Finite-dimensional Distributions Research Articles

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.

Brownian regularity for the Airy line ensemble, and multi-polymer watermelons in Brownian last passage percolation

The Airy line ensemble is a positive-integer indexed system of random continuous curves whose finite dimensional distributions are given by the multi-line Airy process. It is a natural object in the KPZ universality class: for example, its highest curve, the Airy 2 _2 process, describes after the subtraction of a parabola the limiting law of the scaled energy of a geodesic running from the origin to a variable point on an anti-diagonal line in such problems as Poissonian last passage percolation. The ensemble of curves resulting from the Airy line ensemble after the subtraction of the same parabola enjoys a simple and explicit spatial Markov property, the Brownian Gibbs property. In this paper, we employ the Brownian Gibbs property to make a close comparison between the Airy line ensemble’s curves after affine shift and Brownian bridge, proving the finiteness of a superpolynomially growing moment bound on Radon-Nikodym derivatives. We also determine the value of a natural exponent describing in Brownian last passage percolation the decay in probability for the existence of several near geodesics that are disjoint except for their common endpoints, where the notion of ‘near’ refers to a small deficit in scaled geodesic energy, with the parameter specifying this nearness tending to zero. To prove both results, we introduce a technique that may be useful elsewhere for finding upper bounds on probabilities of events concerning random systems of curves enjoying the Brownian Gibbs property. Several results in this article play a fundamental role in a further study of Brownian last passage percolation in three companion papers (Hammond 2017a,b,c), in which geodesic coalescence and geodesic energy profiles are investigated in scaled coordinates.

On the Numbers of Particles in Cells in an Allocation Scheme Having an Even Number of Particles in Each Cell

We consider the usual random allocation model of distinguishable particles into distinct cells in the case when there are an even number of particles in each cell. For inhomogeneous allocations, we study the numbers of particles in the first K cells. We prove that, under some conditions, this K-dimensional random vector with centralised and normalised coordinates converges in distribution to the K-dimensional standard Gaussian law. We obtain both local and integral versions of this limit theorem. The above limit theorem implies a χ2 limit theorem which leads to a χ2-test. The parity bit method does not detect even numbers of errors in binary files; therefore, our model can be applied to describe the distribution of errors in those files. For the homogeneous allocation model, we obtain a limit theorem when both the number of particles and the number of cells tend to infinity. In that case, we prove convergence to the finite dimensional distributions of the Brownian bridge. This result also implies a χ2-test. To handle the mathematical problem, we insert our model into the framework of Kolchin’s generalized allocation scheme.

Markov processes related to the stationary measure for the open KPZ equation

We provide a probabilistic description of the stationary measures for the open KPZ on the spatial interval [0, 1] in terms of a Markov process Y, which is a Doob’s h transform of the Brownian motion killed at an exponential rate. Our work builds on a recent formula of Corwin and Knizel which expresses the multipoint Laplace transform of the stationary solution of the open KPZ in terms of another Markov process $$\mathbb T$$ : the continuous dual Hahn process with Laplace variables taking on the role of time-points in the process. The core of our approach is to prove that the Laplace transforms of the finite dimensional distributions of Y and $$\mathbb T$$ are equal when the time parameters of one process become the Laplace variables of the other process and vice versa.

Pricing variance swaps under subordinated Jacobi stochastic volatility models

In this paper, we first study the pricing of variance swaps under the Jacobi stochastic volatility model, where the squared volatility of the log-price follows a Jacobi process with values in some compact interval. The Jacobi model is versatile and highly tractable and belongs to the class of polynomial diffusions. The attractive property for this class is that the moments of the finite-dimensional distributions of the log-prices can be calculated explicitly. We employ this property and are able to obtain the closed-form solutions to variance swap prices. To introduce jumps in the asset price processes, we further extend the Jacobi model to subordinated Jacobi models where the subordination can come from either Lévy or additive subordinators. Based on an important observation that the spatial and temporal variables are separated in the moment formula of the polynomial processes, we express the moment formula under the subordinated models explicitly, which enables us to calculate the prices of variance swaps in an efficient way. The empirical analysis demonstrates that the Jacobi model can potentially outperform the Heston model and the inclusion of subordination can greatly improve the goodness of fit of the Jacobi model.

