Random fractals and probability metrics

2000 ◽  
Vol 32 (4) ◽  
pp. 925-947 ◽  
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.

scholarly journals On intensities of modulated Cox measures

1998 ◽  
Vol 11 (3) ◽  
pp. 411-423 ◽  
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.

Rarefactions of compound point processes

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.

scholarly journals Multifractal spectra for random self-similar measures via branching processes

2011 ◽  
Vol 43 (01) ◽  
pp. 1-39
Author(s):  
J. D. Biggins ◽  
B. M. Hambly ◽  
O. D. Jones
Keyword(s):  
Random Number ◽  
Branching Processes ◽  
Multifractal Spectrum ◽  
Compact Set ◽  
Similar Measure ◽  
Random Mass ◽  
Random Fractal ◽  
Multifractal Spectra ◽  
Self Similar ◽  
Random Division

Start with a compact setK⊂Rd. This has a random number of daughter sets, each of which is a (rotated and scaled) copy ofKand all of which are insideK. The random mechanism for producing daughter sets is used independently on each of the daughter sets to produce the second generation of sets, and so on, repeatedly. The random fractal setFis the limit, asngoes to ∞, of the union of thenth generation sets. In addition,Khas a (suitable, random) mass which is divided randomly between the daughter sets, and this random division of mass is also repeated independently, indefinitely. This division of mass will correspond to a random self-similar measure onF. The multifractal spectrum of this measure is studied here. Our main contributions are dealing with the geometry of realisations inRdand drawing systematically on known results for general branching processes. In this way we generalise considerably the results of Arbeiter and Patzschke (1996) and Patzschke (1997).

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

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.

scholarly journals On the modified Palm version

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.

Perturbation analysis of functionals of random measures

1995 ◽  
Vol 27 (2) ◽  
pp. 306-325 ◽  
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.

Extremal processes and random measures

1975 ◽  
Vol 12 (2) ◽  
pp. 316-323 ◽  
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.

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

2019 ◽  
Vol 8 (6) ◽  
pp. 70
Author(s):  
Jason Hong Jae Park
Keyword(s):  
Random Measure ◽  
Convolution Product ◽  
Random Measures ◽  
Integral Convolution ◽  
Wiener Processes ◽  
Dot Product

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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