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 [2469 journals] 
 From Irrevocably Modulated Filtrations to Dynamical Equations Over Random
Networks
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Abstract: Abstract We develop a probabilistic information framework via what we call irrevocably modulated filtrations produced by noninvertible matrixvalued 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 switchedoff 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 Lietype operator as a morphism between spaces of sigmaalgebras, 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: 20220929

 On Order Isomorphisms Intertwining Semigroups for Dirichlet Forms

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Abstract: Abstract This paper is devoted to characterizing socalled order isomorphisms intertwining the \(L^2\) semigroups of two quasiregular Dirichlet forms. We first show that every unitary order isomorphism intertwining semigroups is the composition of htransformation and quasihomeomorphism. In addition, under the assumption that the underlying spaces admit socalled irreducible decompositions for Dirichlet forms, every (not necessarily unitary) order isomorphism intertwining semigroups can be expressed as the composition of htransformation, quasihomeomorphism and multiplication by a certain step function.
PubDate: 20220927

 Limit Theorems for Iterates of the Szász–Mirakyan Operator in
Probabilistic View
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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 secondorder 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: 20220913

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

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PubDate: 20220901

 A note on the maximal expected local time of $${\text {L}}_2$$ bounded
martingales
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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: 20220901

 On the Additive Property of Finitely Additive Measures

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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: 20220901

 Estimates of Certain Exit Probabilities for pAdic Brownian Bridges

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Abstract: Abstract For each prime p, a diffusion constant together with a positive exponent specify a Vladimirov operator and an associated padic diffusion equation. The fundamental solution of this pseudodifferential equation gives rise to a measure on the Skorokhod space of padic 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: 20220901

 Lie Point Symmetries of Autonomous Scalar FirstOrder ItÃ´ Stochastic
Delay Ordinary Differential Equations
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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: 20220901

 Local Central Limit Theorem for Multigroup Curie–Weiss Models

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Abstract: Abstract We study a multigroup version of the meanfield Ising model, also called Curie–Weiss model. It is known that, in the hightemperature 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 hightemperature regime.
PubDate: 20220901

 Octonionic Brownian Windings

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Abstract: Abstract We define and study windings along Brownian paths on octonionic, Euclidean, projective, and hyperbolic spaces which are isometric to 8dimensional 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 longtime behavior.
PubDate: 20220901

 On a Certain Operator Related to Blackwell’s Markov Chain

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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: 20220901

 On Smooth Mesoscopic Linear Statistics of the Eigenvalues of Random
Permutation Matrices
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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 scaleinvariant Poisson point process.
PubDate: 20220901

 Large and Moderate Deviations Principles and Central Limit Theorem for the
Stochastic 3D Primitive Equations with GradientDependent Noise
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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: 20220901

 Excursions of the Brox Diffusion

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Abstract: Abstract We describe the excursion point process of the socalled 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: 20220901

 Divergence Criterion for a Class of Random Series Related to the Partial
Sums of I.I.D. Random Variables
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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: 20220901

 Strong Solutions to a BetaWishart Particle System

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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 nonnegative, 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: 20220901

 On a Weak Law of Large Numbers with Regularly Varying Normalizing
Sequences
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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: 20220901

 Positivity of the Density for Rough Differential Equations

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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: 20220901

 Convergence Towards the End Space for Random Walks on Schreier Graphs

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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 nontrivial Poisson boundary. Some assumption on the moment of the measure is necessary as follows from an example by Juschenko and Zheng.
PubDate: 20220901

 Functional Limit Theorems for the Pólya Urn

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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: 20220901
