Subjects -> STATISTICS (Total: 130 journals)
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 Journal of Theoretical ProbabilityJournal Prestige (SJR): 0.981 Citation Impact (citeScore): 1Number of Followers: 3      Hybrid journal (It can contain Open Access articles) ISSN (Print) 1572-9230 - ISSN (Online) 0894-9840 Published by Springer-Verlag  [2469 journals]
• From Irrevocably Modulated Filtrations to Dynamical Equations Over Random
Networks

Abstract: Abstract We develop a probabilistic information framework via what we call irrevocably modulated filtrations produced by non-invertible matrix-valued jump processes acting on multivariate observation processes carrying noisy signals. Under certain conditions, we provide dynamical representations of conditional expectation martingales in systems where signals from randomly changing information networks may get irreversibly amalgamated or switched-off over random time horizons. We apply the framework to scenarios where the flow of information goes through multiple modulations before reaching observing agents. This leads us to introduce a Lie-type operator as a morphism between spaces of sigma-algebras, which quantifies information discrepancy caused by different modulation sequences. As another example, we show how random graphs can be used to generate irrevocably modulated filtrations that lead to pure noise scenarios. Finally, we construct systems that exhibit gradual decay of additional sources of information through the choice of spectral radii of the modulators.
PubDate: 2022-09-29

• On Order Isomorphisms Intertwining Semigroups for Dirichlet Forms

Abstract: Abstract This paper is devoted to characterizing so-called order isomorphisms intertwining the $$L^2$$ -semigroups of two quasi-regular Dirichlet forms. We first show that every unitary order isomorphism intertwining semigroups is the composition of h-transformation and quasi-homeomorphism. In addition, under the assumption that the underlying spaces admit so-called irreducible decompositions for Dirichlet forms, every (not necessarily unitary) order isomorphism intertwining semigroups can be expressed as the composition of h-transformation, quasi-homeomorphism and multiplication by a certain step function.
PubDate: 2022-09-27

• Limit Theorems for Iterates of the Szász–Mirakyan Operator in
Probabilistic View

Abstract: Abstract The Szász–Mirakyan operator is known as a positive linear operator which uniformly approximates a certain class of continuous functions on the half line. The purpose of the present paper is to find out limiting behaviors of the iterates of the Szász–Mirakyan operator in a probabilistic point of view. We show that the iterates of the Szász–Mirakyan operator uniformly converge to a continuous semigroup generated by a second-order degenerate differential operator. A probabilistic interpretation of the convergence in terms of a discrete Markov chain constructed from the iterates and a limiting diffusion process on the half line is captured as well.
PubDate: 2022-09-13

• Correction to: A Functional CLT for Partial Traces of Random Matrices

PubDate: 2022-09-01

• A note on the maximal expected local time of $${\text {L}}_2$$ -bounded
martingales

Abstract: Abstract For an $${\text {L}}_2$$ -bounded martingale starting at 0 and having final variance $$\sigma ^2$$ , the expected local time at $$a \in \text {R}$$ is at most $$\sqrt{\sigma ^2+a^2}- a$$ . This sharp bound is attained by Standard Brownian Motion stopped at the first exit time from the interval $$(a-\sqrt{\sigma ^2+a^2},a+\sqrt{\sigma ^2+a^2})$$ . In particular, the maximal expected local time anywhere is at most $$\sigma$$ , and this bound is sharp. Sharp bounds for the expected maximum, maximal absolute value, maximal diameter and maximal number of upcrossings of intervals have been established by Dubins and Schwarz (Societé Mathématique de France, Astérisque 157(8), 129–145 1988), by Dubins et al. (Ann Probab 37(1), 393–402 2009) and by the authors (2018).
PubDate: 2022-09-01

Abstract: Abstract By additive property, we refer to a condition under which $$L^p$$ spaces over finitely additive measures are complete. In their 2000 paper, Basile and Rao gave a necessary and sufficient condition that a finite sum of finitely additive measures has the additive property. We generalize this result to the case of a countable sum of finitely additive measures. We also apply this result to density measures, the finitely additive probabilities on $$\mathbb {N}$$ which extend asymptotic density (also called natural density), and provide the necessary and sufficient condition that a certain type of density measure has the additive property.
PubDate: 2022-09-01