Asymptotic analysis of higher-order scattering transform of Gaussian processes

We analyze the scattering transform with the quadratic nonlinearity (STQN) of Gaussian processes without depth limitation. STQN is a nonlinear transform that involves a sequential interlacing convolution and nonlinear operators, which is motivated to model the deep convolutional neural network. We prove that with a proper normalization, the output of STQN converges to a chi-square process with one degree of freedom in the finite dimensional distribution sense, and we provide a total variation distance control of this convergence at each time that converges to zero at an exponential rate. To show these, we derive a recursive formula to represent the intricate nonlinearity of STQN by a linear combination of Wiener chaos, and then apply the Malliavin calculus and Stein’s method to achieve the goal.

Probability of Resolution of MUSIC and g-MUSIC: An Asymptotic Approach

In this article, the outlier production mechanism of the conventional Multiple Signal Classification (MUSIC) and the g-MUSIC Direction-of-Arrival (DoA) estimation technique is investigated using tools from Random Matrix Theory (RMT). A general Central Limit Theorem (CLT) is derived that allows to analyze the asymptotic stochastic behavior of eigenvector-based cost functions in the asymptotic regime where the number of snapshots and the number of antennas increase without bound at the same rate. Furthermore, this CLT is used to provide an accurate prediction of the resolution capabilities of the MUSIC and the g-MUSIC DoA estimation method. The finite dimensional distribution of the MUSIC and the g-MUSIC cost function is shown to be asymptotically jointly Gaussian distributed in the asymptotic regime.

A Central Limit Theorem for Predictive Distributions

Let S be a Borel subset of a Polish space and F the set of bounded Borel functions f:S→R. Let an(·)=P(Xn+1∈·∣X1,…,Xn) be the n-th predictive distribution corresponding to a sequence (Xn) of S-valued random variables. If (Xn) is conditionally identically distributed, there is a random probability measure μ on S such that ∫fdan⟶a.s.∫fdμ for all f∈F. Define Dn(f)=dn∫fdan−∫fdμ for all f∈F, where dn>0 is a constant. In this note, it is shown that, under some conditions on (Xn) and with a suitable choice of dn, the finite dimensional distributions of the process Dn=Dn(f):f∈F stably converge to a Gaussian kernel with a known covariance structure. In addition, Eφ(Dn(f))∣X1,…,Xn converges in probability for all f∈F and φ∈Cb(R).

Critical Galton–Watson Processes with Overlapping Generations

Abstract A properly scaled critical Galton–Watson process converges to a continuous state critical branching process ξ ⁢ ( ⋅ ) \xi(\,{\cdot}\,) as the number of initial individuals tends to infinity. We extend this classical result by allowing for overlapping generations and considering a wide class of population counts. The main result of the paper establishes a convergence of the finite-dimensional distributions for a scaled vector of multiple population counts. The set of the limiting distributions is conveniently represented in terms of integrals ( ∫ 0 y ξ ⁢ ( y - u ) ⁢ d u γ \int_{0}^{y}\xi(y-u)\,du^{\gamma} , y ≥ 0 y\geq 0 ) with a pertinent γ ≥ 0 \gamma\geq 0 .

Nonstationary Generalized TASEP in KPZ and Jamming Regimes

We study the model of the totally asymmetric exclusion process with generalized update, which compared to the usual totally asymmetric exclusion process, has an additional parameter enhancing clustering of particles. We derive the exact multiparticle distributions of distances travelled by particles on the infinite lattice for two types of initial conditions: step and alternating once. Two different scaling limits of the exact formulas are studied. Under the first scaling associated to Kardar-Parisi-Zhang (KPZ) universality class we prove convergence of joint distributions of the scaled particle positions to finite-dimensional distributions of the universal Airy$_2$ and Airy$_1$ processes. Under the second scaling we prove convergence of the same position distributions to finite-dimensional distributions of two new random processes, which describe the transition between the KPZ regime and the deterministic aggregation regime, in which the particles stick together into a single giant cluster moving as one particle. It is shown that the transitional distributions have the Airy processes and fully correlated Gaussian fluctuations as limiting cases. We also give the heuristic arguments explaining how the non-universal scaling constants appearing from the asymptotic analysis in the KPZ regime are related to the properties of translationally invariant stationary states in the infinite system and how the parameters of the model should scale in the transitional regime.

