Journal of Theoretical Probability
Journal Prestige (SJR): 0.981 Citation Impact (citeScore): 1 Number of Followers: 3 Hybrid journal (It can contain Open Access articles) ISSN (Print) 15729230  ISSN (Online) 08949840 Published by SpringerVerlag [2467 journals] 
 Equivalence–Singularity Dichotomy in Markov Measures

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Abstract: Abstract We establish an equivalence–singularity dichotomy for a large class of onedimensional Markov measures. Our approach is new in that we deal with onesided and twosided chains simultaneously and in that we do not appeal to a 01 law. In fact, we deduce a new 01 law from the dichotomy.
PubDate: 20221202

 Vector Random Fields on the Probability Simplex with MetricDependent
Covariance Matrix Functions
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Abstract: Abstract This paper constructs a class of isotropic vector random fields on the probability simplex via infinite series expansions involving the ultraspherical polynomials, whose covariance matrix functions are functions of the metric (distance function) on the probability simplex, and introduces the scalar and vector fractional, bifractional, and trifractional Brownian motions over the probability simplex, while the metric is shown to be conditionally negative definite.
PubDate: 20221201

 SecondOrder Tail Behavior for Stochastic Discounted Value of Aggregate
Net Losses in a DiscreteTime Risk Model
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Abstract: Abstract Consider a discretetime risk model, in which an insurer makes both riskfree and risky investments. Within period k, the net loss is denoted by a realvalued random variable \(X_k\) , and the stochastic discount factor is a bounded positive random variable \(Y_k\) . Assume that \((X_k,Y_k), k\in {\mathbb {N}}\) , form a sequence of independent and identically distributed random pairs following a common bivariate Farlie–Gumbel–Morgenstern distribution with marginal distributions F on \({\mathbb {R}}\) and G on [a, b], respectively, for some \(0<a\le b<\infty \) . Under the condition that F is secondorder subexponential, we establish a secondorder expansion for the tail probability of the stochastic discounted value of aggregate net losses. Compared with the firstorder one, our secondorder asymptotic result is more precise, which is demonstrated by numerical studies.
PubDate: 20221201

 HigherOrder Error Estimates of the DiscreteTime Clark–Ocone
Formula
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Abstract: Abstract In this article, we investigate the convergence rate of the discretetime Clark–Ocone formula provided by Akahori–Amaba–Okuma (J Theor Probab 30: 932–960, 2017). In that paper, they mainly focus on the \(L_{2}\) convergence rate of the firstorder error estimate related to the tracking error of the delta hedge in mathematical finance. Here, as two extensions, we estimate “the higher order error” for Wiener functionals with an integrability index 2 and “an arbitrary differentiability index.”
PubDate: 20221201

 Exact Uniform Modulus of Continuity and Chung’s LIL for the Generalized
Fractional Brownian Motion
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Abstract: Abstract The generalized fractional Brownian motion (GFBM) \(X:=\{X(t)\}_{t\ge 0}\) with parameters \(\gamma \in [0, 1)\) and \(\alpha \in \left( \frac{1}{2}+\frac{\gamma }{2}, \, \frac{1}{2}+\frac{\gamma }{2} \right) \) is a centered Gaussian Hselfsimilar process introduced by Pang and Taqqu (2019) as the scaling limit of powerlaw shot noise processes, where \(H = \alpha \frac{\gamma }{2}+\frac{1}{2} \in (0,1)\) . When \(\gamma = 0\) , X is the ordinary fractional Brownian motion. When \(\gamma \in (0, 1)\) , GFBM X does not have stationary increments, and its sample path properties such as Hölder continuity, path differentiability/nondifferentiability, and the functional law of the iterated logarithm (LIL) have been investigated recently by Ichiba et al. (J Theoret Probab 10.1007/s10959020010661, 2021). They mainly focused on sample path properties that are described in terms of the selfsimilarity index H (e.g., LILs at infinity or at the origin). In this paper, we further study the sample path properties of GFBM X and establish the exact uniform modulus of continuity, small ball probabilities, and Chung’s laws of iterated logarithm at any fixed point \(t > 0\) . Our results show that the local regularity properties away from the origin and fractal properties of GFBM X are determined by the index \(\alpha +\frac{1}{2}\) instead of the selfsimilarity index H. This is in contrast with the properties of ordinary fractional Brownian motion whose local and asymptotic properties are determined by the single index H.
PubDate: 20221201

