Wlodzimierz Bryc, Jacek Wesolowski
We show that stochastic processes with linear conditional expectations and quadratic conditional variances are Markov, and their transition probabilities are related to a three-parameter family of orthogonal polynomials which generalize the Meixner polynomials. Special cases of these processes are known to arise from the non-commutative generalizations of the Levy processes.
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We study a time-non-homogeneous Markov process which arose from free probability, and which also appeared in the study of stochastic processes with linear regressions and quadratic conditional variances. Our main result is the explicit expression for the generator of the (non-homogeneous) transition operator acting on functions that extend analytically to complex domain. The paper is self-contained and does not use free probability techniques.
We investigate Laha-Lukacs properties of noncommutative random variables (processes). We prove that some families of free Meixner distributions can be characterized by the conditional moments of polynomial functions of degree 3. We also show that this fact has consequences in describing some free Levy processes. The proof relies on a combinatorial identity. At the end of this paper we show that this result can be extended to a q-Gausian variable.
The free Meixner laws arise as the distributions of orthogonal polynomials with constant-coefficient recursions. We show that these are the laws of the free pairs of random variables which have linear regressions and quadratic conditional variances when conditioned with respect to their sum. We apply this result to describe free Levy processes with quadratic conditional variances, and to prove a converse implication related to asymptotic freeness of random Wishart matrices.
We obtain factorial moment identities for the Charlier, Meixner and Krawtchouk orthogonal polynomial ensembles. Building on earlier results by Ledoux [Elect. J. Probab. 10, (2005)], we find hypergeometric representations for the factorial moments when the reference measure is Poisson (Charlier ensemble) and geometric (a particular case of the Meixner ensemble). In these cases, if the number of particles is suitably randomised, the factorial moments have a polynomial property, and satisfy three-term recurrence relations and differential equations. In particular, the normalised factorial moments…
Classical versions of q-Gaussian processes: conditional moments and Bell's inequality
该研究针对满足q-交换关系的经典q-高斯过程，证明其具备线性回归与二次条件方差特性，并将Bell协方差不等式的适用范围从q=-1拓展至整个-1<q<1区间。
We calculate moments and moment generating functions of two distributions: the so called $q-$Normal and the so called conditional $q-$Normal distributions. These distributions generalize both Normal ($q=1),$ Wigner ($% q=0,$ $q-$Normal) and Kesten-McKay ($q=0,$ conditional $q-$Normal) distributions. As a by product we get asymptotic properties of some expansions in modified Bessel functions.
The metalog distributions represent a convenient way to approach many practical applications. Their distinctive feature is simple closed-form expressions for quantile functions. This paper contributes to further development of the metalog distributions by deriving the closed-form expressions for the Conditional Value at Risk, a risk measure that is closely related to the tail conditional expectations. It also addressed the derivation of the first-order partial moments and shows that they are convex with respect to the vector of the metalog distribution parameters.
A generalized moment approach to sharp bounds for conditional expectations
本文提出广义矩方法的改编版，利用锥对偶性获得条件期望的锐界，并建立基于矩的分布鲁棒优化框架，用于不确定性下的决策。
We study properties of a subclass of Markov processes that have all moments that are continuous functions of the time parameter and more importantly are characterized by the property that say their $n-$th conditional moment given the past is also a polynomial of degree not exceeding $n.$ Of course all processes with independent increments with all moments belong to this class. We give characterization of them within the studied class. We indicate other examples of such process. Besides we indicate families of polynomials that have the property of constituting martingales. We also study conditi…
Asymptotics of higher-order conditional tail moments for convolution-equivalently distributed losses
本文研究系统性崩溃下高阶条件尾部矩的渐近性，针对两个投资组合的相依损失，分别处理轻尾卷积等价分布和重尾正则变化分布，证明渐近行为对相依强度稳健，并通过数值模拟验证。
Noncausal, or anticipative, heavy-tailed processes generate trajectories featuring locally explosive episodes akin to speculative bubbles in financial time series data. For $(X_t)$ a two-sided infinite $α$-stable moving average (MA), conditional moments up to integer order four are shown to exist provided $(X_t)$ is anticipative enough, despite the process featuring infinite marginal variance. Formulae of these moments at any forecast horizon under any admissible parameterisation are provided. Under the assumption of errors with regularly varying tails, closed-form formulae of the predictive d…
In a recent paper of Eichelsbacher and Koenig (2008) the model of ordered random walks has been considered. There it has been shown that, under certain moment conditions, one can construct a k-dimensional random walk conditioned to stay in a strict order at all times. Moreover, they have shown that the rescaled random walk converges to the Dyson Brownian motion. In the present paper we find the optimal moment assumptions for the construction of the conditional random walk and generalise the limit theorem for this conditional process.
