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Optimal transport and Wasserstein distances are flourishing in many scientific fields as a means for comparing and connecting random structures. Here we pioneer the use of an optimal transport distance between Lévy measures to solve a statistical problem. Dependent Bayesian nonparametric models provide flexible inference on distinct, yet related, groups of observations. Each component of a vector of random measures models a group of exchangeable observations, while their dependence regulates the borrowing of information across groups. We derive the first statistical index of dependence in [ 0 , 1 ] for (completely) random measures that accounts for their whole infinite-dimensional distribution, which is assumed to be equal across different groups. This is accomplished by using the geometric properties of the Wasserstein distance to solve a max–min problem at the level of the underlying Lévy measures. The Wasserstein index of dependence sheds light on the models' deep structure and has desirable properties: (i) it is 0 if and only if the random measures are independent; (ii) it is 1 if and only if the random measures are completely dependent; (iii) it simultaneously quantifies the dependence of d ≥ 2 random measures, avoiding the need for pairwise comparisons; (iv) it can be evaluated numerically. Moreover, the index allows for informed prior specifications and fair model comparisons for Bayesian nonparametric models. for this article are available online. [ABSTRACT FROM AUTHOR]

Using an intrinsic approach, we study some properties of random fields which appear as tail fields of regularly varying stationary random fields. The index set is allowed to be a general locally compact Hausdorff Abelian group G . The values are taken in a measurable cone, equipped with a pseudo norm. We first discuss some Palm formulas for the exceedance random measure ξ associated with a stationary (measurable) random field Y = (Y s) s ∈ G . It is important to allow the underlying stationary measure to be σ -finite. Then we proceed to a random field (defined on a probability space) which is spectrally decomposable, in a sense which is motivated by extreme value theory. We characterize mass-stationarity of the exceedance random measure in terms of a suitable version of the classical Mecke equation. We also show that the associated stationary measure is homogeneous, that is a tail measure. We then proceed with establishing and studying the spectral representation of stationary tail measures and with characterizing a moving shift representation. Finally we discuss anchoring maps and the candidate extremal index. [ABSTRACT FROM AUTHOR]

In this paper, we analyze multidimensional generalized backward stochastic differential equations with jumps in a filtration that supports a Brownian motion and an independent integer-valued random measure. Under monotonicity and linear growth assumptions on the coefficients, we give the existence and uniqueness of 핃 2 -solutions provided that the generators and the terminal condition satisfy some suitable integrability conditions. [ABSTRACT FROM AUTHOR]

This work contributes to the theory of Hawkes processes. We introduce and study a new class of Hawkes processes that we call generalized Hawkes processes, and their special subclass – the generalized multivariate Hawkes processes (GMHPs). GMHPs are multivariate marked point processes that add an important feature to the family of the (classical) multivariate Hawkes processes: they allow for explicit modeling of simultaneous occurrence of excitation events coming from different sources, i.e., caused by different coordinates of the multivariate process. [ABSTRACT FROM AUTHOR]

This paper first explains the concept of infinite divisibility for random variables and processes. As a special form of stochastic process, random measures are defined and the most important examples are given. The integral with respect to a random measure is first defined for simple and then for general functions. Necessary and sufficient conditions for the integrand that ensure the integral is well defined are stated. Explicit expressions are given for the characteristic triplet of an infinitely divisible random variable defined by the integral. In the last part, examples of processes defined by the stochastic integral and their most important properties are given. [ABSTRACT FROM AUTHOR]

We present a framework for performing regression when both covariate and response are probability distributions on a compact interval. Our regression model is based on the theory of optimal transportation, and links the conditional Fréchet mean of the response to the covariate via an optimal transport map. We define a Fréchet-least-squares estimator of this regression map, and establish its consistency and rate of convergence to the true map, under both full and partial observations of the regression pairs. Computation of the estimator is shown to reduce to a standard convex optimization problem, and thus our regression model can be implemented with ease. We illustrate our methodology using real and simulated data. [ABSTRACT FROM AUTHOR]

