Joint Cumulants Research Articles

Exponential moments for disk counting statistics at the hard edge of random normal matrices

We consider the multivariate moment generating function of the disk counting statistics of a model Mittag-Leffler ensemble in the presence of a hard wall. Let n be the number of points. We focus on two regimes: (a) the “hard edge regime” where all disk boundaries are at a distance of order \frac{1}{n} from the hard wall, and (b) the “semi-hard edge regime” where all disk boundaries are at a distance of order \frac{1}{\sqrt{n}} from the hard wall. As n \to + \infty , we prove that the moment generating function enjoys asymptotics of the form \exp (C_{1}n + C_{2}\ln n + C_{3} + \frac{C_{4}}{\sqrt{n}} + \mathcal{O}(n^{-\frac{3}{5}})) for the hard edge, \exp (C_{1}n + C_{2}\sqrt{n} + C_{3} + \frac{C_{4}}{\sqrt{n}} + \mathcal{O}(\frac{(\ln n)^{4}}{n})) for the semi-hard edge. In both cases, we determine the constants C_{1},\dots,C_{4} explicitly. We also derive precise asymptotic formulas for all joint cumulants of the disk counting function, and establish several central limit theorems. Surprisingly, and in contrast to the “bulk”, “soft edge”, and “semi-hard edge” regimes, the second and higher order cumulants of the disk counting function in the “hard edge” regime are proportional to n and not to \sqrt{n} .

Properties of the Gradient Squared of the Discrete Gaussian Free Field

In this paper we study the properties of the centered (norm of the) gradient squared of the discrete Gaussian free field in Uε=U/ε∩Zd\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$U_{\\varepsilon }=U/\\varepsilon \\cap \\mathbb {Z}^d$$\\end{document}, U⊂Rd\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$U\\subset \\mathbb {R}^d$$\\end{document} and d≥2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$d\\ge 2$$\\end{document}. The covariance structure of the field is a function of the transfer current matrix and this relates the model to a class of systems (e.g. height-one field of the Abelian sandpile model or pattern fields in dimer models) that have a Gaussian limit due to the rapid decay of the transfer current. Indeed, we prove that the properly rescaled field converges to white noise in an appropriate local Besov-Hölder space. Moreover, under a different rescaling, we determine the k-point correlation function and joint cumulants on Uε\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$U_{\\varepsilon }$$\\end{document} and in the continuum limit as ε→0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\varepsilon \\rightarrow 0$$\\end{document}. This result is related to the analogue limit for the height-one field of the Abelian sandpile (Dürre in Stoch Process Appl 119(9):2725–2743, 2009), with the same conformally covariant property in d=2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$d=2$$\\end{document}.

A scaling limit of the parabolic Anderson model with exclusion interaction

AbstractWe consider the (discrete) parabolic Anderson model , , , where the ξ‐field is ‐valued and plays the role of a dynamic random environment, and Δ is the discrete Laplacian. We focus on the case in which ξ is given by a properly rescaled symmetric simple exclusion process under which it converges to an Ornstein–Uhlenbeck process. Scaling the Laplacian diffusively and restricting ourselves to a torus, we show that in dimension upon considering a suitably renormalised version of the above equation, the sequence of solutions converges in law. As a by‐product of our main result we obtain precise asymptotics for the survival probability of a simple random walk that is killed at a scale dependent rate when meeting an exclusion particle. Our proof relies on the discrete theory of regularity structures of Erhard and Hairer and on novel sharp estimates of joint cumulants of arbitrary large order for the exclusion process. We think that the latter is of independent interest and may find applications elsewhere.

THE USE OF MODEL DISTRIBUTIONS IN THE ANALYSIS OF QUEUING SYSTEMS WITH CORRELATED ARRIVALS

The article considers the task of determining the waiting time for a request in a queue in a general-type single-channel queueing system when time intervals between incoming requests are correlated. For solving the task, it is proposed to carry out decorrelation of specified time intervals using the discrete cosine transform. Generally, to analyze arbitrary distributions included in the expression for a decorrelated sequence, one can suggest an approach based on the approximation of unknown densities by a model distribution. The calculation of cumulants and moments of random variables is very simple compared to the analytical methods. Based on the found final set of cumulants, a model probability density can always be reproduced with a certain error. Generally, to calculate the average waiting time of a request in a queue, it is necessary to determine the correlation properties and one-dimensional probability density of time intervals between incoming requests, synthesize a two-dimensional probability density of a sequence of given intervals that has a measured correlation function, and, further, calculate joint moments and cumulants for correlated values, on which their model density is built.