• Estimates of Certain Exit Probabilities for p-Adic Brownian Bridges

Abstract: Abstract For each prime p, a diffusion constant together with a positive exponent specify a Vladimirov operator and an associated p-adic diffusion equation. The fundamental solution of this pseudo-differential equation gives rise to a measure on the Skorokhod space of p-adic valued paths that is concentrated on the paths originating at the origin. We calculate the first exit probabilities of paths from balls and estimate these probabilities for the Brownian bridges.
PubDate: 2022-09-01

• Lie Point Symmetries of Autonomous Scalar First-Order Itô Stochastic
Delay Ordinary Differential Equations

Abstract: Abstract In this paper, we consider an extension of Lie group theory to the class of constant delay autonomous stochastic differential equations of Itô form. The determining equations are deterministic even though they represent the stochastic process. The Lie algebras obtained are of low dimensions, and they form an Abelian group.
PubDate: 2022-09-01

• Local Central Limit Theorem for Multi-group Curie–Weiss Models

Abstract: Abstract We study a multi-group version of the mean-field Ising model, also called Curie–Weiss model. It is known that, in the high-temperature regime of this model, a central limit theorem holds for the vector of suitably scaled group magnetisations, that is, for the sum of spins belonging to each group. In this article, we prove a local central limit theorem for the group magnetisations in the high-temperature regime.
PubDate: 2022-09-01

• Octonionic Brownian Windings

Abstract: Abstract We define and study windings along Brownian paths on octonionic, Euclidean, projective, and hyperbolic spaces which are isometric to 8-dimensional Riemannian model spaces. In particular, the asymptotic laws of these windings are shown to be Gaussian for flat and spherical geometries while the hyperbolic winding exhibits different long-time behavior.
PubDate: 2022-09-01

• On a Certain Operator Related to Blackwell’s Markov Chain

Abstract: Abstract We present an example of a densely defined, linear operator on the $$l^{1}$$ space with the property that each basis vector of the standard Schauder basis of $$l^{1}$$ does not belong to its domain. Our example is based on the construction of a Markov chain with all states instantaneous given by D. Blackwell in 1958. In addition, it turns out that the closure of this operator is the generator of a strongly continuous semigroup of Markov operators associated with Blackwell’s chain.
PubDate: 2022-09-01

• On Smooth Mesoscopic Linear Statistics of the Eigenvalues of Random
Permutation Matrices

Abstract: Abstract We study the limiting behavior of smooth linear statistics of the spectrum of random permutation matrices in the mesoscopic regime, when the permutation follows one of the Ewens measures on the symmetric group. If we apply a smooth enough test function f to all the determinations of the eigenangles of the permutations, we get a convergence in distribution when the order of the permutation tends to infinity. Two distinct kinds of limit appear: if $$f(0)\ne 0$$ , we have a central limit theorem with a logarithmic variance; and if $$f(0) = 0$$ , the convergence holds without normalization and the limit involves a scale-invariant Poisson point process.
PubDate: 2022-09-01

• Large and Moderate Deviations Principles and Central Limit Theorem for the
Stochastic 3D Primitive Equations with Gradient-Dependent Noise

Abstract: Abstract We establish the large deviations principle (LDP), the moderate deviations principle (MDP), and an almost sure version of the central limit theorem (CLT) for the stochastic 3D viscous primitive equations driven by multiplicative white noise allowing dependence on the spatial gradient of velocity with initial data in $$H^2$$ . We establish the LDP using the weak convergence approach by Budjihara and Dupuis and a uniform version of the stochastic Gronwall lemma. The result corrects a minor technical issue in Dong et al. (J Differ Equ 263(5):3110–3146, 2017) and establishes the result for a more general noise. The MDP is established by a similar argument.
PubDate: 2022-09-01

• Excursions of the Brox Diffusion

Abstract: Abstract We describe the excursion point process of the so-called Brox diffusion together with the characteristic measure. We do so in terms of the excursion point process of a Brownian motion. To relate these two processes, we make use of the Itô–McKean representation for diffusions and the representation of their local time. As a first application of this analysis, we obtain the distribution of random variables regarding the local time at certain hitting times. At the end, we show an application that illustrates the connection between the paths of the Brox diffusion and the local minima of the environment.
PubDate: 2022-09-01

• Divergence Criterion for a Class of Random Series Related to the Partial
Sums of I.I.D. Random Variables

