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.

scholarly journals On modulated random measures

1991 ◽  
Vol 4 (4) ◽  
pp. 305-312 ◽  
Author(s):  
Jewgeni H. Dshalalow
Keyword(s):  
Random Measure ◽  
Regenerative Process ◽  
Random Measures ◽  
Locally Compact ◽  
Compact Spaces ◽  
Special Cases ◽  
Queueing Processes

In this paper the author introduces the notion of a modulated marked random measure, Zξ, on the class of locally compact and σ-compact spaces with countable bases. As special cases, are marked processes modulated by ξ are considered where ξ is a semi-Markov or semi-regenerative process. For either case, the intensities k=limt→∞1tE[Zξ([0,t])] are evaluated in terms of parameters of ξ. Examples and applications to inventories, queueing processes and economics are discussed.

Note on generalizations of Mecke's formula and extensions of H = λG

1995 ◽  
Vol 32 (1) ◽  
pp. 105-122 ◽  
Author(s):  
Masakiyo Miyazawa
Keyword(s):  
Abelian Group ◽  
Invariant Measure ◽  
Compact Abelian Group ◽  
Sample Path ◽  
Random Measure ◽  
Locally Compact Abelian Group ◽  
Random Measures ◽  
Locally Compact ◽  
Stationary Version ◽  
Little's Formula

Mecke's formula is concerned with a stationary random measure and shift-invariant measure on a locally compact Abelian group , and relates integrations concerning them to each other. For , we generalize this to a pair of random measures which are jointly stationary. The resulting formula extends the so-called Swiss Army formula, which was recently obtained as a generalization for Little's formula. The generalized Mecke formula, which is called GMF, can be also viewed as a generalization of the stationary version of H = λG. Under the stationary and ergodic assumptions, we apply it to derive many sample path formulas which have been known as extensions of H = λG. This will make clear what kinds of probabilistic conditions are sufficient to get them. We also mention a further generalization of Mecke's formula.

Note on generalizations of Mecke's formula and extensions of H = λG

1995 ◽  
Vol 32 (01) ◽  
pp. 105-122
Author(s):  
Masakiyo Miyazawa
Keyword(s):  
Abelian Group ◽  
Invariant Measure ◽  
Compact Abelian Group ◽  
Sample Path ◽  
Random Measure ◽  
Locally Compact Abelian Group ◽  
Random Measures ◽  
Locally Compact ◽  
Stationary Version ◽  
Little's Formula

Mecke's formula is concerned with a stationary random measure and shift-invariant measure on a locally compact Abelian group, and relates integrations concerning them to each other. For, we generalize this to a pair of random measures which are jointly stationary. The resulting formula extends the so-called Swiss Army formula, which was recently obtained as a generalization for Little's formula. The generalized Mecke formula, which is called GMF, can be also viewed as a generalization of the stationary version ofH = λG. Under the stationary and ergodic assumptions, we apply it to derive many sample path formulas which have been known as extensions ofH = λG.This will make clear what kinds of probabilistic conditions are sufficient to get them. We also mention a further generalization of Mecke's formula.

Scaling sets and generalized scaling sets on Cantor dyadic group

2020 ◽  
Vol 18 (04) ◽  
pp. 2050019
Author(s):  
Prasadini Mahapatra ◽  
Divya Singh
Keyword(s):  
Abelian Groups ◽  
Real Case ◽  
Locally Compact ◽  
Wavelet Sets ◽  
Locally Compact Abelian Groups ◽  
Dyadic Group ◽  
Compact Abelian Groups ◽  
Cantor Dyadic Group

Scaling and generalized scaling sets determine wavelet sets and hence wavelets. In real case, wavelet sets were proved to be an important tool for the construction of MRA as well as non-MRA wavelets. However, any result related to scaling/generalized scaling sets is not available in case of locally compact abelian groups. This paper gives a characterization of scaling sets and its generalized version along with relevant examples in dual Cantor dyadic group [Formula: see text]. These results can further be generalized to arbitrary locally compact abelian groups.

Singular measures with absolutely continuous convolution squares

1966 ◽  
Vol 62 (3) ◽  
pp. 399-420 ◽  
Author(s):  
Edwin Hewitt ◽  
Herbert S. Zuckerman
Keyword(s):  
Abelian Groups ◽  
Natural Order ◽  
Continuous Measure ◽  
Locally Compact ◽  
Absolutely Continuous ◽  
Character Group ◽  
Locally Compact Abelian Groups ◽  
Singular Measures ◽  
Compact Abelian Groups ◽  
Nikodym Derivative

Introduction. A famous construction of Wiener and Wintner ((13)), later refined by Salem ((11)) and extended by Schaeffer ((12)) and Ivašev-Musatov ((8)), produces a non-negative, singular, continuous measure μ on [ − π,π[ such thatfor every ∈ > 0. It is plain that the convolution μ * μ is absolutely continuous and in fact has Lebesgue–Radon–Nikodým derivative f such that For general locally compact Abelian groups, no exact analogue of (1 · 1) seems possible, as the character group may admit no natural order. However, it makes good sense to ask if μ* μ is absolutely continuous and has pth power integrable derivative. We will construct continuous singular measures μ on all non-discrete locally compact Abelian groups G such that μ * μ is a absolutely continuous and for which the Lebesgue–Radon–Nikodým derivative of μ * μ is in, for all real p > 1.

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