 Generalized Fractional Counting Process

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Abstract: Abstract In this paper, we obtain additional results for a fractional counting process introduced and studied by Di Crescenzo et al. [8]. For convenience, we call it the generalized fractional counting process (GFCP). It is shown that the onedimensional distributions of the GFCP are not infinitely divisible. Its covariance structure is studied, using which its longrange dependence property is established. It is shown that the increments of GFCP exhibit the shortrange dependence property. Also, we prove that the GFCP is a scaling limit of some continuous time random walk. A particular case of the GFCP, namely, the generalized counting process (GCP), is discussed for which we obtain a limiting result and a martingale result and establish a recurrence relation for its probability mass function. We have shown that many known counting processes such as the Poisson process of order k, the PólyaAeppli process of order k, the negative binomial process and their fractional versions etc., are other special cases of the GFCP. An application of the GCP to risk theory is discussed.
PubDate: 20221201

 Convex Order, Quantization and Monotone Approximations of ARCH Models

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Abstract: Abstract We are interested in proposing approximations of a sequence of probability measures in the convex order by finitely supported probability measures still in the convex order. We propose to alternate two types of operators: transition according to a onestep martingale Markov kernel mapping a probability measure in the sequence to its successor and spatial discretization through dual (also called Delaunay) quantization. In the case of autoregressive conditional heteroskedasticity (ARCH) models and in particular of the Euler scheme of a driftless Brownian diffusion, the noise has to be truncated to enable the dual quantization step. We analyze the error between the original ARCH model and its approximation with truncated noise and exhibit conditions under which the latter is dominated by the former in the convex order at the level of sample paths. Last, we analyze the error of the scheme combining the dual quantization steps with truncation of the noise according to primal quantization.
PubDate: 20221201

 Convergence Rates in Uniform Ergodicity by Hitting Times and $$L^2$$
Exponential Convergence Rates
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Abstract: Abstract Generally, the convergence rate in \(L^2\) exponential ergodicity \(\lambda \) is an upper bound for the convergence rate \(\kappa \) in uniform ergodicity for a Markov process, that is, \(\lambda \geqslant \kappa \) . In this paper, we prove that \(\kappa \geqslant \inf \left\{ \lambda ,1/M_H\right\} \) , where \(M_H\) is a uniform bound on the moment of the hitting time to a “compact” set H. In the case where \(M_H\) can be made arbitrarily small for H large enough. we obtain that \(\lambda =\kappa \) . The general results are applied to Markov chains, diffusion processes and solutions to stochastic differential equations driven by symmetric stable processes.
PubDate: 20221201

 muBrownian Motion, Dualities, Diffusions, Transforms, and Reproducing
Kernel Hilbert Spaces
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Abstract: Abstract Replacing the Lebesgue measure on an interval by a Stieltjes positive nonatomic measure, we study the corresponding counterpart of the Brownian motion. We introduce a new heat equation associated with the measure and make connections with stationaryincrements Gaussian processes. We introduce a new transform analysis, and heat equation, associated with the measure, and make connections here too with stationaryincrements and stationary Gaussian processes. In the main result of this paper (Theorem 7.2), we use white noise space analysis to derive a new heat equation associated with a (wide class of) stationaryincrements Gaussian processes.
PubDate: 20221201

 Amalgamated Free Lévy Processes as Limits of Sample Covariance
Matrices
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Abstract: Abstract We prove the existence of joint limiting spectral distributions for families of random sample covariance matrices modeled on fluctuations of discretized Lévy processes. These models were first considered in applications of random matrix theory to financial data, where datasets exhibit both strong multicollinearity and nonnormality. When the underlying Lévy process is nonGaussian, we show that the limiting spectral distributions are distinct from Marčenko–Pastur. In the context of operatorvalued free probability, it is shown that the algebras generated by these families are asymptotically free with amalgamation over the diagonal subalgebra. This framework is used to construct operatorvalued \(^*\) probability spaces, where the limits of sample covariance matrices play the role of noncommutative Lévy processes whose increments are free with amalgamation.
PubDate: 20221201