Abstract: Abstract Let $$\{X, X_{n};~n \ge 1 \}$$ be a sequence of independent and identically distributed Banach space valued random variables. This paper is devoted to providing a divergence criterion for a class of random series of the form $$\sum _{n=1}^{\infty } f_{n}\left( \left\ S_{n} \right\ \right)$$ where $$S_{n} = X_{1} + \cdots + X_{n}, ~n \ge 1$$ and $$\left\{ f_{n}(\cdot ); n \ge 1 \right\}$$ is a sequence of nonnegative nondecreasing functions defined on $$[0, \infty )$$ . More specifically, it is shown that (i) the above random series diverges almost surely if $$\sum _{n=1}^{\infty } f_{n} \left( cn^{1/2} \right) = \infty$$ for some $$c > 0$$ and (ii) the above random series converges almost surely if $$\sum _{n=1}^{\infty } f_{n} \left( cn^{1/2} \right) < \infty$$ for some $$c > 0$$ provided additional conditions are imposed involving X, the sequences $$\left\{ S_{n};~n \ge 1 \right\}$$ and $$\left\{ f_{n}(\cdot ); n \ge 1 \right\}$$ , and c. A special case of this criterion is a divergence/convergence criterion for the random series $$\sum _{n=1}^{\infty } a_{n} \left\ S_{n} \right\ ^{q}$$ based on the series $$\sum _{n=1}^{\infty } a_{n} n^{q/2}$$ where $$\left\{ a_{n};~n \ge 1 \right\}$$ is a sequence of nonnegative numbers and $$q > 0$$ .
PubDate: 2022-09-01

• Strong Solutions to a Beta-Wishart Particle System

Abstract: Abstract The purpose of this paper is to study the existence and uniqueness of solutions to a stochastic differential equation (SDE) coming from the eigenvalues of Wishart processes. The coordinates are non-negative, evolve as Cox–Ingersoll–Ross (CIR) processes and repulse each other according to a Coulombian like interaction force. We show the existence of strong and pathwise unique solutions to the system until the first multiple collision and give a necessary and sufficient condition on the parameters of the SDEs for this multiple collision not to occur in finite time.
PubDate: 2022-09-01

• On a Weak Law of Large Numbers with Regularly Varying Normalizing
Sequences

Abstract: Abstract The Kolmogorov–Feller weak law of large numbers for i.i.d. random variables has been extended by Gut (J. Theoret. Probab. 17,  769–779, 2004) to the case where the normalizing sequence is regularly varying with index $$1/\rho$$ for some $$\rho \in ]0,1]$$ . In this paper, we show that the sufficiency part in Gut’s theorem is valid without any restriction on the dependence structure of the underlying sequence, provided that $$\rho \ne 1$$ . We also prove the necessity part in Gut’s weak law of large numbers when the summands are pairwise negatively dependent.
PubDate: 2022-09-01

• Positivity of the Density for Rough Differential Equations

Abstract: Abstract Due to recent developments of Malliavin calculus for rough differential equations, it is now known that, under natural assumptions, the law of a unique solution at a fixed time has a smooth density function. Therefore, it is quite natural to ask whether or when the density is strictly positive. In this paper we study this problem from the viewpoint of Aida–Kusuoka–Stroock’s general theory.
PubDate: 2022-09-01

• Convergence Towards the End Space for Random Walks on Schreier Graphs

Abstract: Abstract We consider a transitive action of a finitely generated group G and the Schreier graph $$\varGamma$$ defined by this action for some fixed generating set. For a probability measure $$\mu$$ on G with a finite first moment, we show that if the induced random walk is transient, it converges towards the space of ends of $$\varGamma$$ . As a corollary, we obtain that for a probability measure with a finite first moment on Thompson’s group F, the support of which generates F as a semigroup, the induced random walk on the dyadic numbers has a non-trivial Poisson boundary. Some assumption on the moment of the measure is necessary as follows from an example by Juschenko and Zheng.
PubDate: 2022-09-01

• Functional Limit Theorems for the Pólya Urn

Abstract: Abstract For the plain Pólya urn with two colors, black and white, we prove a functional central limit theorem for the number of white balls, assuming that the initial number of black balls is large. Depending on the initial number of white balls, the limit is either a pure birth process or a diffusion.
PubDate: 2022-09-01

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