B. L. S. Prakasa Rao
Prakasa Rao (Studia Sci. Math. Hungar. 11 (1976) 277-282) studied a characterization of probability distributions for linear functions of independent generalized random fields. These results are extended to Q-independent generalized random fields. It is known that independence implies Q-independence but the converse is not true.
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Properties of the beta functions are investigated. We define the generalized arcsine probability distribution with bounded support. The properties of the beta functions prove some results for this distribution.
Generalized Divergence Measures and Weak Convergence for the Sets of Probability Measures
该研究将Kullback-Leibler散度和Jensen-Shannon散度推广到概率测度集情形，推导其基本性质与收敛结果，还得到次线性期望下的弱收敛结论。
On some extensions of generalized counting processes
本文研究了泊松过程和广义计数过程的分数扩展，通过引入由稳定分布和温度稳定子ordinator求和的逆变换时间变化，推导了概率分布和生成函数的控制方程，并给出了几种模型的概率分布和生成函数的闭式表达。
To a higher Landau Level corresponds a generalization of the Poisson distribution arising from generalized coherent states. In this paper, we write down the atomic decomposition of this probability measure and expressed its weights through 2F2 hypergeometric polynomials. Then, we prove that it is not infinitely divisible in opposite to the Poisson distribution corresponding to the lowest Landau level. We also derive the Levy-Kintchine representation of its characteristic function when the latter does not vanish and deduce that the representative measure is signed. By considering the total vari…
We consider analytic continuations of Fourier transforms and Stieltjes transforms. This enables us to define what we call complex moments for some class of probability measures which do not have moments in the usual sense. There are two ways to generalize moments accordingly to Fourier and Stieltjes transforms; however these two turn out to coincide. As applications, we give short proofs of the convergence of probability measures to Cauchy distributions with respect to tensor, free, Boolean and monotone convolutions.
We study existence of probability measure valued jump-diffusions described by martingale problems. We develop a simple device that allows us to embed Wasserstein spaces and other similar spaces of probability measures into locally compact spaces where classical existence theory for martingale problems can be applied. The method allows for general dynamics including drift, diffusion, and possibly infinite-activity jumps. We also develop tools for verifying the required conditions on the generator, including the positive maximum principle and certain continuity and growth conditions. To illustra…
It is known that the class $\mathcal{U}_β$, of generalized s-selfdecom-posable probability distributions, can be viewed as an image via random integral mapping $\mathcal{J}^β$ of the class $ID$ of all infinitely divisible measures. We prove that a composition of the mappings $\mathcal{J}^{β_1}, \mathcal{J}^{β_2}, ..., \mathcal{J}^{β_n}$ is again random integral mapping but with a new inner time. In a proof some form of Lagrange interpolation formula is needed. Moreover, some elementary formulas concerning the distributions of products of powers of independent uniformly distributed random varia…
We extend the result of Nualart and Schoutens on chaotic decomposition of the $L^2$-space of a Lévy process to the case of a generalized stochastic processes with independent values.
We are interested in constructing concrete independent events in purely atomic probability spaces with geometric distribution. Among other facts we prove that there are uncountable many sequences of independent events.
Exponential twist of probability measures: drift correction in term of a generalized gradient
本文研究了马尔可夫参考概率测度的指数扭折，即路径积分的指数测度变换。在非常一般的马尔可夫假设下，完全刻画了指数扭折概率测度作为鞅问题的解，并证明其继承了参考测度的马尔可夫性质。鞅问题的生成元显示漂移依赖于某个合适价值函数的广义梯度。
The Proof of Convergence with Probability 1 in the Method of Expansion of Iterated Ito Stochastic Integrals Based on Generalized Multiple Fourier Series
本文证明了基于广义多傅里叶级数的迭代伊藤随机积分展开方法以概率1收敛的定理，并分析了多傅里叶-勒杰恩德系列和多三角傅里叶级数的收敛性。
This note gathers what is known about, and provides some new results concerning the operations of intersection, of ``generated $σ$-field'', and of ``complementation'' for (independent) complete $σ$-fields on probability spaces.
