arXiv subjects

Peter Kern

Publications and source records attributed to Peter Kern.

At least 19 recordsLinked to original sources

arXiv2026

Max-semistable extremal behavior of AR(1)-processes connected with Bernoulli convolutions

We consider simple autoregressive processes of type AR(1), whose stationary distribution is supported on a subset of the unit interval and is an affine transformation of a Bernoulli convolution. A new structural representation of the stationary distribution as a product of a power function with a log-periodic function near the origin is given, which gives structural insight to the stationary distribution on the whole unit interval by using a characteristic functional equation. This enables to prove that the stationary distribution of the AR(1)-process belongs to the domain of geometric partial attraction of a max-semistable law. We further prove uniform convergence of the distribution function of normalized maxima of the AR(1)-process to a certain power of the max-semistable law in the spirit of a merge theorem and point out connections to deterministic and random dynamical systems.

math.PR↗

arXiv2024

On H\"older continuity and $p^\mathrm{th}$-variation function of Weierstrass-type functions

We study H\"older continuity, $p^\mathrm{th}$-variation function and Riesz variation of Weierstrass-type functions along the sequence of $b$-adic partitions, where $b>1$ is an integer. By a Weierstrass-type function, we mean that in the definition of the well-known Weierstrass function, the power function is replaced by a submultiplicative function, and the Lipschitz continuous cosine and sine functions are replaced by a general periodic H\"older continuous function.

math.CA↗

arXiv2023

Parabolic Fractal Geometry of Stable L\'evy Processes with Drift

We explicitly calculate the Hausdorff dimension of the graph and range of an isotropic stable L\'{e}vy process $X$ plus deterministic drift function $f$. For that purpose we use a restricted version of the genuine Hausdorff dimension which is called the parabolic Hausdorff dimension. It turns out that covers by parabolic cylinders are optimal for treating self-similar processes, since their distinct non-linear scaling between time and space geometrically matches the self-similarity of the processes. We provide explicit formulas for the Hausdorff dimension of the graph and the range of $X+f$. In sum the parabolic Hausdorff dimension of the drift term $f$ alone contributes to the Hausdorff dimension of $X+f$. Further, we derive some formulas and bounds for the parabolic Hausdorff dimension.

math.PR↗

arXiv2021

Cutting a Cake Is Not Always a 'Piece of Cake': A Closer Look at the Foundations of Cake-Cutting Through the Lens of Measure Theory

Cake-cutting is a playful name for the fair division of a heterogeneous, divisible good among agents, a well-studied problem at the intersection of mathematics, economics, and artificial intelligence. The cake-cutting literature is rich and edifying. However, different model assumptions are made in its many papers, in particular regarding the set of allowed pieces of cake that are to be distributed among the agents and regarding the agents' valuation functions by which they measure these pieces. We survey the commonly used definitions in the cake-cutting literature, highlight their strengths and weaknesses, and make some recommendations on what definitions could be most reasonably used when looking through the lens of measure theory.

cs.GT↗

arXiv2021

On self-similar Bernstein functions and corresponding generalized fractional derivatives

We use the theory of Bernstein functions to analyze power law tail behavior with log-periodic perturbations which corresponds to self-similarity of the Bernstein functions. Such tail behavior appears in the context of semistable L\'evy processes. The Bernstein approach enables us to solve some open questions concerning semi-fractional derivatives recently introduced in {\it Fract. Calc. Appl. Anal.} {\bf 22}(2), pp. 326--357, by means of the generator of certain semistable L\'evy processes. In particular it is shown that semi-fractional derivatives can be seen as generalized fractional derivatives in the sense of Kochubei ({\it Integr. Equ. Oper. Theory} {\bf 71}, pp. 583--600).

math.PR↗

arXiv2019

Space-time duality for semi-fractional diffusions

Almost sixty years ago Zolotarev proved a duality result which relates an $\alpha$-stable density for $\alpha\in(1,2)$ to the density of a $\frac1{\alpha}$-stable distribution on the positive real line. In recent years Zolotarev duality was the key to show space-time duality for fractional diffusions stating that certain heat-type fractional equations with a fractional derivative of order $\alpha$ in space are equivalent to corresponding time-fractional differential equations of order $\frac1{\alpha}$. We review on this space-time duality and take it as a recipe for a generalization from the stable to the semistable situation.

math.PR↗

arXiv2018

Semi-fractional diffusion equations

It is well known that certain fractional diffusion equations can be solved by the densities of stable L\'evy motions. In this paper we use the classical semigroup approach for L\'evy processes to define semi-fractional derivatives, which allows us to generalize this statement to semistable L\'evy processes. A Fourier series approach for the periodic part of the corresponding L\'evy exponents enables us to represent semi-fractional derivatives by a Gr\"unwald-Letnikov type formula. We use this formula to calculate semi-fractional derivatives and solutions to semi-fractional diffusion equations numerically. In particular, by means of the Gr\"unwald-Letnikov type formula we provide a numerical algorithm to compute semistable densities.

math.PR↗

arXiv2017

On the carrying dimension of occupation measures for self-affine random fields

Hausdorff dimension results are a classical topic in the study of path properties of random fields. This article presents an alternative approach to Hausdorff dimension results for the sample functions of a large class of self-affine random fields. We present a close relationship between the carrying dimension of the corresponding self-affine random occupation measure introduced by U. Z\"ahle and the Hausdorff dimension of the graph of self-affine fields. In the case of exponential scaling operators, the dimension formula can be explicitly calculated by means of the singular value function. This also enables to get a lower bound for the Hausdorff dimension of the range of general self-affine random fields under mild regularity assumptions.