A Note on Cumulant Technique in Random Matrix Theory.

We discuss the cumulant approach to spectral properties of large random matrices. In particular, we study in detail the joint cumulants of high traces of large unitary random matrices and prove Gaussian fluctuation for pair-counting statistics with non-smooth test functions.

Exponential moments for disk counting statistics of random normal matrices in the critical regime

We obtain large n asymptotics for the m-point moment generating function of the disk counting statistics of the Mittag–Leffler ensemble, where n is the number of points of the process and m is arbitrary but fixed. We focus on the critical regime where all disk boundaries are merging at speed , either in the bulk or at the edge. As corollaries, we obtain two central limit theorems and precise large n asymptotics of all joint cumulants (such as the covariance) of the disk counting function. Our results can also be seen as large n asymptotics for n × n determinants with merging planar discontinuities.

Time delay statistics for finite number of channels in all symmetry classes

Within a random matrix theory approach, we obtain spectral statistics of the Wigner time delay matrix Q, for arbitrary channels number M and for all symmetry classes, in fact for the general Dyson parameter β. We also put forth two conjectures: one is related to the large-M expansion of joint cumulants of traces of powers of Q, which generalizes and implies a previous conjecture of Cunden, Mezzadri, Vivo and Simm; the other concerns the tail of the distribution of traces of powers of Q.

An Algorithm for the Computation of Joint Hawkes Moments with Exponential Kernel

The purpose of this paper is to present a recursive algorithm and its implementation in Maple and Mathematica for the computation of joint moments and cumulants of Hawkes processes with exponential kernels. Numerical results and computation times are also discussed. Obtaining closed form expressions can be computationally intensive, as joint fifth cumulant and moment formulas can be respectively expanded into up to 3,288 and 27,116 summands.

Geometric and arithmetic realized comoments

The author investigates realized comoments that overcome the drawback of conventional ones and derive the following findings. First, the author proves that (even generalized) geometric implied lower-order comoments yield neither geometric realized third comoment nor fourth moment. This is in contrast to previous studies that produce geometric realized third moment and arithmetic realized higher-order moments through lower-order implied moments. Second, arithmetic realized joint cumulants are obtained through complete Bell polynomials of lower-order joint cumulants. This study’s realized measures are unbiased estimators and they can, therefore, overcome the drawbacks of conventional realized measures.

Gaussian Asymptotics of Jack Measures on Partitions From Weighted Enumeration of Ribbon Paths

Abstract In this paper, we determine two asymptotic results for Jack measures $M(v^{\textrm {out}}, v^{\textrm {in}})$, a measure on partitions defined by two specializations $v^{\textrm {out}}, v^{\textrm {in}}$ of Jack polynomials proposed by Borodin–Olshanski in [10]. Assuming $v^{\textrm {out}} = v^{\textrm {in}}$, we derive limit shapes and Gaussian fluctuations for the anisotropic profiles of these random partitions in three asymptotic regimes associated to vanishing, fixed, and diverging values of the Jack parameter. To do so, we introduce a generalization of Motzkin paths we call “ribbon paths,” show for arbitrary $v^{\textrm {out}}, v^{\textrm {in}}$ that certain Jack measure joint cumulants ${\kappa _n}$ are weighted sums of connected ribbon paths on $n$ sites with $n-1+g$ pairings, and derive our two results from the contributions of $(n,g)=(1,0)$ and $(2,0)$, respectively. Our analysis makes use of Nazarov–Sklyanin’s spectral theory for Jack polynomials. As a consequence, we give new proofs of several results for Schur measures, Plancherel measures, and Jack–Plancherel measures. In addition, we relate our weighted sums of ribbon paths to the weighted sums of ribbon graphs of maps on non-oriented real surfaces recently introduced by Chapuy–Dołęga.