We investigate the universality of singular value and eigenvalue distributions of matrix valued functions of independent random matrices and apply these general results in several examples. In particular we determine the limit distribution and prove universality under general conditions for singular value and eigenvalue distributions of products of independent matrices from spherical ensembles.
We analyze a class of continuous time random walks in $\mathbb R^d,d\geq 2,$ with uniformly distributed directions. The steps performed by these processes are distributed according to a generalized Dirichlet law. Given the number of changes of orientation, we provide the analytic form of the probability density function of the position $\{\underline{\bf X}_d(t),t>0\}$ reached, at time $t>0$, by the random motion. In particular, we analyze the case of random walks with two steps. In general, it is an hard task to obtain the explicit probability distributions for the process $\{\underline{\bf X}…
In this paper, we introduce a generalization of the Wasserstein barycenter, to a case where the initial probability measures live on different subspaces of R^d. We study the existence and uniqueness of this barycenter, we show how it is related to a larger multi-marginal optimal transport problem, and we propose a dual formulation. Finally, we explain how to compute numerically this generalized barycenter on discrete distributions, and we propose an explicit solution for Gaussian distributions.
We give sharp, uniform estimates for the probability that the empirical distribution function for n uniform-[0,1] random variables stays to one side of a given line.
This paper obtains an asymptotic formula for the finite-time ruin probability of the compound nonhomogeneous Poisson risk model with a constant interest force, in which the claims are conditionally independent random variables with a common subexponential distribution. The paper also obtains some asymptotic relations of randomly weighted sums $\sum_{i=1}^n θ_iX_i$, in which the weights $θ_i$ $i=1,2,\cdots, n$ are positive random variables which are bounded above and the primary random variables $X_i$, $i=1,2,\cdots,n$ are conditionally independent and follow subexponential distributions.
Generalized Birth-Death Process on Finite Lattice
本文研究有限格上的广义生灭过程，证明其转移概率矩阵在适当条件下可对角化从而高效计算$k$步转移概率，并推导出多维网格下转移概率矩阵可交换的充要条件及最小约束数。
A family of generalized binomial probability distributions attached to Landau levels on the Riemann sphere is introduced by constructing a kind of generalized coherent states. Their main statistical parameters are obtained explicitly. As application, photon number statistics related to coherent states under consideration are discussed.
The Continuous Rank Probability Score of a Generalized Beta-Prime Distribution and Some Special Cases
本文推导了广义贝塔素分布及 Dagum 和 Singh-Maddala 分布的连续等级概率分数，并与蒙特卡洛估计进行比较。
Generalized Bratteli diagrams without probability tail invariant measures
本文研究广义Bratteli图尾等价关系无概率不变测度的条件，给出基于关联矩阵的充分条件，并提出新谱方法解决开放问题。
A collection of $n$ random events is said to be $(n - 1)$-wise independent if any $n - 1$ events among them are mutually independent. We characterise all probability measures with respect to which $n$ random events are $(n - 1)$-wise independent. We provide sharp upper and lower bounds on the probability that at least $k$ out of $n$ events with given marginal probabilities occur over these probability measures. The bounds are shown to be computable in polynomial time.
The method of \emph{random integral representation}, that is, the method of representing a given probability measure as the probability distribution of some random integral, was quite successful in the past few decades. In this note we will find such a representation for generalized s-selfdecomposable and selfdecomposable distributions that have the \emph{factorization property}. These are the classes $\mathcal{U}^f_{\be}$ and $L^f$, respectively.
The method of \emph{random integral representation}, that is, the method of representing a given probability measure as the probability distribution of some random integral, was quite successful in the past few decades. In this note we will find such a representation for generalized s-selfdecomposable and selfdecomposable distributions that have the \emph{factorization property}. These are the classes $\mathcal{U}^f_{\be}$ and $L^f$, respectively
In this paper, we define the generalized Wasserstein distance for sets of Borel probability measures and demonstrate that the weak convergence of sublinear expectations can be characterized by means of this distance.