Stationary, Markov, stochastic processes with polynomial conditional moments and continuous paths
本文研究具有条件多项式矩的平稳马尔可夫随机过程，给出用边缘分布偏度和超额峰度表示的连续路径修正存在性判据，并涵盖 Ornstein-Uhlenbeck 过程、Gamma 过程等实例。
Infinitesimal moments in free and c-free probability and Motzkin paths
本文研究了自由概率与c-自由概率中的无穷小矩量，通过Motzkin路径分析，揭示了无穷小自由性和条件自由性的矩特征及几何意义。
The conditional higher moment risk measure: second-order asymptotics with FGM contagion
研究FGM相依结构下条件高阶矩相干风险度量的二阶渐近展开，利用极值理论等推导展开式，相比一阶结果能更准确捕捉高阶行为和相依效应，经模拟和实证应用验证了二阶方法的优势。
We study a class of stationary Markov processes with marginal distributions identifiable by moments such that every conditional moment of degree say $m$ is a polynomial of degree at most $m\;\text{.}\;$ We show that then under some additional, natural technical assumption there exists a family of orthogonal polynomial martingales. More precisely we show that such a family of processes is completely characterized by the sequence $\{(α_{n} ,p_{n})\}_{n \geq 0}$ where $α_{n}^{ \prime } s$ are some positive reals while $p_{n}^{ \prime } s$ are some monic orthogonal polynomials. Paper of Bakry\&Maz…
Conditional Upper Bounds for Large Deviations and Moments of the Riemann Zeta Function
在黎曼假设下，研究了ζ函数在临界线上大偏差的上界，证明了矩的上界与对数项相关，恢复了Harper的界。
For a linear combination of random variables, fix some confidence level and consider the quantile of the combination at this level. We are interested in the partial derivatives of the quantile with respect to the weights of the random variables in the combination. It turns out that under suitable conditions on the joint distribution of the random variables the derivatives exist and coincide with the conditional expectations of the variables given that their combination just equals the quantile. Moreover, using this result, we deduce formulas for the derivatives with respect to the weights of t…
We address the weak numerical solution of stochastic differential equations driven by independent Brownian motions (SDEs for short). This paper develops a new methodology to design adaptive strategies for determining automatically the step-sizes of the numerical schemes that compute the mean values of smooth functions of the solutions of SDEs. First, we introduce a general method for constructing variable step-size weak schemes for SDEs, which is based on controlling the match between the first conditional moments of the increments of the numerical integrator and the ones corresponding to an a…
This paper deals with rates of convergence in the strong law of large numbers, in the Baum-Katz form, for partial sums of Banach space valued random variables. The results are then applied to solve similar problems for weighted partial sums of conditional expectations. They are further used to treat partial sums of powers of a reversible Markov chain operator. The method of proof is based on martingale approximation. The conditions are expressed in terms moments of the individual summands.
Variational proof of conditional expectations
本文通过变分方法证明，给定$σ$-代数时具有有限二阶矩的随机变量的条件期望是Hilbert空间$L^2$中能量泛函的唯一临界点，并利用稠密性将结果推广至所有可积随机变量。
In this paper, we introduce the concept of hyperbolic valued random variables, their expectation and moments. We develop the hyperbolic analogue of Binomial and Poisson distributions. We study some of the properties of expectation on the basis of decomposition of a hyperbolic number into idempotent components. Finally we define conditional expectation of hyperbolic valued random variable and study some of its basic properties. Our random variable can take values which are zero divisors and this is the important part of this study.
Estimating sample paths of Gauss-Markov processes from noisy data
本文针对含噪观测下的高斯-马尔可夫过程，推导了任意样本路径点的条件均值与方差公式，并以布朗运动及布朗桥为例进行验证。
We investigate the problem of representing information measures in terms of the moments of the underlying random variables. First, we derive polynomial approximations of the conditional expectation operator. We then apply these approximations to bound the best mean-square error achieved by a polynomial estimator -- referred to here as the PMMSE. In Gaussian channels, the PMMSE coincides with the minimum mean-square error (MMSE) if and only if the input is either Gaussian or constant, i.e., if and only if the conditional expectation of the input of the channel given the output is a polynomial o…
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