Representations of max-stable processes via exponential tilting

The recent contribution (Dieker and Mikosch, 2015) obtained representations of max-stable stationary Brown–Resnick process ${\zeta }_Z\left(t\right),t\in {\mathbb{R}}^d$ with spectral process $Z$ being Gaussian. With motivations from Dieker and Mikosch (2015) we derive for general $Z$, representations for ${\zeta }_Z$ via exponential tilting of $Z$. Our findings concern Dieker–Mikosch representations of max-stable processes, two-sided extensions of stationary max-stable processes, inf-argmax representation of max-stable distributions, and new formulas for generalised Pickands constants. Our applications include conditions for the stationarity of ${\zeta }_Z$, a characterisation of Gaussian distributions and an alternative proof of Kabluchko’s characterisation of Gaussian processes with stationary increments.     

On Generalised Piterbarg Constants

We investigate generalised Piterbarg constants

$$\mathcal{P}_{\alpha, \delta}^{h}=\lim\limits_{T \rightarrow \infty} \mathbb{E}\left\{ \sup\limits_{t\in \delta \mathbb{Z} \cap [0,T]} e^{\sqrt{2}B_{\alpha}(t)-|t|^{\alpha}- h(t)}\right\} $$

determined in terms of a fractional Brownian motion B _α with Hurst index α/2∈(0,1], the non-negative constant δ and a continuous function h. We show that these constants, similarly to generalised Pickands constants, appear naturally in the tail asymptotic behaviour of supremum of Gaussian processes. Further, we derive several bounds for \(\mathcal {P}_{\alpha , \delta }^{h}\) and in special cases explicit formulas are obtained.

     

Excursion probabilities of isotropic and locally isotropic Gaussian random fields on manifolds

Let X = {X(p), p ∈ M} be a centered Gaussian random field, where M is a smooth Riemannian manifold. For a suitable compact subset \(D\subset M\), we obtain approximations to the excursion probabilities \(\mathbb {P}\{\sup _{p\in D} X(p) \ge u \}\), as \(u\to \infty \), for two cases: (i) X is smooth and isotropic; (ii) X is non-smooth and locally isotropic. For case (i), the expected Euler characteristic approximation is formulated explicitly; while for case (ii), it is shown that the asymptotics is similar to Pickands’ approximation on Euclidean space which involves Pickands’ constant and the volume of D. These extend the results in Cheng and Xiao (Bernoulli 22, 1113–1130 2016) from spheres to general Riemannian manifolds.     

On infinitely divisible semimartingales

Stricker’s theorem states that a Gaussian process is a semimartingale in its natural filtration if and only if it is the sum of an independent increment Gaussian process and a Gaussian process of finite variation, see Stricker (Z Wahrsch Verw Geb 64(3):303–312, 1983). We consider extensions of this result to non Gaussian infinitely divisible processes. First we show that the class of infinitely divisible semimartingales is so large that the natural analog of Stricker’s theorem fails to hold. Then, as the main result, we prove that an infinitely divisible semimartingale relative to the filtration generated by a random measure admits a unique decomposition into an independent increment process and an infinitely divisible process of finite variation. Consequently, the natural analog of Stricker’s theorem holds for all strictly representable processes (as defined in this paper). Since Gaussian processes are strictly representable due to Hida’s multiplicity theorem, the classical Stricker’s theorem follows from our result. Another consequence is that the question when an infinitely divisible process is a semimartingale can often be reduced to a path property, when a certain associated infinitely divisible process is of finite variation. This gives the key to characterize the semimartingale property for many processes of interest. Along these lines, using Basse-O’Connor and Rosiński (Stoch Process Appl 123(6):1871–1890, 2013a), we characterize semimartingales within a large class of stationary increment infinitely divisible processes; this class includes many infinitely divisible processes of interest, including linear fractional processes, mixed moving averages, and supOU processes, as particular cases. The proof of the main theorem relies on series representations of jumps of càdlàg infinitely divisible processes given in Basse-O’Connor and Rosiński (Ann Probab 41(6):4317–4341, 2013b) combined with techniques of stochastic analysis.     