In this article, we propose a way to study sample path properties of processes indexed by a general poset (T , ≼). This framework encompasses large classes of vector spaces, manifolds and continuous R -trees. We define a Wiener-type integral Y t = ∫ ≼ t f d X for all t ∈ T , a deterministic function f : T → R and a set-indexed Lévy process X. Bounds for the Hölder regularity of Y are given which indicate how the regularities of f and X contributes to that of Y. This work is a continuation of Herbin and Richard (2016) and extends those of Jaffard (1999) and Balança and Herbin (2012). [ABSTRACT FROM AUTHOR]

We study a backward SDE driven by a jump Markov process (BSDEJ for short) whose generator may be locally Lipschitz or of logarithmic growth in (y , z (⋅)) -variables. The existence, uniqueness and stability theorems to such BSDEJs are established. We essentially approximate the initial problem by constructing a suitable sequence of BSDEJs with globally Lipschitz generators for which the existence and uniqueness of solutions hold. By passing to the limits, we show the existence and uniqueness of solutions to the original problems. We apply our main results to prove the existence of a unique solution to the Kolmogorov equation of the Markov process. [ABSTRACT FROM AUTHOR]

Metatimes constitute an extension of time-change to general measurable spaces, defined as mappings between two σ-algebras. Equipping the image σ-algebra of a metatime with a measure and defining the composition measure given by the metatime on the domain σ-algebra, we identify metatimes with bounded linear operators between spaces of square integrable functions. We also analyse the possibility to define a metatime from a given bounded linear operator between Hilbert spaces, which we show is possible for invertible operators. Next we establish a link between orthogonal random measures and cylindrical random variables following a classical construction. This enables us to view metatime-changed orthogonal random measures as cylindrical random variables composed with linear operators, where the linear operators are induced by metatimes. In the paper we also provide several results on the basic properties of metatimes as well as some applications towards trawl processes.

We continue our study of a new boundedness condition for affine permutations, motivated by the fruitful concept of periodic boundary conditions in statistical physics. We focus on bounded affine permutations of size N that avoid the monotone decreasing pattern of fixed size m. We prove that the number of such permutations is asymptotically equal to (m - 1) 2 N N (m - 2) / 2 times an explicit constant as N → ∞ . For instance, the number of bounded affine permutations of size N that avoid 321 is asymptotically equal to 4 N (N / 4 π) 1 / 2 . We also prove a permuton-like result for the scaling limit of random permutations from this class, showing that the plot of a typical bounded affine permutation avoiding m ⋯ 1 looks like m - 1 random lines of slope 1 whose y intercepts sum to 0. [ABSTRACT FROM AUTHOR]

Metatimes constitute an extension of time-change to general measurable spaces, defined as mappings between two σ-algebras. Equipping the image σ-algebra of a metatime with a measure and defining the composition measure given by the metatime on the domain σ-algebra, we identify metatimes with bounded linear operators between spaces of square integrable functions. We also analyse the possibility to define a metatime from a given bounded linear operator between Hilbert spaces, which we show is possible for invertible operators. Next we establish a link between orthogonal random measures and cylindrical random variables following a classical construction. This enables us to view metatime-changed orthogonal random measures as cylindrical random variables composed with linear operators, where the linear operators are induced by metatimes. In the paper we also provide several results on the basic properties of metatimes as well as some applications towards trawl processes. [ABSTRACT FROM AUTHOR]

In this paper we explain how the notion of weak Dirichlet process is the suitable generalization of the one of semimartingale with jumps. For such a process we provide a unique decomposition: in particular we introduce characteristics for weak Dirichlet processes. We also introduce a weak concept (in law) of finite quadratic variation. We investigate a set of new useful chain rules and we discuss a general framework of (possibly path-dependent with jumps) martingale problems with a set of examples of SDEs with jumps driven by a distributional drift. [ABSTRACT FROM AUTHOR]

