E. Ostrovsky, L. Sirota
We deduce the non-asymptotical bilateral estimates for moment inequalities for sums of non-negative independent random variables, based on the correspondent estimates for the so-called Bell functions and the Poisson distribution.
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Probabilistic bivariate Bell polynomials
本文研究了矩母函数存在的随机变量相关的概率二元Bell多项式与概率二元r-Bell多项式，并推导出对应于Zheng和Li所得递推关系的递推关系。
Probabilistic heterogeneous Stirling numbers and Bell polynomials
本文提出概率异质斯特林数和贝尔多项式，统一了多个经典和概率性的数学家族，并推导了显式公式和递推关系，应用于泊松和伯努利分布。
We question the commonly accepted statement that random numbers certified by Bell's theorem carry some special sort of randomness, so to say, quantum randomness or intrinsic randomness. We show that such numbers can be easily generated by classical random generators.
Probabilistic Lah numbers and Lah-Bell polynomials
本文研究了矩母函数存在的随机变量Y所对应的概率Lah数与概率Lah-Bell多项式，推导了其性质、显式表达式、递推关系和恒等式，并探讨了Poisson与Bernoulli随机变量的特殊情形。
Probabilistic degenerate poly-Bell polynomials associated with random variables
本文研究与随机变量相关的概率退化多贝尔多项式，推导其显式表达式与恒等式，并讨论伯努利和伽马随机变量的特殊情形。
For a centered $d$-dimensional Gaussian random vector $ξ=(ξ_1,\ldots,ξ_d)$ and a homogeneous function $h:R^d\to R$ we derive asymptotic expansions for the tail of the Gaussian chaos $h(ξ)$ given the function $h$ is sufficiently smooth. Three challenging instances of the Gaussian chaos are the determinant of a Gaussian matrix, the Gaussian orthogonal ensemble and the diameter of random Gaussian clouds. Using a direct probabilistic asymptotic method, we investigate both the asymptotic behaviour of the tail distribution of $h(ξ)$ and its density at infinity and then discuss possible extensions fo…
We study concentration properties of vertex degrees of $n$-dimensional Erdos-Rényi random graphs with the edge probability $ρ/n$ by means of high moments of these random variables in the limit when $n$ and $ρ$ tend to infinity. These moments are asymptotically close to one-variable Bell polynomials ${\cal B}_k(ρ), k\in {\bf N}$ that represent moments of the Poisson probability distribution ${\cal P}(ρ)$. We study asymptotic behavior of the Bell polynomials and modified Bell polynomials for large values of $k$ and $ρ$ with the help of the local limit theorem for auxiliary random variables. Usin…
We present a new probabilistic analysis of distributed algorithms. Our approach relies on the theory of quasi-stationary distributions (QSD) recently developped by Champagnat and Villemonais. We give properties on the deadlock time and the distribution of the model before deadlock, both for discrete and diffusion models. Our results are non-asymptotic since they apply to any finite values of the involved parameters (time, numbers of resources, number of processors, etc.) and reflect the real behavior of these algorithms, with potential applications to deadlock prevention, which are very import…
A theorem of Meinardus provides asymptotics of the number of weighted partitions under certain assumptions on associated ordinary and Dirichlet generating functions. The ordinary generating functions are closely related to Euler's generating function $\prod_{k=1}^\infty S(z^k)$ for partitions, where $S(z)=(1-z)^{-1}$. By applying a method due to Khintchine, we extend Meinardus' theorem to find the asymptotics of the coefficients of generating functions of the form $\prod_{k=1}^\infty S(a_kz^k)^{b_k}$ for sequences $a_k$, $b_k$ and general $S(z)$. We also reformulate the hypotheses of the theor…
Our aim is to emphasize the role of mathematical models in physics, especially models of geometry and probability. We briefly compare developments of geometry and probability by pointing to similarities and differences: from Euclid to Lobachevsky and from Kolmogorov to Bell. In probability Bell played the same role as Lobachevsky in geometry. In fact, violation of Bell's inequality implies the impossibility to apply the classical probability model of Kolmogorov (1933) to quantum phenomena. Thus quantum probabilistic model (based on Born's rule) is an example of non-Kolmogorovian model of proba…
Asymptotics deviation probabilities of the sum S n = X 1 + $\times$ $\times$ $\times$ + X n of independent and identically distributed real-valued random variables have been extensively investigated, in particular when X 1 is not exponentially integrable. For instance, A.V. Nagaev formulated exact asymptotics results for P(S n > x n) when x n > n 1/2 (see, [13, 14]). In this paper, we derive rough asymptotics results (at logarithmic scale) with shorter proofs relying on classical tools of large deviation theory and expliciting the rate function at the transition.
