Probability metrics and recursive algorithms

1995 ◽  
Vol 27 (3) ◽  
pp. 770-799 ◽  
Author(s):  
S. T. Rachev ◽  
L. Rüschendorf
Keyword(s):  
Asymptotic Behaviour ◽  
Recursive Algorithms ◽  
Probability Metrics ◽  
Contraction Method ◽  
Probability Metric ◽  
Wide Range

It is shown by means of several examples that probability metrics are a useful tool to study the asymptotic behaviour of (stochastic) recursive algorithms. The basic idea of this approach is to find a ‘suitable' probability metric which yields contraction properties of the transformations describing the limits of the algorithm. In order to demonstrate the wide range of applicability of this contraction method we investigate examples from various fields, some of which have already been analysed in the literature.

Probability metrics and recursive algorithms

1995 ◽  
Vol 27 (03) ◽  
pp. 770-799 ◽  
Author(s):  
S. T. Rachev ◽  
L. Rüschendorf
Keyword(s):  
Asymptotic Behaviour ◽  
Recursive Algorithms ◽  
Probability Metrics ◽  
Contraction Method ◽  
Probability Metric ◽  
Wide Range

It is shown by means of several examples that probability metrics are a useful tool to study the asymptotic behaviour of (stochastic) recursive algorithms. The basic idea of this approach is to find a ‘suitable' probability metric which yields contraction properties of the transformations describing the limits of the algorithm. In order to demonstrate the wide range of applicability of this contraction method we investigate examples from various fields, some of which have already been analysed in the literature.

scholarly journals Pólya Urns Via the Contraction Method

2014 ◽  
Vol 23 (6) ◽  
pp. 1148-1186 ◽  
Author(s):  
MARGARETE KNAPE ◽  
RALPH NEININGER
Keyword(s):  
Asymptotic Behaviour ◽  
Discrete Time ◽  
Normal Limit ◽  
Rooted Trees ◽  
Limit Laws ◽  
Contraction Method ◽  
Limit Distributions

We propose an approach to analysing the asymptotic behaviour of Pólya urns based on the contraction method. For this, a new combinatorial discrete-time embedding of the evolution of the urn into random rooted trees is developed. A decomposition of these trees leads to a system of recursive distributional equations which capture the distributions of the numbers of balls of each colour. Ideas from the contraction method are used to study such systems of recursive distributional equations asymptotically. We apply our approach to a couple of concrete Pólya urns that lead to limit laws with normal limit distributions, with non-normal limit distributions and with asymptotic periodic distributional behaviour.

Stochastic Analysis of ‘Simultaneous Merge–Sort'

1997 ◽  
Vol 29 (03) ◽  
pp. 669-694
Author(s):  
M. Cramer
Keyword(s):  
Asymptotic Behaviour ◽  
Stochastic Analysis ◽  
Challenging Problem ◽  
Contraction Method ◽  
Merge Sort ◽  
Kolmogorov Metric

The asymptotic behaviour of the recursion is investigated; Yk describes the number of comparisons which have to be carried out to merge two sorted subsequences of length 2k –1 and Mk can be interpreted as the number of comparisons of ‘Simultaneous Merge–Sort'. The challenging problem in the analysis of the above recursion lies in the fact that it contains a maximum as well as a sum. This demands different ideal properties for the metric in the contraction method. By use of the weighted Kolmogorov metric it is shown that an exponential normalization provides the recursion's convergence. Furthermore, one can show that any sequence of linear normalizations of Mk must converge towards a constant if it converges in distribution at all.

Stochastic Analysis of ‘Simultaneous Merge–Sort'

1997 ◽  
Vol 29 (3) ◽  
pp. 669-694 ◽  
Author(s):  
M. Cramer
Keyword(s):  
Asymptotic Behaviour ◽  
Stochastic Analysis ◽  
Challenging Problem ◽  
Contraction Method ◽  
Merge Sort ◽  
Kolmogorov Metric

The asymptotic behaviour of the recursion is investigated; Yk describes the number of comparisons which have to be carried out to merge two sorted subsequences of length 2k–1 and Mk can be interpreted as the number of comparisons of ‘Simultaneous Merge–Sort'. The challenging problem in the analysis of the above recursion lies in the fact that it contains a maximum as well as a sum. This demands different ideal properties for the metric in the contraction method. By use of the weighted Kolmogorov metric it is shown that an exponential normalization provides the recursion's convergence. Furthermore, one can show that any sequence of linear normalizations of Mk must converge towards a constant if it converges in distribution at all.

The contraction method for recursive algorithms

Algorithmica ◽  
2001 ◽  
Vol 29 (1-2) ◽  
pp. 3-33 ◽  
Author(s):  
U. Rösler ◽  
L. Rüschendorf
Keyword(s):  
Recursive Algorithms ◽  
Contraction Method

Recent Applications of High Resolution Electron Microscopy to Crystalline Arrays of Virus Particles

1978 ◽  
Vol 36 (3) ◽  
pp. 470-482 ◽  
Author(s):  
R.W. Horne
Keyword(s):  
Electron Microscopy ◽  
Image Processing ◽  
Analytical Techniques ◽  
Staining Technique ◽  
High Resolution Electron Microscopy ◽  
Optical Diffraction ◽  
Full Potential ◽  
Virus Particles ◽  
Resolution Electron ◽  
Wide Range

The technique of surrounding virus particles with a neutralised electron dense stain was described at the Fourth International Congress on Electron Microscopy, Berlin 1958 (see Home & Brenner, 1960, p. 625). For many years the negative staining technique in one form or another, has been applied to a wide range of biological materials. However, the full potential of the method has only recently been explored following the development and applications of optical diffraction and computer image analytical techniques to electron micrographs (cf. De Hosier & Klug, 1968; Markham 1968; Crowther et al., 1970; Home & Markham, 1973; Klug & Berger, 1974; Crowther & Klug, 1975). These image processing procedures have allowed a more precise and quantitative approach to be made concerning the interpretation, measurement and reconstruction of repeating features in certain biological systems.

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