Random Measure Algebras Under O-dot Product and Morse-Transue Integral Convolution

In this article, we consider two operations of random measures: O-dot product and the convolution product by Morse-Transue integral. With these two operations, we construct algebras of random measures. Also we investigate further on the explicit forms of the products of Wiener processes by O-dot operation and by Morse-Transue integral convolution.

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Random measure algebras

Topology, Geometry, and Dynamics - Contemporary Mathematics ◽

10.1090/conm/774/15573 ◽

2021 ◽

pp. 195-204

Author(s):

Jason Park

Keyword(s):

Vector Space ◽

Random Measure ◽

Random Measures ◽

Content Type ◽

Dot Product

In this article, we introduce algebras of random measures. Algebra is a vector space V V over a field F F with a multiplication satisfying the property: 1) distribution and 2) c ( x ⋅ y ) = ( c x ) ⋅ y = x ⋅ ( c y ) c(x\cdot y) = (cx)\cdot y = x\cdot (cy) for every c ∈ F c \in F and x , y ∈ V x, y \in V . The first operation is a trivial addition operation. For the second operation, we present three different methods 1) a convolution by covariance method, 2) O-dot product, 3) a convolution of bimeasures by Morse-Transue integral. With those operations, it is possible to build three different algebras of random measures.

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On Product Measures Associated with Stationary Processes

10.1093/oxfordhb/9780199568444.013.15 ◽

2018 ◽

Author(s):

Alain Boudou ◽

Yves Romain

Keyword(s):

Type Theorem ◽

Random Measure ◽

Continuous Homomorphism ◽

Stationary Processes ◽

Convolution Product ◽

Random Measures ◽

Unknown Quantity ◽

Shift Operators ◽

Potential Applications ◽

Product Measures

This article considers the connections between product measures and stationary processes. It first provides an overview of historical facts and relevant terminology, basic concepts and the mathematical approach. In particular, it discusses random measures, the projection-valued spectral measure (PVSM), convolution products, and the association between shift operators and PVSMs. It then presents the main results and their first potential applications, focusing on stochastic integrals, the image of a random measure under measurable mapping, the existence of a transport-type theorem, and the transpose of a continuous homomorphism between groups. It also describes the PVSM associated with a unitary operator, the convolution product of two PVSMs, the unitary operators generated by a PVSM, extension of the convolution product of two PVSMs, an equation where the unknown quantity is a PVSM, and the convolution product of two random measures. The article concludes with an analysis of mathematical developments related to the previous results.

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On intensities of modulated Cox measures

Journal of Applied Mathematics and Stochastic Analysis ◽

10.1155/s1048953398000343 ◽

1998 ◽

Vol 11 (3) ◽

pp. 411-423 ◽

Cited By ~ 2

Author(s):

Jewgeni H. Dshalalow

Keyword(s):

Stochastic Process ◽

Stochastic Models ◽

Queueing Systems ◽

Random Measure ◽

Random Measures ◽

Explicit Formulas ◽

State Dependent

In this paper we introduce and study functionals of the intensities of random measures modulated by a stochastic process ξ, which occur in applications to stochastic models and telecommunications. Modulation of a random measure by ξ is specified for marked Cox measures. Particular cases of modulation by ξ as semi-Markov and semiregenerative processes enabled us to obtain explicit formulas for the named intensities. Examples in queueing (systems with state dependent parameters, Little's and Campbell's formulas) demonstrate the use of the results.

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Rarefactions of compound point processes

Journal of Applied Probability ◽

10.1017/s0021900200037426 ◽

1984 ◽

Vol 21 (04) ◽

pp. 710-719

Author(s):

Richard F. Serfozo

Keyword(s):

Poisson Process ◽

Point Process ◽

Classical Result ◽

Point Processes ◽

Rare Events ◽

Random Measure ◽

Bernoulli Trials ◽

Random Measures ◽

Infinitely Divisible ◽

Independent Increments

The Poisson process is regarded as a point process of rare events because of the classical result that the number of successes in a sequence of Bernoulli trials is asymptotically Poisson as the probability of a success tends to 0. It is shown that this rareness property of the Poisson process is characteristic of any infinitely divisible point process or random measure with independent increments. These processes and measures arise as limits of certain rarefactions of compound point processes: purely atomic random measures with uniformly null atom sizes. Examples include thinnings and partitions of point processes.