Real class sizes

Following Wiener, we consider the zeroes of Gaussian analytic functions in a strip in the complex plane, with translation-invariant distribution. We show that the variance of the number of zeroes in a long horizontal rectangle [?T,T] × [a, b] is asymptotically between cT and CT², with positive constants c and C. We also supply with conditions (in terms of the spectral measure) under which the variance grows asymptotically linearly with T, as a quadratic function of T, or has intermediate growth. The results are compared with known results for real stationary Gaussian processes and other models.     

Tempering stable processes

A tempered stable Lévy process combines both the α$\alpha$-stable and Gaussian trends. In a short time frame it is close to an α$\alpha$-stable process while in a long time frame it approximates a Brownian motion. In this paper we consider a general and robust class of multivariate tempered stable distributions and establish their identifiable parametrization. We prove short and long time behavior of tempered stable Lévy processes and investigate their absolute continuity with respect to the underlying α$\alpha$-stable processes. We find probabilistic representations of tempered stable processes which specifically show how such processes are obtained by cutting (tempering) jumps of stable processes. These representations exhibit α$\alpha$-stable and Gaussian tendencies in tempered stable processes and thus give probabilistic intuition for their study. Such representations can also be used for simulation. We also develop the corresponding representations for Ornstein–Uhlenbeck-type processes.     

The principal bundles over an inverse semigroup

This paper is a contribution to the development of the theory of representations of inverse semigroups in toposes. It continues the work initiated by Funk and Hofstra (Theory Appl Categ 24(7):117–147, 2010). For the topos of sets, we show that torsion-free functors on Loganathan’s category L(S) of an inverse semigroup S are equivalent to a special class of non-strict representations of S, which we call connected. We show that the latter representations form a proper coreflective subcategory of the category of all non-strict representations of S. We describe the correspondence between directed and pullback preserving functors on L(S) and transitive and effective representations of S, as well as between filtered such functors and universal representations introduced by Lawson, Margolis and Steinberg. We propose a definition of a universal representation, or, equivalently, an S-torsor, of an inverse semigroup S in the topos of sheaves \({\mathsf {Sh}}(X)\) on a topological space X. We prove that the category of filtered functors from L(S) to the topos \({\mathsf {Sh}}(X)\) is equivalent to the category of universal representations of S in \({\mathsf {Sh}}(X)\). We finally propose a definition of an inverse semigroup action in an arbitrary Grothendieck topos, which arises from a functor on L(S).     

Bahadur–Kiefer representations for time dependent quantile processes

We define a time dependent empirical process based on n independent fractional Brownian motions and describe strong approximations to it by Gaussian processes. They lead to strong approximations and functional laws of the iterated logarithm for the quantile or inverse of this empirical process. They are obtained via time dependent Bahadur–Kiefer representations.     

Explicit determinantal representation formulas for the solution of the two-sided restricted quaternionic matrix equation

Weighted singular value decomposition of a quaternion matrix and with its help determinantal representations of the quaternion weighted Moore–Penrose inverse have been derived recently by the author. In this paper, using these determinantal representations, explicit determinantal representation formulas for the solution of the restricted quaternionic matrix equations, \(\mathbf{A}{} \mathbf{X}{} \mathbf{B}=\mathbf{D}\), and consequently, \(\mathbf{A}{} \mathbf{X}=\mathbf{D}\) and \(\mathbf{X}{} \mathbf{B}=\mathbf{D}\) are obtained within the framework of the theory of column–row determinants. We consider all possible cases depending on weighted matrices.     

On Gaussian Processes Equivalent in Law to Fractional Brownian Motion

We consider Gaussian processes that are equivalent in law to the fractional Brownian motion and their canonical representations. We prove a Hitsuda type representation theorem for the fractional Brownian motion with Hurst index H1/2. For the case H>1/2 we show that such a representation cannot hold. We also consider briefly the connection between Hitsuda and Girsanov representations. Using the Hitsuda representation we consider a certain special kind of Gaussian stochastic equation with fractional Brownian motion as noise.     