Let $LA_{n}(τ)$ be the length of the longest alternating subsequence of a uniform random permutation $τ\in[n]$. Classical probabilistic arguments are used to rederive the asymptotic mean, variance and limiting law of $LA_{n}(τ)$. Our methodology is robust enough to tackle similar problems for finite alphabet random words or even Markovian sequences in which case our results are mainly original. A sketch of how some cases of pattern restricted permutations can also be tackled with probabilistic methods is finally presented.
A Framework for Asymptotic Limit Problems of Probabilistic Nature
本文提出以有限多重集关联原子概率测度并利用依分布收敛统一处理各类概率型渐近极限问题的通用框架，并通过示例说明其用法。
Cyclic systems of dichotomous random variables have played a prominent role in contextuality research, describing such experimental paradigms as the Klyachko-Can-Binicoglu-Shumovky, Einstein-Podolsky-Rosen-Bell, and Leggett-Garg ones in physics, as well as conjoint binary choices in human decision making. Here, we understand contextuality within the framework of the Contextuality-by-Default (CbD) theory, based on the notion of probabilistic couplings satisfying certain constraints. CbD allows us to drop the commonly made assumption that systems of random variables are consistently connected. C…
We consider the value function of a stochastic optimal control of degenerate diffusion processes in a domain $D$. We study the smoothness of the value function, under the assumption of the non-degeneracy of the diffusion term along the normal to the boundary and an interior condition weaker than the non-degeneracy of the diffusion term. When the diffusion term, drift term, discount factor, running payoff and terminal payoff are all in the class of $C^{1,1}(\bar D)$, the value function turns out to be the unique solution in the class of $C_{loc}^{1,1}(D)\cap C^{0,1}(\bar D)$ to the associated d…
Contextuality, Complementarity, Signaling, and Bell tests
本文综述了互补性与语境性的相互作用及其与贝尔不等式的联系，指出贝尔不等式是对语境性的统计检验，并探讨了信号传递的影响及默认语境性理论。
Quantum violation of Bell inequalities is now used in many quantum information applications and it is important to analyze it both quantitatively and conceptually. In the present paper, we analyze violation of multipartite Bell inequalities via the local probability model - the LqHV (local quasi hidden variable) model [Loubenets, J. Math. Phys. 53, 022201 (2012)], incorporating the LHV model only as a particular case and correctly reproducing the probabilistic description of every quantum correlation scenario, more generally, every nonsignaling scenario. The LqHV probability framework allows u…
"Local Realism", Bell's Theorem and Quantum "Locally Realistic" Inequalities
基于新的一般框架的概率描述的实验，介绍了在quant-ph/0305126，quant-ph/0312199，我们分析了在数学方面的有效性贝尔型不等式联合实验后，任何类型的系统和物理概念的“本地现实主义”之间的联系。我们证明了量子情形下Bell型不等式的破坏与“局域实在论”的破坏无关。在一般情况下，我们制定了数学术语的条件下，“局部现实主义”的联合实验，并考虑量子“局部现实主义”的联合实验的例子。我们，特别是，表明量子联合实验的爱丽丝/鲍勃类型是“本地现实”。对于任意的二分量子态，我们导出了原来的贝尔不等式的量子类似。根据我们的结果，我们认为，贝尔型不等式在量子情况下的破坏不能是一个有效的参数在讨论局域性或非局域性的量子相互作用。
New probabilistic methods for physics
本文提出了一种新的概率方法，用于描述迭代的渐近行为，并应用于常微分方程和力学问题的逼近。
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