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The identification problem for BSDEs driven by possibly non-quasi-left-continuous random measures

Stochastics and Dynamics ◽

10.1142/s0219493720400110 ◽

2020 ◽

Vol 20 (06) ◽

pp. 2040011

Author(s):

Elena Bandini ◽

Francesco Russo

Keyword(s):

Random Measure ◽

Jump Process ◽

Identification Problem ◽

Local Martingale ◽

Random Measures ◽

Underlying Process ◽

Process Solution ◽

Deterministic Function ◽

Singular Coefficients

In this paper, we focus on the so-called identification problem for a BSDE driven by a continuous local martingale and a possibly non-quasi-left-continuous random measure. Supposing that a solution [Formula: see text] of a BSDE is such that [Formula: see text] where [Formula: see text] is an underlying process and [Formula: see text] is a deterministic function, solving the identification problem consists in determining [Formula: see text] and [Formula: see text] in terms of [Formula: see text]. We study the over-mentioned identification problem under various sets of assumptions and we provide a family of examples including the case when [Formula: see text] is a non-semimartingale jump process solution of an SDE with singular coefficients.

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On the modified Palm version

Advances in Applied Probability ◽

10.1017/apr.2018.85 ◽

2018 ◽

Vol 50 (A) ◽

pp. 271-280

Author(s):

Hermann Thorisson

Keyword(s):

Limit Theorem ◽

Abelian Groups ◽

Random Measure ◽

Random Measures ◽

Locally Compact ◽

Standard Version ◽

Biased Coin

Abstract The interpretation of the ‘standard’ Palm version of a stationary random measure ξ is that it behaves like ξ conditioned on containing the origin in its mass. The interpretation of the ‘modified’ Palm version is that it behaves like ξ seen from a typical location in its mass. In this paper we shall focus on the modified Palm version, comparing it with the standard version in the transparent case of mixed biased coin tosses, and then establishing a limit theorem that motivates the above interpretation in the case of random measures on locally compact second countable Abelian groups possessing Følner averaging sets.

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Perturbation analysis of functionals of random measures

Advances in Applied Probability ◽

10.2307/1427827 ◽

1995 ◽

Vol 27 (2) ◽

pp. 306-325 ◽

Cited By ~ 4

Author(s):

François Baccelli ◽

Maurice Klein ◽

Sergei Zuyev

Keyword(s):

Phase Space ◽

Random Measure ◽

Random Processes ◽

Stationary Processes ◽

Light Traffic ◽

Random Measures ◽

First Order ◽

Order Expansion ◽

Palm Measure ◽

New Algorithms

We use the fact that the Palm measure of a stationary random measure is invariant to phase space change to generalize the light traffic formula initially obtained for stationary processes on a line to general spaces. This formula gives a first-order expansion for the expectation of a functional of the random measure when its intensity vanishes. This generalization leads to new algorithms for estimating gradients of functionals of geometrical random processes.

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Extremal processes and random measures

Journal of Applied Probability ◽

10.2307/3212445 ◽

1975 ◽

Vol 12 (2) ◽

pp. 316-323 ◽

Cited By ~ 9

Author(s):

R. W. Shorrock

Keyword(s):

Random Measure ◽

Record Values ◽

Poisson Processes ◽

Two Dimensional ◽

Random Measures ◽

Record Times ◽

Upper Record Values ◽

Extremal Processes ◽

Upper Record ◽

Gamma Processes

Discrete time extremal processes with a continuous underlying c.d.f. are random measures which can be viewed as two-dimensional Poisson processes and this representation is used to obtain the conditional law of the sequence of states the process passes through (upper record values) given the sequence of holding times in states (inter-record times). In addition the Gamma processes (which lead to the Ferguson Dirichlet processes) and a random measure that arises in sampling from a biological population are discussed as two-dimensional Poisson processes.

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Random fractals and probability metrics

Advances in Applied Probability ◽

10.1017/s0001867800010375 ◽

2000 ◽

Vol 32 (4) ◽

pp. 925-947 ◽

Cited By ~ 14

Author(s):

John E. Hutchinson ◽

Ludger Rüschendorf

Keyword(s):

Probability Distributions ◽

Random Measure ◽

Iterative Sequence ◽

Probability Metrics ◽

Random Measures ◽

Random Mass ◽

Uniqueness Results ◽

Regularity Properties ◽

Self Similar ◽

Mass Case

New metrics are introduced in the space of random measures and are applied, with various modifications of the contraction method, to prove existence and uniqueness results for self-similar random fractal measures. We obtain exponential convergence, both in distribution and almost surely, of an iterative sequence of random measures (defined by means of the scaling operator) to a unique self-similar random measure. The assumptions are quite weak, and correspond to similar conditions in the deterministic case.The fixed mass case is handled in a direct way based on regularity properties of the metrics and the properties of a natural probability space. Proving convergence in the random mass case needs additional tools, such as a specially adapted choice of the space of random measures and of the space of probability distributions on measures, the introduction of reweighted sequences of random measures and a comparison technique.

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