Recursive computation of the Hawkes cumulants

We propose a recursive method for the computation of the cumulants of self-exciting point processes of Hawkes type, based on standard combinatorial tools such as Bell polynomials. This closed-form approach is easier to implement on higher-order cumulants in comparison with existing methods based on differential equations, tree enumeration or martingale arguments. The results are corroborated by Monte Carlo simulations, and also apply to the computation of joint cumulants generated by multidimensional self-exciting processes.

Systematic analysis of flow distributions

The information of the event-by-event fluctuations is extracted from flow harmonic distributions and cumulants, which can be done experimentally. In this work, we employ the standard method of Gram-Charlier series with the normal kernel to find such distribution, which is the generalization of recently introduced flow distributions for the studies of the event-by-event fluctuations. Also, we introduce a new set of cumulants $j_n\{2k\}$ which have more information about the fluctuations compared with other known cumulants. The experimental data imply that not only all of the information about the event-by-event fluctuations of collision zone properties and different stages of the heavy-ion process are not encoded in the radial flow distribution $p(v_n)$, but also the observables describing harmonic flows can generally be given by the joint distribution $\mathcal{P}(v_1,v_2,...)$. In such a way, we first introduce a set of joint cumulants $\mathcal{K}_{nm}$, and then we find the flow joint distribution using these joint cumulants. Finally, we show that the Symmetric Cumulants $SC(2,3)$ and $SC(2,4)$ obtained from ALICE data are explained by the combinations $\mathcal{K}_{22}+\frac{1}{2}\mathcal{K}_{04}-\mathcal{K}_{31}$ and $\mathcal{K}_{22}+4\mathcal{K}_{11}^2$.

Realized higher-order comoments

We propose a new realized third-order comoment and new realized fourth-order joint cumulants, which are standardized comoments. They are obtained from sub-period returns and lower-order comoments and satisfy A. Neuberger’s (Realized skewness. Rev. Financ. Stud., 2012, 25(11), 3423–3455) aggregation property. Different from other realized higher-order comoments obtained from sub-period returns only, those in this study reflect characteristics of the volatility of volatility as well as jump contributions. As a result, our realized kurtosis and coskewness can reflect well-known phenomena such as the positive autocorrelation of volatility or negative correlation between returns and covariances.

Probabilistic active distribution network equivalence with correlated uncertain injections for grid analysis

Equivalent modelling for active distribution networks (ADNs) is essential for improving the efficiency of analysing transmission networks. Current equivalent modelling methods for ADNs neglect the probabilistic characteristics of renewable energy sources (RESs) and loads. To address this issue, this study proposes a probabilistic equivalent modelling method (PEMM) for ADNs considering the uncertainty of RESs and loads. The uncertainty of the RESs and loads is transferred to the equivalent boundary bus injection using the properties of cumulant and power transfer matrices. The PEMM is extended to incorporate the correlations of RESs through an orthogonal transformation. A sampling method using the Gaussian copula function is employed to generate the correlated samples and the joint cumulants, providing the input data for the PEMM. The comparative results of the case studies on two different test systems demonstrate the effectiveness of the PEMM. The equivalent model developed in this study is a practical solution for analysing the transmission network efficiently and taking the uncertainty of RESs and loads in the ADNs into account simultaneously.

On the probability of nonexistence in binomial subsets

Given a hypergraph $\Gamma =(\Omega ,\mathcal{X})$ and a sequence $\mathbf{p}=(p_{\omega })_{\omega \in \Omega }$ of values in $(0,1)$, let $\Omega_{\mathbf{p}}$ be the random subset of $\Omega $ obtained by keeping every vertex $\omega $ independently with probability $p_{\omega }$. We investigate the general question of deriving fine (asymptotic) estimates for the probability that $\Omega_{\mathbf{p}}$ is an independent set in $\Gamma $, which is an omnipresent problem in probabilistic combinatorics. Our main result provides a sequence of upper and lower bounds on this probability, each of which can be evaluated explicitly in terms of the joint cumulants of small sets of edge indicator random variables. Under certain natural conditions, these upper and lower bounds coincide asymptotically, thus giving the precise asymptotics of the probability in question. We demonstrate the applicability of our results with two concrete examples: subgraph containment in random (hyper)graphs and arithmetic progressions in random subsets of the integers.