Asymptotic Behavior of Factorization and Projection Constants

This paper is concerned with estimates of important factorization constants that appear in Banach space theory. We prove upper bounds of the Hilbertian norm of projections on finite-dimensional spaces of interpolation spaces generated by certain abstract interpolation functors and show applications to Calderón–Lozanovskii spaces. We also prove estimates of the p-factorization norm and projection constants for finite-dimensional Banach lattices. We show as a consequence of our results that in a large class of n-dimensional Banach sequence lattices \(E_n\) the projection constants \(\lambda (E_n)\) satisfy \(\lim _{n\rightarrow \infty }\lambda (E_n)/\sqrt{n} = c\), where \(c=\sqrt{2/\pi }\) in the real case and \(c= \sqrt{\pi }/2\) in the complex case. Applications are given to vector-valued sequence spaces.     

On consistency of the likelihood moment estimators for a linear process with regularly varying innovations

In 1975 James Pickands III showed that the excesses over a high threshold are approximatly Generalized Pareto distributed. Since then, a variety of estimators for the parameters of this cdf have been studied, but always assuming the underlying data to be independent. In this paper we consider the special case where the underlying data arises from a linear process with regularly varying (i.e. heavy-tailed) innovations. Using this setup, we then show that the likelihood moment estimators introduced by Zhang Aust. N.Z. J. Stat. 49, 69–77 (2007) are consistent estimators for the parameters of the Generalized Pareto distribution.     

The model theory of modules of a C*-algebra

We study the theory of a Hilbert space H as a module for a unital C*-algebra ${\mathcal{A}}$ from the point of view of continuous logic. We give an explicit axiomatization for this theory and describe the structure of all the representations which are elementary equivalent to it. Also, we show that this theory has quantifier elimination and we characterize the model companion of the incomplete theory of all non-degenerate representations of ${\mathcal{A}}$ . Finally, we show that there is an homeomorphism between the space of types of norm less than 1 in this model companion, and the space of quasistates of the C*-algebra ${\mathcal{A}}$ .     

Memory properties of transformations of linear processes

In this paper, we study the memory properties of transformations of linear processes. Dittmann and Granger (J Econ 110:113–133, 2002) studied the polynomial transformations of Gaussian FARIMA(0, d, 0) processes by applying the orthonormality of the Hermite polynomials under the measure for the standard normal distribution. Nevertheless, the orthogonality does not hold for transformations of non-Gaussian linear processes. Instead, we use the decomposition developed by Ho and Hsing (Ann Stat 24:992–1024, 1996; Ann Probab 25:1636–1669, 1997) to study the memory properties of nonlinear transformations of linear processes, which include the FARIMA(p, d, q) processes, and obtain consistent results as in the Gaussian case. In particular, for stationary processes, the transformations of short-memory time series still have short-memory and the transformation of long-memory time series may have different weaker memory parameters which depend on the power rank of the transformation. On the other hand, the memory properties of transformations of non-stationary time series may not depend on the power ranks of the transformations. This study has application in econometrics and financial data analysis when the time series observations have non-Gaussian heavy tails. As an example, the memory properties of call option processes at different strike prices are discussed in details.     

On Subordinated Multivariate Gaussian Lévy Processes

We discuss criteria for the selfdecomposability of multivariate Lévy processes. We consider in detail Thorin subordinated multivariate Gaussian Lévy processes. Partially on the basis of the author’s recent results (MII preprint No. 2004-33, 2004), in this paper, we consider the properties of the Pólya subordinated multivariate Gaussian Lévy processes. We define, as a special class, the multivariate generalized z-processes. The one-dimensional case was investigated in (Grigelionis, B.: Liet. Mat. Rink. 41(3), 303–309, 2001).     

Representations of Gaussian Random Fields and Approximation of Elliptic PDEs with Lognormal Coefficients