math.PR↗

arXiv2016

On exact Hausdorff measure functions of operator semistable L\'evy processes

Let $X=\{X(t)\}_{t\geq0}$ be an operator semistable L\'evy process on $\mathbb{R}^d$ with exponent $E$, where $E$ is an invertible linear operator on $\mathbb{R}^d$. In this paper we determine exact Hausdorff measure functions for the range of $X$ over the time interval $[0,1]$ under certain assumptions on the principal spectral component of $E$. As a byproduct we also present Tauberian results for semistable subordinators and sharp bounds for the asymptotic behavior of the expected sojourn times of $X$.

math.PR↗

arXiv2016

Asymptotic behavior of semistable L\'evy exponents and applications to fractal path properties

This paper proves sharp bounds on the tails of the L\'evy exponent of an operator semistable law on $\mathbb R^d$. These bounds are then applied to explicitly compute the Hausdorff and packing dimensions of the range, graph, and other random sets describing the sample paths of the corresponding operator semi-selfsimilar L\'evy processes. The proofs are elementary, using only the properties of the L\'evy exponent, and certain index formulae.

math.PR↗

arXiv2016

An integral representation of dilatively stable processes with independent increments

Dilative stability generalizes the property of selfsimilarity for infinitely divisible stochastic processes by introducing an additional scaling in the convolution exponent. Inspired by results of Igl\'oi, we will show how dilatively stable processes with independent increments can be represented by integrals with respect to time-changed L\'evy processes. Via a Lamperti-type transformation these representations are shown to be closely connected to translatively stable processes of Ornstein-Uhlenbeck-type, where translative stability generalizes the notion of stationarity. The presented results complement corresponding representations for selfsimilar processes with independent increments known from the literature.

math.PR↗

arXiv2014

Dilatively semistable stochastic processes

Dilative semistability extends the notion of semi-selfsimilarity for infinitely divisible stochastic processes by introducing an additional scaling in the convolution exponent. It is shown that this scaling relation is a natural extension of dilative stability and some examples of dilatively semistable processes are given. We further characterize dilatively stable and dilatively semistable processes as limits for certain rescaled aggregations of independent processes.

math.PR↗

arXiv2014

Gauss-Markov processes as space-time scaled stationary Ornstein-Uhlenbeck processes

We present a class of Gauss-Markov processes which can be represented as space-time scaled stationary Ornstein-Uhlenbeck processes defined on the real line. We give several explicit examples of the representation for certain Gauss bridge processes. As an application, we derive a formula for the density function of the supremum location of certain standardized Gauss-Markov processes on compact time intervals. We also present some sufficient conditions under which mean centered Gauss-Markov processes take zero at a fixed time with probability one.

math.PR↗

arXiv2014

On some compound distributions with Borel summands

The generalized Poisson distribution is well known to be a compound Poisson distribution with Borel summands. As a generalization we present closed formulas for compound Bartlett and Delaporte distributions with Borel summands and a recursive structure for certain compound shifted Delaporte mixtures with Borel summands. Our models are introduced in an actuarial context as claim number distributions and are derived only with probabilistic arguments and elementary combinatorial identities. In the actuarial context related compound distributions are of importance as models for the total size of insurance claims for which we present simple recursion formulas of Panjer type.

math.PR↗

arXiv2014

Dilatively stable stochastic processes and aggregate similarity

Dilatively stable processes generalize the class of infinitely divisible self-similar processes. We reformulate and extend the definition of dilative stability introduced by Igl\'oi (2008) using characteristic functions. We also generalize the concept of aggregate similarity introduced by Kaj (2005). It turns out that these two notions are essentially the same for infinitely divisible processes. Examples of dilatively stable generalized fractional L\'evy processes are given and we point out that certain limit processes in aggregation models are dilatively stable.

math.PR↗

arXiv2014

A link between Bougerol's identity and a formula due to Donati-Martin, Matsumoto and Yor

We point out an easy link between two striking identities on exponential functionals of the Wiener process and the Wiener bridge originated by Bougerol, and Donati-Martin, Matsumoto and Yor, respectively. The link is established using a continuous one-parameter family of Gaussian processes known as $\alpha$-Wiener bridges or scaled Wiener bridges, which in case $\alpha=0$ coincides with a Wiener process and for $\alpha=1$ is a version of the Wiener bridge.

math.PR↗

arXiv2013

Operator scaled Wiener bridges

We introduce operator scaled Wiener bridges by incorporating a matrix scaling in the drift part of the SDE of a multidimensional Wiener bridge. A sufficient condition for the bridge property of the SDE solution is derived in terms of the eigenvalues of the scaling matrix. We analyze the asymptotic behavior of the bridges and briefly discuss the question whether the scaling matrix determines uniquely the law of the corresponding bridge.

math.PR↗

arXiv2013

The dimension of the St. Petersburg game

Let $S_n$ be the total gain in $n$ repeated St.\ Petersburg games. It is known that $n^{-1}(S_n-n\log_2n)$ converges in distribution to a random element $Y(t)$ along subsequences of the form $k(n)=2^{p(n)}t(n)$ with $p(n)=\lceil\log_2k(n)\rceil\to\infty$ and $t(n)\to t\in[\frac12,1]$. We determine the Hausdorff and box-counting dimension of the range and the graph for almost all sample paths of the stochastic process $\{Y(t)\}_{t\in[1/2,1]}$. The results are compared to the fractal dimension of the corresponding limiting objects when gains are given by a deterministic sequence initiated by Hugo Steinhaus.

math.PR↗