On a characterization of probability distribution based on maxima of independent or max-independent random variables
本文拓展了Kotlarski的经典分布识别结果，将基于单共享变量最大值的分布刻画推广到含两个同分布共享变量、带已知正常数系数的最大值情形，覆盖独立与max-独立随机变量两类设定。
Multivariate Generalized Counting Process via Gamma Subordination
本文提出了一种通过伽马子ordination的多元广义计数过程，研究了其数学性质及在冲击模型中的应用。
For two families of beta distributions, we show that the generalized Stieltjes transforms of their elements may be written as elementary functions (powers and fractions) of the Stieltjes transform of the Wigner distribution. In particular, we retrieve the examples given by the author in a previous paper and relating generalized Stieltjes transforms of special beta distributions to powers of (ordinary) Stieltjes ones. We also provide further examples of similar relations which are motivated by the representation theory of symmetric groups. Remarkably, the power of the Stieltjes transform of the…
This paper studies the properties of the probability density function $p_{α,ν, n}(\mathbf{x})$ of the $n$-variate generalized Linnik distribution whose characteristic function $φ_{α,ν,n}(\boldsymbol{t})$ is given by φ_{α,ν,n}(\boldsymbol{t})=\frac{1} {(1+\Vert\boldsymbol{t}\Vert^α)^ν}, α\in (0,2], ν>0, \boldsymbol{t}\in \mathbb{R}^n, where $\Vert\boldsymbol{t}\Vert$ is the Euclidean norm of $\boldsymbol{t}\in\mathbb{R}^n$. Integral representations of $p_{α,ν, n}(\mathbf{x})$ are obtained and used to derive the asymptotic expansions of $p_{α,ν, n}(\mathbf{x})$ when $\Vert\mathbf{x}\Vert\to 0$ a…
Let $V(k)$ denote the waiting time, the number of trials needed to get a consecutive $k$ ones. We propose recurrence algorithms for the probability distribution function (pdf) and the probability generating function (pgf) of $V(k)$ in sequences of independent and Markov dependent Bernoulli trials using generalized Fibonacci sequences of order $k$. Maximum likelihood estimation (MLE) methods for the probability distributions are presented in both cases with simulation examples.
Unlike classical and free independence, the boolean and monotone notions of independence lack of the property of independent constants. In the scalar case, this leads to restrictions for the central limit theorems, as observed by F. Oravecz. We characterize the property of independent constants from a combinatorial point of view, based on cumulants and set partitions. This characterization also holds for the operator-valued extension. Our considerations lead rather directly to very mild variations of boolean and monotone cumulants, where constants are now independent. These alternative probabi…
A family of probability distributions attached to a class of generalized weighted Bergman spaces on the Poincaré disk are introduced by constructing a kind of generalized coherent states. Their main statistical parameters are obtained explicitly. As application, photon number statistics related to coherent states under consideration are discussed.
We consider the problem of finding the optimal upper bound for the tail probability of a sum of $k$ nonnegative, independent and identically distributed random variables with given mean $x$. For $k=1$ the answer is given by Markov's inequality and for $k=2$ the solution was found by Hoeffding and Shrikhande in 1955. We solve the problem for $k=3$ as well as for general $k$ and $x\leq1/(2k-1)$ by showing that it follows from the fractional version of an extremal graph theory problem of Erdős on matchings in hypergraphs.
In the context of bounding probability of small deviation, there are limited general tools. However, such bounds have been widely applied in graph theory and inventory management. We introduce a common approach to substantially sharpen such inequality bounds by combining the semidefinite optimization approach of moments problem and the Berry-Esseen theorem. As an application, we improve the lower bound of Feige's conjecture from 0.14 to 0.1798.
