Finite-dimensional Distributions Research Articles

Finite-dimensional probability distributions in the random Burgers-Riemann problem

We fully quantify the uncertainty for the Burgers’ partial differential equation with random Riemann initial states, by obtaining the generalized probability density functions associated to the finite-dimensional distributions of the response. The entropy solution is a mixed random field, decomposed into discrete and continuous parts, therefore the formalism of the Dirac delta function is employed in a novel way to derive closed-form expressions for the generalized densities, both for non-independent and independent initial conditions. This work extends the recent papers that focused on the first and the second finite-dimensional laws.

A crossinggram for random fields on lattices

The modeling of risk situations that occur in a space framework can be done using max-stable random fields on lattices. Although the summary coefficients for the spatial behavior do not characterize the finite-dimensional distributions of the random field, they have the advantage of being immediate to interpret and easier to estimate. The coefficients that we propose give us information about the tendency of a random field for local oscillations of its values in relation to real valued high levels. It is not the magnitude of the oscillations that is being evaluated, but rather the greater or lesser number of oscillations, that is, the tendency of the trajectories to oscillate. We can observe surface trajectories more smooth over a region according to higher crossinggram value. It takes value in [0,1] and increases with the concordance of the variables of the random field.

Asymptotic Properties of Extremal Markov Processes Driven by Kendall Convolution

This paper is devoted to the analysis of the finite-dimensional distributions and asymptotic behavior of extremal Markov processes connected with the Kendall convolution. In particular, we provide general formulas for the finite dimensional distributions of the random walk driven by the Kendall convolution for a large class of step size distributions. Moreover, we prove limit theorems for random walks and associated continuous-time stochastic processes.

ALGORITHM FOR ASSESSING THE QUALITY OF MULTIDIMENSIONAL EXPERIMENTAL DATA BASED ON STATISTICAL ANALYSIS OF MARKOV PROCESSES

The article describes a real-time piecewise-stationary vector process with several states stipulated by the appearance of one or more components (research intervals) of the “intense signal” – a low-frequency trend. We have found a separable function equivalent to the process under study in order to determine the state of a random vector process by investigating the character and structure of relationships between the process components, when the finite-dimensional distributions of this process were a priori unknown. This function is represented as a projection of a random process defined in a probability space on a subspace of sample functions located at the smallest distance from the process. Then, according to projection theorem with the account of Markov property, it is possible to construct a H0: Pt(t + h) = Pt(t) statistical hypothesis, assuming that statistical characteristics of (t + h) components interconnection get no significant changes until the detection of an “intense signal” in t + h interval. It is known that the problem of finding the projection corresponds to the problem of finding the mean square regression coefficients. Statistical analysis allowed to find the procedure and get the Bayesian algorithm in order to determine the state of Markov vector process by regression analysis methods. The efficiency of proposed approach is demonstrated on a simple example.

Externalities in Queues as Stochastic Processes: The Case of FCFS M/G/1

Externalities are the costs that a user of a common resource imposes on others. In the context of an FCFS M/G/1 queue, where a customer with service demand [Formula: see text] arrives when the workload level is [Formula: see text], the externality [Formula: see text] is the total waiting time that could be saved if this customer gave up on their service demand. In this work, we analyze the externalities process [Formula: see text]. It is shown that this process can be represented by an integral of a (shifted in time by v) compound Poisson process with a positive discrete jump distribution, so that [Formula: see text] is convex. Furthermore, we compute the Laplace-Stieltjes transform of the finite-dimensional distributions of [Formula: see text] and its mean and auto-covariance functions. We also identify conditions under which a sequence of normalized externalities processes admits a weak convergence on [Formula: see text] equipped with the uniform metric to an integral of a (shifted in time by v) standard Wiener process. Finally, we also consider the extended framework when v is a general nonnegative random variable which is independent from the arrival process and the service demands. Our analysis leads to substantial generalizations of the results presented in the existing literature. Funding: This research was supported by the European Union’s Horizon 2020 research and innovation programme [Marie Skłodowska-Curie Grant Agreement 945045] and the NWO Gravitation project NETWORKS [Grant 024.002.003].

Exact Solution of Interacting Particle Systems Related to Random Matrices

We consider one-dimensional diffusions, with polynomial drift and diffusion coefficients, so that in particular the motion can be space-inhomogeneous, interacting via one-sided reflections. The prototypical example is the well-known model of Brownian motions with one-sided collisions, also known as Brownian TASEP, which is equivalent to Brownian last passage percolation. We obtain a formula for the finite dimensional distributions of these particle systems, starting from arbitrary initial condition, in terms of a Fredholm determinant of an explicit kernel. As far as we can tell, in the space-inhomogeneous setting and for general initial condition this is the first time such a result has been proven. We moreover consider the model of non-colliding diffusions, again with polynomial drift and diffusion coefficients, which includes the ones associated to all the classical ensembles of random matrices. We prove that starting from arbitrary initial condition the induced point process has determinantal correlation functions in space and time with an explicit correlation kernel. A key ingredient in our general method of exact solution for both models is the application of the backward in time diffusion flow on certain families of polynomials constructed from the initial condition.

