Random elements in separable metric spaces

1973 ◽  
pp. 15-30
Author(s):  
W. J. Padgett ◽  
R. L. Taylor
Keyword(s):  
Metric Spaces ◽  
Random Elements

scholarly journals Expectation in metric spaces and characterizations of Banach spaces

2009 ◽  
Vol 42 (4) ◽  
Author(s):  
Artur Bator ◽  
Wiesław Zięba
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Metric Space ◽  
Metric Spaces ◽  
Bochner Integral ◽  
Random Elements

AbstractWe consider different definitions of expectation of random elements taking values in metric spaces. All such definitions are valid also in Banach spaces and in this case the results coincide with the Bochner integral. There may exist an isometry between considered metric space and some Banach space and in this case one can use the Bochner integral instead of expectation in metric space. We give some conditions which ensure existence of such isometry, for two different definitions of expectation in metric space.

Convergence of elements in random normed spaces

1975 ◽  
Vol 12 (1) ◽  
pp. 31-47 ◽  
Author(s):  
Robert Lee Taylor
Keyword(s):  
Normed Linear Space ◽  
Metric Spaces ◽  
Normed Spaces ◽  
Linear Topology ◽  
Laws Of Large Numbers ◽  
Convergence In Measure ◽  
Large Numbers ◽  
Random Elements ◽  
Random Normed Spaces ◽  
Necessary And Sufficient

For a random normed space of mappings into a separable normed linear space, convergence of identically distributed elements in i the random norm (norm distribution) is shown to be equivalent to convergence in measure in the weak linear topology. Convergence in measure in each coordinate of a Schauder basis is also shown to be a necessary and sufficient condition for convergence in the random norm topology. These results have laws of large numbers for random elements in separable normed linear spaces as almost immediate corollaries and illustrate some of the recently obtained laws of large numbers for random elements. Similar results are also given for elements which need not have the same norm distributions, and the results are extended to linear metric spaces. Finally, applications of the results to stochastic processes are considered.

scholarly journals Convergence to Stable Laws in the Space D

2015 ◽  
Vol 52 (01) ◽  
pp. 1-17 ◽  
Author(s):  
François Roueff ◽  
Philippe Soulier
Keyword(s):  
Point Process ◽  
Regular Variation ◽  
Metric Spaces ◽  
Empirical Process ◽  
Stable Distributions ◽  
Stable Laws ◽  
Random Elements ◽  
Process Convergence ◽  
Reward Process ◽  
Renewal Reward Process

We study the convergence of centered and normalized sums of independent and identically distributed random elements of the spaceDof càdlàg functions endowed with Skorokhod'sJ1topology, to stable distributions inD. Our results are based on the concept of regular variation on metric spaces and on point process convergence. We provide some applications; in particular, to the empirical process of the renewal-reward process.

scholarly journals Convergence to Stable Laws in the SpaceD

2015 ◽  
Vol 52 (1) ◽  
pp. 1-17 ◽  
Author(s):  
François Roueff ◽  
Philippe Soulier
Keyword(s):  
Point Process ◽  
Regular Variation ◽  
Metric Spaces ◽  
Empirical Process ◽  
Stable Distributions ◽  
Stable Laws ◽  
Random Elements ◽  
Process Convergence ◽  
Reward Process ◽  
Renewal Reward Process

We study the convergence of centered and normalized sums of independent and identically distributed random elements of the spaceDof càdlàg functions endowed with Skorokhod'sJ1topology, to stable distributions inD. Our results are based on the concept of regular variation on metric spaces and on point process convergence. We provide some applications; in particular, to the empirical process of the renewal-reward process.

Crystallography in two-dimensional metric spaces*

1969 ◽  
Vol 130 (1-6) ◽  
pp. 277-303 ◽  
Author(s):  
Aloysio Janner ◽  
Edgar Ascher
Keyword(s):  
Metric Spaces ◽  
Two Dimensional
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