 Concentration Inequalities on the Multislice and for Sampling Without
Replacement
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Abstract: Abstract We present concentration inequalities on the multislice which are based on (modified) logSobolev inequalities. This includes bounds for convex functions and multilinear polynomials. As an application, we show concentration results for the triangle count in the G(n, M) Erdős–Rényi model resembling known bounds in the G(n, p) case. Moreover, we give a proof of Talagrand’s convex distance inequality for the multislice. Interpreting the multislice in a sampling without replacement context, we furthermore present concentration results for n out of N sampling without replacement. Based on a bounded difference inequality involving the finitesampling correction factor \(1  (n / N)\) , we present an easy proof of Serfling’s inequality with a slightly worse factor in the exponent, as well as a subGaussian right tail for the Kolmogorov distance between the empirical measure and the true distribution of the sample.
PubDate: 20221201

 General SelfSimilarity Properties for Markov Processes and Exponential
Functionals of Lévy Processes
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Abstract: Abstract Positive selfsimilar Markov processes are positive Markov processes that satisfy the scaling property and it is known that they can be represented as the exponential of a timechanged Lévy process via Lamperti representation. In this work, we are interested in what happens if we consider Markov processes in dimension 1 or 2 that satisfy selfsimilarity properties of a more general form than a scaling property. We characterize them by proving a generalized Lamperti representation. Our results show that, in dimension 1, the classical Lamperti representation only needs to be slightly generalized. However, in dimension 2, our generalized Lamperti representation is much more different and involves the exponential functional of a bivariate Lévy process. We briefly discuss the complications that occur in higher dimensions. We present examples in dimensions 1, 2 and 3 that are built from growthfragmentation, selfsimilar fragmentation and Continuousstate Branching processes in Random Environment. Some of our arguments apply in the context of a general state space and show that we can exhibit a topological group structure on the state space of a Markov process that satisfies general selfsimilarity properties, which allows to write a Lampertitype representation for this process in terms of a Lévy process on the group.
PubDate: 20221201

 Stochastic Forcing in Hydrodynamic Models with Nonlocal Interactions

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Abstract: Abstract The hydrodynamical model of the collective behavior of animals consists of the Euler equation with additional nonlocal forcing terms representing the repulsive and attractive forces among individuals. This paper deals with the system endowed with an additional whitenoise forcing and an artificial viscous term. We provide a proof of the existence of a dissipative martingale solution—a cornerstone for a subsequent analysis of the system with stochastic forcing.
PubDate: 20221201

 Stratonovich Solution for the Wave Equation

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Abstract: Abstract In this article, we construct a Stratonovich solution for the stochastic wave equation in spatial dimension \(d \le 2\) , with timeindependent noise and linear term \(\sigma (u)=u\) multiplying the noise. The noise is spatially homogeneous and its spectral measure satisfies an integrability condition which is stronger than Dalang’s condition. We give a probabilistic representation for this solution, similar to the Feynman–Kactype formula given in Dalang et al. (Trans Am Math Soc 360:4681–4703, 2008) for the solution of the stochastic wave equation with spatially homogeneous Gaussian noise, that is white in time. We also give the chaos expansion of the Stratonovich solution and we compare it with the chaos expansion of the Skorohod solution from Balan et al. (Exact asymptotics of the stochastic wave equation with time independent noise, 2020. arXiv:2007.10203).
PubDate: 20221201

 Local Times for Continuous Paths of Arbitrary Regularity

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Abstract: Abstract We study a pathwise local time of even integer order \(p \ge 2\) , defined as an occupation density, for continuous functions with finite pth variation along a sequence of time partitions. With this notion of local time and a new definition of the Föllmer integral, we establish Tanakatype changeofvariable formulas in a pathwise manner. We also derive some identities involving this highorder pathwise local time, each of which generalizes a corresponding identity from the theory of semimartingale local time. We then use collision local times between multiple functions of arbitrary regularity to study the dynamics of ranked continuous functions.
PubDate: 20221201