Weak convergence of non-neutral genealogies to Kingman’s coalescent

Interacting particle systems undergoing repeated mutation and selection steps model genetic evolution, and also describe a broad class of sequential Monte Carlo methods. The genealogical tree embedded into the system is important in both applications. Under neutrality, when fitnesses of particles are independent from those of their parents, rescaled genealogies are known to converge to Kingman’s coalescent. Recent work has established convergence under non-neutrality, but only for finite-dimensional distributions. We prove weak convergence of non-neutral genealogies on the space of càdlàg paths under standard assumptions, enabling analysis of the whole genealogical tree.

Kernel Estimation of Bivariate Time-Varying Coefficient Model for Longitudinal Data with Terminal Event

We propose a nonparametric bivariate time-varying coefficient model for longitudinal measurements with the occurrence of a terminal event that is subject to right censoring. The time-varying coefficients capture the longitudinal trajectories of covariate effects along with both the followup time and the residual lifetime. The proposed model extends the parametric conditional approach given terminal event time in recent literature, and thus avoids potential model misspecification. We consider a kernel smoothing method for estimating regression coefficients in our model and use cross-validation for bandwidth selection, applying undersmoothing in the final analysis to eliminate the asymptotic bias of the kernel estimator. We show that the kernel estimates follow a finite-dimensional normal distribution asymptotically under mild regularity conditions, and provide an easily computed sandwich covariance matrix estimator. We conduct extensive simulations that show desirable performance of the proposed approach, and apply the method to analyzing the medical cost data for patients with end-stage renal disease. Supplementary materials for this article are available online.

Tightness of discrete Gibbsian line ensembles

A discrete Gibbsian line ensemble L=(L1,…,LN) consists of N independent random walks on the integers conditioned not to cross one another, i.e., L1≥⋯≥LN. In this paper we provide sufficient conditions for convergence of a sequence of suitably scaled discrete Gibbsian line ensembles fN=(f1N,…,fNN) as the number of curves N tends to infinity. Assuming log-concavity and a KMT-type coupling for the random walk jump distribution, we prove that under mild control of the one-point marginals of the top curves with a global parabolic shift, the full sequence (fN) is tight in the topology of uniform convergence over compact sets, and moreover any weak subsequential limit possesses the Brownian Gibbs property. If in addition the top curves converge in finite-dimensional distributions to the parabolic Airy2 process, then a result of Dimitrov (2021) implies that (fN) converges to the parabolically shifted Airy line ensemble. These results apply to a broad class of discrete jump distributions, including geometric as well as any log-concave distribution whose support forms a compact integer interval.

Critical branching as a pure death process coming down from infinity

AbstractWe consider the critical Galton–Watson process with overlapping generations stemming from a single founder. Assuming that both the variance of the offspring number and the average generation length are finite, we establish the convergence of the finite-dimensional distributions, conditioned on non-extinction at a remote time of observation. The limiting process is identified as a pure death process coming down from infinity.This result brings a new perspective on Vatutin’s dichotomy, claiming that in the critical regime of age-dependent reproduction, an extant population either contains a large number of short-living individuals or consists of few long-living individuals.

G-Gaussian processes under sublinear expectations and <inline-formula><tex-math id="M1">$ q $</tex-math></inline-formula>-Brownian motion in quantum mechanics

An important observation of this paper is that a non-trivial $ G $-Brownian motion is not a Gaussian process, e.g., finite dimensional distributions of $ G $-Brownian motion is not G-normal, or G-Gaussian. We then have to start from the very beginning, to establish the foundation of $ G $-Gaussian processes which is more suitable for space-parameter systems. It is known that the notion of classical Brownian motion is not suitable to model the random propagation of a quantum particle. In this paper we have rigorously defined a new stochastic process called $ q $-Brownian who's propagator exactly coincides with the one proposed by Feynman based on the solution of Schrödinger equation. The notion of expectation plays a fundamental base of the above results. This paper was originally published on [38].

Generalization of the Glivenko–Cantelli theorem to finite-dimensional distributions of ergodic homogeneous random fields

Let X(t),t∈Zm×Rk, be an ergodic stationary random processes or an ergodic homogeneous random field (k+m≥1). We prove that the distribution function of each random vector (X(t1),…,X(ts)) can be a.s. arbitrary fine uniformly approximated by the empirical distribution functions, if, e.g., in their construction increasing bounded convex sets Tn with infinitely increasing intrinsic diameters are used. We consider also the case when X is observed on a mixing homogeneous countable random set Sm(ω)⊂Rm (e.g., on a Poisson random set). These results open a way to consistent statistical inference on finite-dimensional distribution functions of ergodic processes and fields.