Approximation of elliptic PDEs with random diffusion coefficients typically requires a representation of the diffusion field in terms of a sequence \(y=(y_j)_{j\ge 1}\) of scalar random variables. One may then apply high-dimensional approximation methods to the solution map \(y\mapsto u(y)\). Although Karhunen–Loève representations are commonly used, it was recently shown, in the relevant case of lognormal diffusion fields, that multilevel-type expansions may yield better approximation rates. Motivated by these results, we construct wavelet-type representations of stationary Gaussian random fields defined on arbitrary bounded domains. The size and localization properties of these wavelets are studied, and used to obtain polynomial approximation results for the related elliptic PDE which outperform those achievable when using Karhunen–Loève representations. Our construction is based on a periodic extension of the stationary random field, and the expansion on the domain is then obtained by simple restriction. This makes the approach easily applicable even when the computational domain of the PDE has a complicated geometry. In particular, we apply this construction to the class of Gaussian processes defined by the family of Matérn covariances. The proposed periodic continuation technique has other relevant applications such as fast simulation of trajectories. It can be regarded as a continuous analog of circulant embedding techniques introduced for Toeplitz matrices. One of its specific features is that the rate of decay of the eigenvalues of the covariance operator of the periodized process provably matches that of the Fourier transform of the covariance function of the original process.     

On the Equivalence of Probability Spaces

For a general class of Gaussian processes W, indexed by a sigma-algebra \({\mathscr {F}}\) of a general measure space \((M,{\mathscr {F}}, \sigma )\), we give necessary and sufficient conditions for the validity of a quadratic variation representation for such Gaussian processes, thus recovering \(\sigma (A)\), for \(A\in {\mathscr {F}}\), as a quadratic variation of W over A. We further provide a harmonic analysis representation for this general class of processes. We apply these two results to: (i) a computation of generalized Ito integrals and (ii) a proof of an explicit and measure-theoretic equivalence formula, realizing an equivalence between the two approaches to Gaussian processes, one where the choice of sample space is the traditional path space, and the other where it is Schwartz’ space of tempered distributions.     

Generalized sub-Gaussian fractional Brownian motion queueing model

It is well known that often the one-dimensional distribution of a queue content is not Gaussian but its tails behave like a Gaussian. We propose to consider a general class of processes, namely the class of $\varphi $ -sub-Gaussian random processes, which is more general than the Gaussian one and includes non-Gaussian processes. The class of sub-Gaussian random processes contains Gaussian processes also and therefore is of special interest. In this paper we provide an estimate for the queue content distribution of a fluid queue fed by $N$ independent strictly $\varphi $ -sub-Gaussian generalized fractional Brownian motion input processes. We obtain an upper estimate of buffer overflow probability in a finite buffer system defined on any finite time interval $[a,b]$ or infinite interval $[0,\infty )$ . The derived estimate captures more accurately the performance of the queueing system for a wider-range of input processes.     

Prediction of catastrophes in space over time

Predicting rare events, such as high level up-crossings, for spatio-temporal processes plays an important role in the analysis of the occurrence and impact of potential catastrophes in, for example, environmental settings. Designing a system which predicts these events with high probability, but with few false alarms, is clearly desirable. In this paper an optimal alarm system in space over time is introduced and studied in detail. These results generalize those obtained by de Maré (Ann. Probab. 8, 841–850, 1980) and Lindgren (Ann. Probab. 8, 775–792, 1980, Ann. Probab. 13, 804–824, 1985) for stationary stochastic processes evolving in continuous time and are applied here to stationary Gaussian random fields.     

Sur la densité des représentations cristallines de \text{ Gal}(\overline{\mathbb Q }_p/\mathbb Q _p)

Let ${\mathfrak X }_d$ be the $p$ -adic analytic space classifying the semisimple continuous representations $\text{ Gal}(\overline{\mathbb Q }_p/\mathbb Q _p) \rightarrow \mathrm GL _d(\overline{\mathbb Q }_p)$ . We show that the crystalline representations are Zarski-dense in many irreducible components of ${\mathfrak X }_d$ , including the components made of residually irreducible representations. This extends to any dimension $d$ previous results of Colmez and Kisin for $d = 2$ . For this we construct an analogue of the infinite fern of Gouvêa–Mazur in this context, based on a study of analytic families of trianguline $(\varphi ,\Gamma )$ -modules over the Robba ring. We show in particular the existence of a universal family of (framed, regular) trianguline $(\varphi ,\Gamma )$ -modules, as well as the density of the crystalline $(\varphi ,\Gamma )$ -modules in this family. These results may be viewed as a local analogue of the theory of $p$ -adic families of finite slope automorphic forms and they are new already in dimension $2$ . The technical heart of the paper is a collection of results about the Fontaine–Herr cohomology of families of trianguline $(\varphi ,\Gamma )$ -modules.