On simultaneous limits for aggregation of stationary randomized INAR(1) processes with poisson innovations

Abstract We investigate joint temporal and contemporaneous aggregation of N independent copies of strictly stationary INteger-valued AutoRegressive processes of order 1 (INAR(1)) with random coefficient α ∈ (0, 1) and with idiosyncratic Poisson innovations. Assuming that α has a density function of the form ψ(x) (1 − x) β , x ∈ (0, 1), with β ∈ (−1, ∞) and lim x ↑ 1 ψ ( x ) = ψ 1 ∈ ( 0 , ∞ ) $\lim\limits_{x\uparrow 1} \psi(x) = \psi_1 \in (0, \infty)$ , different limits of appropriately centered and scaled aggregated partial sums are shown to exist for β ∈ (−1, 0] in the so-called simultaneous case, i.e., when both N and the time scale n increase to infinity at a given rate. The case β ∈ (0, ∞) remains open. We also give a new explicit formula for the joint characteristic functions of finite dimensional distributions of the appropriately centered aggregated process in question.

Choquet random sup-measures with aggregations

A variation of Choquet random sup-measures is introduced. These random sup-measures are shown to arise as the scaling limits of empirical random sup-measures of a general aggregated model. Because of the aggregations, the finite-dimensional distributions of introduced random sup-measures do not necessarily have classical extreme-value distributions. Examples include the recently introduced stable-regenerative random sup-measures as a special case.

Liouville’s equations for random systems

Given a random system, a Liouville’s equation is an exact partial differential equation that describes the evolution of the probability density function of the solution. In this article, we derive Liouville’s equations for the first-order homogeneous semilinear random partial differential equation. This is done for all finite-dimensional distributions of the random field solution, starting with dimension one, then dimension two, and finally generalizing to any dimension. Several examples, including the linear advection equation with random coefficients, are treated. As a corollary, we deduce Liouville’s equations for path-wise stochastic integrals and nonlinear random ordinary differential equations.

The Large Deviation Principle for Finite-Dimensional Distributions of Multidimensional Renewal Processes

The Large Deviation Principle for Finite-Dimensional Distributions of Multidimensional Renewal Processes

Multivariate Normal Approximation on the Wiener Space: New Bounds in the Convex Distance

AbstractWe establish explicit bounds on the convex distance between the distribution of a vector of smooth functionals of a Gaussian field and that of a normal vector with a positive-definite covariance matrix. Our bounds are commensurate to the ones obtained by Nourdin et al. (Ann Inst Henri Poincaré Probab Stat 46(1):45–58, 2010) for the (smoother) 1-Wasserstein distance, and do not involve any additional logarithmic factor. One of the main tools exploited in our work is a recursive estimate on the convex distance recently obtained by Schulte and Yukich (Electron J Probab 24(130):1–42, 2019). We illustrate our abstract results in two different situations: (i) we prove a quantitative multivariate fourth moment theorem for vectors of multiple Wiener–Itô integrals, and (ii) we characterize the rate of convergence for the finite-dimensional distributions in the functional Breuer–Major theorem.

Limit theorems for integral functionals of Hermite-driven processes

Consider a moving average process X of the form X(t)=∫−∞tφ(t−u)dZu, t≥0, where Z is a (non Gaussian) Hermite process of order q≥2 and φ:R+→R is sufficiently integrable. This paper investigates the fluctuations, as T→∞, of integral functionals of the form t↦∫0TtP(X(s))ds, in the case where P is any given polynomial function. It extends a study initiated in (Stoch. Dyn. 18 (2018) 1850028, 18), where only the quadratic case P(x)=x2 and the convergence in the sense of finite-dimensional distributions were considered.