A type of forward‐backward doubly stochastic differential equations driven by Brownian motions and the Poisson process (FBDSDEP) is studied. Under some monotonicity assumptions, the existence and uniqueness results for measurable solutions of FBDSDEP are established via a method of continuation. Then the continuity and differentiability of the solutions to FBDSDEP depending on parameters is discussed. Furthermore, these results were applied to backward doubly stochastic linear quadratic (LQ) nonzero sum differential games with random jumps to get the explicit form of the open‐loop Nash equilibrium point by the solution of the FBDSDEP. [ABSTRACT FROM AUTHOR]

We give asymptotics for the cumulative distribution function (CDF) for degrees of large dense random graphs sampled from a graphon. The proof is based on precise asymptotics for binomial random variables. This result is a first step for giving a nonparametric test for identifying the degree function of a large random graph. Replacing the indicator function in the empirical CDF by a smoother function, we get general asymptotic results for functionals of homomorphism densities for partially labeled graphs. This general setting allows to recover recent results on asymptotics for homomorphism densities of sampled graphon. [ABSTRACT FROM AUTHOR]

We present a complete characterization of the metric compactification of L p spaces for 1 ≤ p < ∞ . Each element of the metric compactification of L p is represented by a random measure on a certain Polish space. By way of illustration, we revisit the L p -mean ergodic theorem for 1 < p < ∞ , and Alspach's example of an isometry on a weakly compact convex subset of L 1 with no fixed points. [ABSTRACT FROM AUTHOR]

This work is divided in three parts. First, we introduce quasi-infinitely divisible (QID) random measures and formulate spectral representations. Second, we introduce QID stochastic integrals and present integrability conditions, continuity properties and spectral representations. Finally, we introduce QID processes, i.e. stochastic processes with QID finite dimensional distributions. For example, a process X is QID if there exist two ID processes Y and Z such that X + Y = d Z with Y independent of X. The class of QID processes is strictly larger than the class of ID processes. We provide spectral representations and Lévy–Khintchine formulations for potentially all QID processes. Many examples are presented. [ABSTRACT FROM AUTHOR]

We consider an adversarial Bayesian signal processing problem involving “us” and an “adversary”. The adversary observes our state in noise; updates its posterior distribution of our state and then chooses an action based on this posterior. Given knowledge of “our” state and sequence of adversary's actions observed in noise, we consider three problems: (i) How can the adversary's posterior distribution be estimated? Estimating the posterior is an inverse filtering problem involving a random measure - we formulate and solve several versions of this problem in a Bayesian setting. (ii) How can the adversary's observation likelihood be estimated? This tells us how accurate the adversary's sensors are. We compute the maximum likelihood estimator for the adversary's observation likelihood given our measurements of the adversary's actions where the adversary's actions are in response to estimating our state. (iii) How can the state be chosen by us to minimize the covariance of the estimate of the adversary's observation likelihood? “Our” state can be viewed as a probe signal which causes the adversary to act; so choosing the optimal state sequence is an input design problem. The above questions are motivated by the design of counter-autonomous systems: given measurements of the actions of a sophisticated autonomous adversary, how can our counter-autonomous system estimate the underlying belief of the adversary, predict future actions and therefore guard against these actions. [ABSTRACT FROM AUTHOR]

In this paper, we provide sufficient conditions ensuring that a ѱ-mixing property holds for the sequence of empirical cumulative distribution functions associated with a conjugate process. Numerical examples are also provided to illustrate our results. [ABSTRACT FROM AUTHOR]

This work provides a semi-analytic approximation method for decoupled forward-backward SDEs (FBSDEs) with jumps. In particular, we construct an asymptotic expansion method for FBSDEs driven by the random Poisson measures with σ-finite compensators as well as the standard Brownian motions around the small-variance limit of the forward SDE. We provide a semi-analytic solution technique as well as its error estimate for which we only need to solve essentially a system of linear ODEs. In the case of a finite jump measure with a bounded intensity, the method can also handle state-dependent and hence non-Poissonian jumps, which are quite relevant for many practical applications. [ABSTRACT FROM AUTHOR]