Peng.It turns out that these theorems are natural and fairly neat extensions of the classical Kolmogorov’s strong law of large numbers to the case where probability measures are no longer additive. An important feature of these strong laws of large numbers is to provide a frequentist perspective on capacities.     

Ergodic properties of some piecewise-deterministic Markov process with application to gene expression modelling

We investigate a piecewise-deterministic Markov process with a Polish state space, whose deterministic behaviour between random jumps is governed by a finite number of semiflows. We provide tractable conditions ensuring a form of exponential ergodicity and the strong law of large numbers for the chain given by the post-jump locations. Further, we establish a one-to-one correspondence between invariant measures of the chain and those of the continuous-time process. These results enable us to derive the strong law of large numbers for the latter. The studied dynamical system is inspired by certain models of gene expression, which are also discussed here.     

Uniform Strong Law of Large Numbers

Methodology and Computing in Applied Probability - We prove the strong law of large numbers for random signed measures. The result is uniform over a family of subsets under mild assumptions.     

On the Critical Dimension and AC Entropy for Markov Odometers

Markov odometers are natural models for non-homogeneous Markov chains, and are natural generalisations of infinite product measures. We show how to calculate the critical dimension of these measures: this is an invariant which describes the asymptotic growth rate of sums of Radon-Nikodym derivatives. This interesting invariant appears to give a kind of entropy for non-singular odometer actions. The techniques require a law of large numbers for inhomogeneous Markov chains.     

On the Critical Dimension and AC Entropy for Markov Odometers

Markov odometers are natural models for non-homogeneous Markov chains, and are natural generalisations of infinite product measures. We show how to calculate the critical dimension of these measures: this is an invariant which describes the asymptotic growth rate of sums of Radon-Nikodym derivatives. This interesting invariant appears to give a kind of entropy for non-singular odometer actions. The techniques require a law of large numbers for inhomogeneous Markov chains.     

Limit Theorems for Non-Commutative Random Variables

We study the weak law of large numbers and the central limit theorem for non-commutative random variables. We first define the concepts of variance and expectation for probability measures on homogeneous spaces, and formulate the weak law of large numbers and the central limit theorem for probability measures on locally compact groups. Then, we consider the non-commutative case, where the homogeneous space is replaced by a C^*-algebra that is equipped with a locally compact group G of automorphisms. We define the concepts of variance and expectation in the non-commutative situation. Furthermore, we prove that the weak law of large numbers and the central limit theorem hold for non-commutative random variables on if they hold on the group G of automorphisms.     

正相协列乘积和的重对数律与强大数律

证明了强平稳正相协列乘积和的重对数律与不同分布正相协列乘积和的强大数律,指出了部分和服从强大数律但乘积和未必服从强大数律这一事实,并讨论了定理2中一个条件的必要性．     

随机变量序列的大数定律及常返性定理

对一类有界独立或相依的随机变量序列|ξn|,获得了它的伯努利大数定律、波雷尔强大数定律及常返性定理.作为应用,得出了Loève专著[1]中的推广的伯努利大数定律、常返性定理,改进了[1]中的推广的波雷尔强大数定律.     

关于渐近循环马氏链泛函的强大数定律

研究了一类非齐次马氏链———渐近循环马氏链泛函的强大数定律,首先引出了渐近循环马氏链的概念,然后给出了若干引理.利用了渐近循环马氏链关于状态序偶出现频率的强大数定理给出并证明了关于渐近循环马氏链泛函的强大数定律,所得定理作为推论可得到已有的结果.     

A generalized strong law of large numbers

Strong laws of large numbers have been stated in the literature for measurable functions taking on values on different spaces. In this paper, a strong law of large numbers which generalizes some previous ones (like those for real-valued random variables and compact random sets) is established. This law is an example of a strong law of large numbers for Borel measurable nonseparably valued elements of a metric space. Received: 24 February 1998 / Revised version: 3 January 1999     

Derived random measures

A derived random measure is constructed by integration of a random process with respect to a random measure independent of that process. Basic distributional properties, a continuity theorem, sample path properties, a strong law of large numbers, and a central limit theorem for derived random measures are established. Applications are given to compounding and thinning of point processes and the measure of a random set.     

Banach空间Teicher强大数定律的一般形式

本文研究取值在Banach空间中的随机变量序列的强收敛问题,在假设随机变量序列弱大数定律成立的条件下,证明了一般形式的Teicher强大数定律成立.     

（0,1）矩阵,组合竞赛与Ramsey问题

本文提出用（０，１）矩阵的乘法来研究Ｒａｍｓｅｙ问题，并结合组合竞赛观点给出一大类经典Ｒａｍｓｅｙ数的构造性（算法性）下界。     

φ混合序列的强大数律

讨论了不同分布φ混合序列的强大数律,推广了Kolmogorov强大数定律和Marcinkiewicz强大数定律.     

■混合序列的强大数律

讨论了不同分布■混合序列的强大数律,推广了Kolmogorov强大数定律和Marcinkiewicz强大数定律.     

An Extension of the Kolmogorov–Feller Weak Law of Large Numbers with an Application to the St. Petersburg Game

The Kolmogorov–Feller weak law of large numbers for i.i.d. random variables without finite mean is extended to a larger class of distributions, requiring regularly varying normalizing sequences. As an application we show that the weak law of large numbers for the St. Petersburg game is an immediate consequence of our result.     

On One Sufficient Condition for the Validity of the Strong Law of Large Numbers for Martingales

We prove a theorem on the strong law of large numbers for martingales. The existence of higher moments is not assumed. From the theorem proved, we deduce numerous well-known results on the strong law of large numbers both for martingales and for sequences of sums of independent random variables.