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.

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.

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.

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.

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 A martingale view of Blackwell’s renewal theorem and its extensions to a general counting process

2019 ◽  
Vol 56 (2) ◽  
pp. 602-623
Author(s):  
Daryl J. Daley ◽  
Masakiyo Miyazawa
Keyword(s):  
Asymptotic Behaviour ◽  
Renewal Process ◽  
Stochastic Analysis ◽  
Counting Process ◽  
Intensity Function ◽  
Counting Processes ◽  
Renewal Theorem ◽  
Basic Tool ◽  
Semimartingale Representation ◽  
Blackwell's Renewal Theorem

AbstractMartingales constitute a basic tool in stochastic analysis; this paper considers their application to counting processes. We use this tool to revisit a renewal theorem and give extensions for various counting processes. We first consider a renewal process as a pilot example, deriving a new semimartingale representation that differs from the standard decomposition via the stochastic intensity function. We then revisit Blackwell’s renewal theorem, its refinements and extensions. Based on these observations, we extend the semimartingale representation to a general counting process, and give conditions under which asymptotic behaviour similar to Blackwell’s renewal theorem holds.

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