This paper studies variance functions of Cauchy-Stieltjes Kernel families generated by compactly supported centered probability measures. We describe several operations that allow us to construct additional variance functions from known ones. We construct a class of examples which exhausts all cubic variance functions, and provide examples of polynomial variance functions of arbitrary degree. We also relate Cauchy-Stieltjes Kernel families with polynomial variance functions to generalized orthogonality. Our main results are stated solely in terms of classical probability; some proofs rely on a…
On some characterizations of probability distributions based on maxima or minima of some families of dependent random variables
本文研究 max-independent、min-independent 与 quasi-independent 随机变量极值性质对概率分布的刻画，推广了已有独立随机变量情形下的结果。
In this Book we argue that the fruitful interaction of computer vision and belief calculus is capable of stimulating significant advances in both fields. From a methodological point of view, novel theoretical results concerning the geometric and algebraic properties of belief functions as mathematical objects are illustrated and discussed in Part II, with a focus on both a perspective 'geometric approach' to uncertainty and an algebraic solution to the issue of conflicting evidence. In Part III we show how these theoretical developments arise from important computer vision problems (such as ar…
Splitting AVF method for generalized Langevin equations: probability density function and geometric ergodicity
本文提出一种结构保持的分裂平均向量场方法，用于求解广义 Langevin 方程，该方法在保持指数可积性、Malliavin 可微性和ergodic性的同时，保证了概率密度函数的存在性和光滑性，并通过数值实验验证了几何ergodic性。
A New Family of Fractional Counting Probability Distributions
该研究提出基于三参数广义Mittag-Leffler函数的新型分数阶计数过程族，拓展出多种分数阶组合数与量子相干态，其特例可复现多种经典及分数阶分布、组合数与相干态的已知结果。
A k-wise independent distribution on n bits is a joint distribution of the bits such that each k of them are independent. In this paper we consider k-wise independent distributions with identical marginals, each bit has probability p to be 1. We address the following question: how high can the probability that all the bits are 1 be, for such a distribution? For a wide range of the parameters n,k and p we find an explicit lower bound for this probability which matches an upper bound given by Benjamini et al., up to multiplicative factors of lower order. The question we investigate can be seen a…
The goal of this paper is to analyse the asymptotic behavior of the cycle process and the total number of cycles of weighted and generalized weighted random permutations which are relevant models in physics and which extend the Ewens measure. We combine tools from combinatorics and complex analysis (e.g. singularity analysis of generating functions) to prove that under some analytic conditions (on relevant generating functions) the cycle process converges to a vector of independent Poisson variables and to establish a central limit theorem for the total number of cycles. Our methods allow us t…
In this work we present expansions of intersection local times of fractional Brownian motions in $\R^d$, for any dimension $d\geq 1$, with arbitrary Hurst coefficients in $(0,1)^d$. The expansions are in terms of Wick powers of white noises (corresponding to multiple Wiener integrals), being well-defined in the sense of generalized white noise functionals. As an application of our approach, a sufficient condition on $d$ for the existence of intersection local times in $L^2$ is derived, extending the results of D. Nualart and S. Ortiz-Latorre in "Intersection Local Time for Two Independent Frac…
We study the eigenvectors of generalized Wigner matrices with subexponential entries and prove that they delocalize at the optimal rate with overwhelming probability. We also prove high probability delocalization bounds with sharp constants. Our proof uses an analysis of the eigenvector moment flow introduced by Bourgade and Yau (2017) to bound logarithmic moments of eigenvector entries for random matrices with small Gaussian components. We then extend this control to all generalized Wigner matrices by comparison arguments based on a framework of regularized eigenvectors, level repulsion, and …
We generalize and improve results of Andrews, Gravner, Holroyd, Liggett, and Romik on metastability thresholds for generalized two-dimensional bootstrap percolation models, and answer several of their open problems and conjectures. Specifically, we prove slow convergence and localization bounds for Holroyd, Liggett, and Romik's k-percolation models, and in the process provide a unified and improved treatment of existing results for bootstrap, modified bootstrap, and Frobose percolation. Furthermore, we prove improved asymptotic bounds for the generating functions of partitions without k-gaps, …
In this paper, we introduce a generalized composite fading distribution (termed extended generalized-K (EGK)) to model the envelope and the power of the received signal in millimeter wave (60 GHz or above) and free-space optical channels. We obtain the first and the second-order statistics of the received signal envelope characterized by the EGK composite fading distribution. In particular, expressions for probability density function, cumulative distribution function, level crossing rate and average fade duration, and fractional moments are derived. In addition performance measures such as am…
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