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A Markov process associated with plotsize distribution in Czech Land Registry and its numbertheoretic properties
 J. Phys. A: Math. Theor
, 2008
"... The size distribution of land plots is a result of land allocation processes in the past. In the absence of regulation this is a Markov process leading an equilibrium described by a probabilistic equation used commonly in the insurance and financial mathematics. We support this claim by analyzing th ..."
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The size distribution of land plots is a result of land allocation processes in the past. In the absence of regulation this is a Markov process leading an equilibrium described by a probabilistic equation used commonly in the insurance and financial mathematics. We support this claim by analyzing the distribution of two plot types, garden and buildup areas, in the Czech Land Registry pointing out the coincidence with the distribution of prime number factors described by Dickman function in the first case. The distribution of commodities is an important research topic in economy – see [CC07] for an extensive literature overview. In this letter we focus on a particular case, the allocation of land representing a nonconsumable commodity, and a way in which the distribution is reached. Generally speaking, it results from a process of random commodity exchanges between agents in the situation when the aggregate commodity volume is conserved, in other words, one deals with pure trading which leads commodity redistribution. Models of this type were recently intensively discussed [SGG06] and are usually referred to as kinetic exchange models. Our approach here will be different being based on the concept known as
Appendix to “Approximating perpetuities”
, 2012
"... An algorithm for perfect simulation from the unique solution of the distributional fixed point equation Y =d UY +U(1−U) is constructed, where Y and U are independent and U is uniformly distributed on [0, 1]. This distribution comes up as a limit distribution in the probabilistic analysis of the Quic ..."
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An algorithm for perfect simulation from the unique solution of the distributional fixed point equation Y =d UY +U(1−U) is constructed, where Y and U are independent and U is uniformly distributed on [0, 1]. This distribution comes up as a limit distribution in the probabilistic analysis of the Quickselect algorithm. Our simulation algorithm is based on coupling from the past with a multigamma coupler. It has four lines of code.
Exact Simulation of Random Variables that are Solutions of FixedPoint Equations
, 2001
"... An algorithm is developed for the exact simulation from distributions that are defined as fixedpoints of maps between spaces of probability measures. The fixedpoints of the class of maps under consideration include examples of limit distributions of random variables studied in the probabilistic an ..."
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An algorithm is developed for the exact simulation from distributions that are defined as fixedpoints of maps between spaces of probability measures. The fixedpoints of the class of maps under consideration include examples of limit distributions of random variables studied in the probabilistic analysis of algorithms. The sampling algorithm relies on a modified rejection method.
A Gaussian limit process for optimal FIND algorithms
, 2013
"... We consider versions of the FIND algorithm where the pivot element used is the median of a subset chosen uniformly at random from the data. For the median selection we assume that subsamples of size asymptotic to c · nα are chosen, where 0 < α ≤ 1 2, c> 0 and n is the size of the data set to b ..."
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We consider versions of the FIND algorithm where the pivot element used is the median of a subset chosen uniformly at random from the data. For the median selection we assume that subsamples of size asymptotic to c · nα are chosen, where 0 < α ≤ 1 2, c> 0 and n is the size of the data set to be split. We consider the complexity of FIND as a process in the rank to be selected and measured by the number of key comparisons required. After normalization we show weak convergence of the complexity to a centered Gaussian process as n → ∞, which depends on α. The proof relies on a contraction argument for probability distributions on càdlàg functions. We also identify the covariance function of the Gaussian limit process and discuss path and tail properties. AMS 2010 subject classifications. Primary 60F17, 68P10; secondary 60G15, 60C05, 68Q25. Key words. FIND algorithm, Quickselect, complexity, key comparisons, functional limit theorem,
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"... An dieser Stelle möchte ich mich bei all jenen bedanken, die mir diese Diplomarbeit ermöglicht und zu ihrer Erstellung beigetragen haben. Herrn Prof. Dr. R. Neininger danke ich für das interessante Thema und die hervorragende Betreuung. Sein Interesse am Fortgang der Arbeit, seine konstruktiven ..."