 A New Life of Pearson’s Skewness

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Abstract: Abstract In this work, we show how coupling and stochastic dominance methods can be successfully applied to a classical problem of rigorizing Pearson’s skewness. Here, we use Fréchet means to define generalized notions of positive and negative skewness that we call truly positive and truly negative. Then, we apply a stochastic dominance approach in establishing criteria for determining whether a continuous random variable is truly positively skewed. Intuitively, this means that the scaled right tail of the probability density function exhibits strict stochastic dominance over the equivalently scaled left tail. Finally, we use the stochastic dominance criteria and establish some basic examples of true positive skewness, thus demonstrating how the approach works in general.
PubDate: 20221201

 A Simple Method to Find All Solutions to the Functional Equation of the
Smoothing Transform
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Abstract: Abstract Given a nonincreasing null sequence \(T = (T_j)_{j \geqslant 1}\) of nonnegative random variables satisfying some classical integrability assumptions and \({\mathbb {E}}(\sum _{j}T_{j}^{\alpha })=1\) for some \(\alpha >0\) , we characterize the solutions of the wellknown functional equation $$\begin{aligned} f(t)\,=\,\textstyle {\mathbb {E}}\left( \prod _{j\geqslant 1}f(tT_{j})\right) ,\quad t\geqslant 0, \end{aligned}$$ related to the socalled smoothing transform and its mintype variant. In order to do so within the class of nonnegative and nonincreasing functions, we provide a new threestep method whose merits are that it simplifies earlier approaches in some relevant aspects; it works under weaker, close to optimal conditions in the socalled boundary case when \({\mathbb {E}}\big (\sum _{j\geqslant 1}T_{j}^{\alpha }\log T_{j}\big )=0\) ; it can be expected to work as well in more general setups like random environment. At the end of this article, we also give a onetoone correspondence between those solutions that are Laplace transforms and thus correspond to the fixed points of the smoothing transform and certain fractal random measures. The latter are defined on the boundary of a weighted tree related to an associated branching random walk.
PubDate: 20221201

 Strong Approximation of the Anisotropic Random Walk Revisited

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Abstract: Abstract We study the path behavior of the anisotropic random walk on the twodimensional lattice \(\mathbb {Z}^2\) . Simultaneous strong approximation of its components are given.
PubDate: 20221201

 A Limit Theorem for Bernoulli Convolutions and the $$\Phi $$ Variation of
Functions in the Takagi Class
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Abstract: Abstract We consider a probabilistic approach to compute the Wiener–Young \(\Phi \) variation of fractal functions in the Takagi class. Here, the \(\Phi \) variation is understood as a generalization of the quadratic variation or, more generally, the pth variation of a trajectory computed along the sequence of dyadic partitions of the unit interval. The functions \(\Phi \) we consider form a very wide class of functions that are regularly varying at zero. Moreover, for each such function \(\Phi \) , our results provide in a straightforward manner a large and tractable class of functions that have nontrivial and linear \(\Phi \) variation. As a corollary, we also construct stochastic processes whose sample paths have nontrivial, deterministic, and linear \(\Phi \) variation for each function \(\Phi \) from our class. The proof of our main result relies on a limit theorem for certain sums of Bernoulli random variables that converge to an infinite Bernoulli convolution.
PubDate: 20221201

 Embrechts–Goldie’s Problem on the Class of Lattice Convolution
Equivalent Distributions
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Abstract: Abstract We show that the class of lattice convolution equivalent distributions is not closed under convolution roots. We prove that the class of lattice convolution equivalent distributions is closed under convolution roots under the assumption of the exponentially asymptotic decreasing condition. This result is extended to the class \(\mathcal {S}_{\Delta }\) of \(\Delta \) subexponential distributions. As a corollary, we show that the class \(\mathcal {S}_{\Delta }\) is closed under convolution roots in the class \(\mathcal {L}_{\Delta }\) . Moreover, we prove that the class of lattice convolution equivalent distributions is not closed under convolutions. Finally, we give a survey on the closure under convolution roots of the other distribution classes.
PubDate: 20221201