Exchangeable min-id sequences: Characterization, exponent measures and non-decreasing id-processes

We establish a one-to-one correspondence between (i) exchangeable sequences of random variables whose finite-dimensional distributions are minimum (or maximum) infinitely divisible and (ii) non-negative, non-decreasing, infinitely divisible stochastic processes. The exponent measure of an exchangeable minimum infinitely divisible sequence is shown to be the sum of a very simple “drift measure” and a mixture of product probability measures, which uniquely corresponds to the Lévy measure of a non-negative and non-decreasing infinitely divisible process. The latter is shown to be supported on non-negative and non-decreasing functions. In probabilistic terms, the aforementioned infinitely divisible process is equal to the conditional cumulative hazard process associated with the exchangeable sequence of random variables with minimum (or maximum) infinitely divisible marginals. Our results provide an analytic umbrella which embeds the de Finetti subfamilies of many interesting classes of multivariate distributions, such as exogenous shock models, exponential and geometric laws with lack-of-memory property, min-stable multivariate exponential and extreme-value distributions, as well as reciprocal Archimedean copulas with completely monotone generator and Archimedean copulas with log-completely monotone generator.

Functional Central Limit Theorems for Stick-Breaking Priors

We obtain the strong law of large numbers, Glivenko-Cantelli theorem, central limit theorem, functional central limit theorem for various Bayesian nonparametric priors which include the stick-breaking process with general stick-breaking weights, the two-parameter Poisson-Dirichlet process, the normalized inverse Gaussian process, the normalized generalized gamma process, and the generalized Dirichlet process. For the stick-breaking process with general stick-breaking weights, we introduce two general conditions such that the central limit theorem and functional central limit theorem hold. Except in the case of the generalized Dirichlet process, since the finite dimensional distributions of these processes are either hard to obtain or are complicated to use even they are available, we use the method of moments to obtain the convergence results. For the generalized Dirichlet process we use its marginal distributions to obtain the asymptotics although the computations are highly technical.

On Convergence of Multidimensional Workload with Dominant Service Duration to a Stable Process

A service system model introduced by I. Kai and M. S. Takku is considered. A limit theorem on the convergence of finite-dimensional distributions of the total workload process with multidimensional resource to the corresponding distributions of multivariate stable process is proved. The situation is considered when the service durations prevails over the values of multidimensional resources.

Multivariate Gaussian Random Fields over Generalized Product Spaces involving the Hypertorus

The paper deals with multivariate Gaussian random fields defined over generalized product spaces that involve the hypertorus. The assumption of Gaussianity implies the finite dimensional distributions to be completely specified by the covariance functions, being in this case matrix valued mappings. We start by considering the spectral representations that in turn allow for a characterization of such covariance functions. We then provide some methods for the construction of these matrix valued mappings. Finally, we consider strategies to evade radial symmetry (called isotropy in spatial statistics) and provide representation theorems for such a more general case.

Whittle estimation based on the extremal spectral density of a heavy-tailed random field

We consider a strictly stationary random field on the two-dimensional integer lattice with regularly varying marginal and finite-dimensional distributions. Exploiting the regular variation, we define the spatial extremogram which takes into account only the largest values in the random field. This extremogram is a spatial autocovariance function. We define the corresponding extremal spectral density and its estimator, the extremal periodogram. Based on the extremal periodogram, we consider the Whittle estimator for suitable classes of parametric random fields including the Brown–Resnick random field and regularly varying max-moving averages.

Stable fluctuations of iterated perturbed random walks in intermediate generations of a general branching process tree*

Consider a general branching process, a.k.a. Crump–Mode–Jagers process, generated by a perturbed random walk η1, ξ1 + η2, ξ1 + ξ2 + η3, …, where (ξ1, η1), (ξ2, η2), … are independent identically distributed random vectors with arbitrarily dependent positive components. Denote by Nj(t) the number of the jth generation individuals with birth times ⩽ t. Assume that j = j(t) → ∞ and j(t) = o(ta) as t → ∞ for some explicitly given a > 0 (to be specified in the paper). The corresponding jth generation belongs to the set of intermediate generations. We provide sufficient conditions under which the finite-dimensional distributions of the process (N⌊j(t)u⌋(t))u > 0, properly normalized and centered, converge weakly to those of an integral functional of a stable Lévy process with finite mean.

Description of random fields by systems of conditional distributions

In this paper, we consider the direct and inverse problems of the description of lattice positive random fields by various systems of finite-dimensional (as well as one-point) probability distributions parameterized by boundary conditions. In the majority of cases, we provide necessary and sufficient conditions for the system to be a conditional distribution of a (unique) random field. The exception is Dobrushin-type systems for which only sufficient conditions are known. Also, we discuss possible applications of the considered systems.

Convergence of partial sum processes to stable processes with application for aggregation of branching processes

We provide a generalization of Theorem 1 in Bartkiewicz et al. (2011) in the sense that we give sufficient conditions for weak convergence of finite dimensional distributions of the partial sum processes of a strongly stationary sequence to the corresponding finite dimensional distributions of a non-Gaussian stable process instead of weak convergence of the partial sums themselves to a non-Gaussian stable distribution. As an application, we describe the asymptotic behaviour of finite dimensional distributions of aggregation of independent copies of a strongly stationary subcritical Galton–Watson branching process with regularly varying immigration having index in (0,1)∪(1,4/3) in a so-called iterated case, namely when first taking the limit as the time scale and then the number of copies tend to infinity.