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An dieser Stelle möchte ich mich bei all jenen bedanken, die mir diese Diplomarbeit ermöglicht und zu ihrer Erstellung beigetragen haben. Herrn Prof. Dr. R. Neininger danke ich für das interessante Thema und die hervorragende Betreuung. Sein Interesse am Fortgang der Arbeit, seine konstruktiven
Analysis of the Expected Number . . .
, 2009
"... When algorithms for sorting and searching are applied to keys that are represented as bit strings, we can quantify the performance of the algorithms not only in terms of the number of key comparisons required by the algorithms but also in terms of the number of bit comparisons. Some of the standard ..."
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When algorithms for sorting and searching are applied to keys that are represented as bit strings, we can quantify the performance of the algorithms not only in terms of the number of key comparisons required by the algorithms but also in terms of the number of bit comparisons. Some of the standard sorting and searching algorithms have been analyzed with respect to key comparisons but not with respect to bit comparisons. In this paper, we investigate the expected number of bit comparisons required by Quickselect (also known as Find). We develop exact and asymptotic formulae for the expected number of bit comparisons required to find the smallest or largest key by Quickselect and show that the expectation is asymptotically linear with respect to the number of keys. Similar results are obtained for the average case. For finding keys of arbitrary rank, we derive an exact formula for the expected number of bit comparisons that (using rational arithmetic) requires only finite summation (rather than such operations as numerical integration) and use it to compute the expectation for each target rank.
Methodol Comput Appl Probab (2008) 10:507–529 DOI 10.1007/s110090079059x Approximating Perpetuities
, 2007
"... Abstract We propose and analyze an algorithm to approximate distribution functions and densities of perpetuities. Our algorithm refines an earlier approach based on iterating discretized versions of the fixed point equation that defines the perpetuity. We significantly reduce the complexity of the e ..."
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Abstract We propose and analyze an algorithm to approximate distribution functions and densities of perpetuities. Our algorithm refines an earlier approach based on iterating discretized versions of the fixed point equation that defines the perpetuity. We significantly reduce the complexity of the earlier algorithm. Also one particular perpetuity arising in the analysis of the selection algorithm Quickselect is studied in more detail. Our approach works well for distribution functions. For densities we have weaker error bounds although computer experiments indicate that densities can also be approximated well.
CONVERGENCE TO TYPE I DISTRIBUTION OF THE EXTREMES OF SEQUENCES DEFINED BY RANDOM DIFFERENCE EQUATION
"... Abstract. We study the extremes of a sequence of random variables (Rn) defined by the recurrence Rn = MnRn−1 + q, n ≥ 1, where R0 is arbitrary, (Mn) are iid copies of a non– degenerate random variable M, 0 ≤ M ≤ 1, and q> 0 is a constant. We show that under mild and natural conditions on M the su ..."
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Abstract. We study the extremes of a sequence of random variables (Rn) defined by the recurrence Rn = MnRn−1 + q, n ≥ 1, where R0 is arbitrary, (Mn) are iid copies of a non– degenerate random variable M, 0 ≤ M ≤ 1, and q> 0 is a constant. We show that under mild and natural conditions on M the suitably normalized extremes of (Rn) converge in distribution to a double exponential random variable. This partially complements a result of de Haan, Resnick, Rootzén, and de Vries who considered extremes of the sequence (Rn) under the assumption that P(M> 1)> 0. 1.
Discrete Mathematics and Theoretical Computer Science (subm.), by the authors, 1–rev A Note on the Approximation of Perpetuities
"... We propose and analyze an algorithm to approximate distribution functions and densities of perpetuities. Our algorithm refines an earlier approach based on iterating discretized versions of the fixed point equation that defines the perpetuity. We significantly reduce the complexity of the earlier al ..."
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We propose and analyze an algorithm to approximate distribution functions and densities of perpetuities. Our algorithm refines an earlier approach based on iterating discretized versions of the fixed point equation that defines the perpetuity. We significantly reduce the complexity of the earlier algorithm. Also one particular perpetuity arising in the analysis of the selection algorithm Quickselect is studied in more detail. Our approach works well for distribution functions. For densities we have weaker error bounds although computer experiments indicate that densities can also be well approximated.