Change Point Analysis of Correlation in Non-stationary Time Series

A restrictive assumption in change point analysis is stationarity under the null hypothesis of no change-point, which is crucial for asymptotic theory but not very realistic from a practical point of view. For example, if change point analysis for correlations is performed, it is not necessarily clear that the mean, marginal variance or higher order moments are constant, even if there is no change in the correlation. This paper develops change point analysis for the correlation structures under less restrictive assumptions. In contrast to previous work, our approach does not require that the mean, variance and fourth order joint cumulants are constant under the null hypothesis. Moreover, we also address the problem of detecting relevant change points.

General Form for Interaction Measures and Framework for Deriving Higher-Order Emergent Effects

Interactions are ubiquitous and have been extensively studied in many ecological, evolutionary, and physiological systems. A variety of measures—ANOVA, covariance, epistatic additivity, mutual information, joint cumulants, Bliss independence—exist that compute interactions across fields. However, these are not discussed and derived within a single, general framework. This missing framework likely contributes to the confusion about proper formulations and interpretations of higher-order interactions. Intriguingly, despite higher-order interactions having received little attention, they have been recently discovered to be highly prevalent and to likely impact the dynamics of complex biological systems. Here, we introduce a single, explicit mathematical framework that simultaneously encompasses all of these measures of pairwise interactions. The generality and simplicity of this framework allows us to establish a rigorous method for deriving higher-order interaction measures based on any of the pairwise interactions listed above. These generalized higher-order interaction measures enable the exploration of emergent phenomena across systems such as multiple predator effects, gene epistasis, and environmental stressors. These results provide a mechanistic basis to better account for how interactions affect biological systems. Our theoretical advance provides a foundation for understanding multi-component interactions in complex systems such as evolving populations within ecosystems or communities.

Moments and cumulants of the extremes of a sample from a uniform distribution

We give expansions in inverse powers of the sample size n for the joint moments and cumulants of the extremes when sampling from a uniform distribution. Estimates of low bias are given for smooth functions of the end points and their performance illustrated by simulations. Some extensions are given for general bivariate extreme estimates.

Summability of Connected Correlation Functions of Coupled Lattice Fields

We consider two nonindependent random fields $\psi$ and $\phi$ defined on a countable set $Z$. For instance, $Z={\mathbb Z}^d$ or $Z={\mathbb Z}^d\times I$, where $I$ denotes a finite set of possible "internal degrees of freedom" such as spin. We prove that, if the cumulants of both $\psi$ and $\phi$ are $\ell_1$-clustering up to order $2 n$, then all joint cumulants between $\psi$ and $\phi$ are $\ell_2$-summable up to order $n$, in the precise sense described in the text. We also provide explicit estimates in terms of the related $\ell_1$-clustering norms, and derive a weighted $\ell_2$-summation property of the joint cumulants if the fields are merely $\ell_2$-clustering. One immediate application of the results is given by a stochastic process $\psi_t(x)$ whose state is $\ell_1$-clustering at any time $t$: then the above estimates can be applied with $\psi=\psi_t$ and $\phi=\psi_0$ and we obtain uniform in $t$ estimates for the summability of time-correlations of the field. The above clustering assumption is obviously satisfied by any $\ell_1$-clustering stationary state of the process, and our original motivation for the control of the summability of time-correlations comes from a quest for a rigorous control of the Green-Kubo correlation function in such a system. A key role in the proof is played by the properties of non-Gaussian Wick polynomials and their connection to cumulants.

Signal Detection in Correlated Non-Gaussian Noise Using Higher-Order Statistics

The authors of this paper study the synthesis of new models and methods for signal detection in additive correlated non-Gaussian noise. A new moment quality criterion decision making is proposed based on a random process description using moments and a formation of polynomial decision rules. Taking into account parameters of non-Gaussian distribution of random variables (such as the moments of third and higher orders and joint cumulants), it is shown that nonlinear processing of samples can increase the signal processing efficiency. A synthesis of effective methods and algorithms of data processing in non-Gaussian noise is also presented in this work.