Related papers: On the Law of Large Numbers for non-equally distri…
Let $\Omega$ be a countable infinite product $\Omega^\N$ of copies of the same probability space $\Omega_1$, and let ${\Xi_n}$ be the sequence of the coordinate projection functions from $\Omega$ to $\Omega_1$. Let $\Psi$ be a possibly…
We establish a strong law of large numbers for one-dimensional continuous-time random walks in dynamic random environments under two main assumptions: the environment is required to satisfy a decoupling inequality that can be interpreted as…
This paper introduces new notions of Fubini independence and Exponential independence of random variables under capacities to fit Ellsberg's model, and finds out the relations between Fubini independence, Exponential independence,…
We consider a population with non-overlapping generations, whose size goes to infinity. It is described by a discrete genealogy which may be time non-homogeneous and we pay special attention to branching trees in varying environments. A…
The sub-linear expectation space is a nonlinear expectation space having advantages of modelling the uncertainty of probability and distribution. In the sub-linear expectation space, we use capacity and sub-linear expectation to replace…
We obtain non-uniform Berry-Esseen type estimates for several classes of weakly dependent sequences of random variables, including uniformly elliptic inhomogeneous Markov chains, random and time-varying (partially) hyperbolic or expanding…
This paper introduces the notion of pseudo-independence on the sublinear expectation space $(\Omega,\mathcal{F},\mathcal{P})$ via the classical conditional expectation, and the relations between pseudo-independence and Peng's independence…
We give the proof of a tight lower bound on the probability that a binomial random variable exceeds its expected value. The inequality plays an important role in a variety of contexts, including the analysis of relative deviation bounds in…
The purpose of this paper is to establish a general strong law of large numbers (SLLN) for arbitrary sequences of random variables (rv's) based on the squared indice method and to provide applications to SLLN of associated sequences. This…
We present a Law of Large Numbers principle for uniformly continuous random quantum dynamical semigroups. Random iterates of independent copies of these semigroups are shown to be Chernoff equivalent to the quantum dynamical semigroup by…
We discuss the (twisted) weak positivity theorem. We also treat some applications.
We extend the random characteristics approach to Wigner matrices whose entries are not required to have a normal distribution. As an application, we give a simple and fully dynamical proof of the weak local semicircle law in the bulk.
In this paper, we investigate the law of large numbers for strictly stationary random fields, that is, we provide sufficient conditions on the moments and the dependence of the random field in order to guarantee the almost sure convergence…
A model is proposed for the statistical analysis of arbitrary-strength quantum measurements, based on a picture of "sampling weak values" from different configurations of the system. The model is comprised of two elements: a "local weak…
We present a new proof rule for verifying lower bounds on quantities of probabilistic programs. Our proof rule is not confined to almost-surely terminating programs -- as is the case for existing rules -- and can be used to establish…
We study the statistical properties of random numbers under the Martin-L\"of definition of randomness, proving that random numbers obey analogues of Strong Law of Large Numbers, the Law of the Iterated Logarithm, and that they are normal.…
In this paper, we obtain some results on precise large deviations for non-random and random sums of widely dependent random variables with common dominatedly varying tail distribution or consistently varying tail distribution on…
In this note, three Lagrange multiplier rules introduced in the literature for set valued optimization problems are compared. A generalization of all three results is given which proves that under rather mild assumptions, $x$ is a weak…
Limit theorems of strong law of large numbers and central limit theorem types are obtained for the compositions of independent identically distributed random unitary channels.
We formulate conditions for convergence of Laws of Large Numbers and show its links with of the parts of mathematical analysis such as summation theory, convergence of orthogonal series. We present also applications of the Law of Large…