Probability Approximations via the Poisson Clumping Heuristic

Aldous, David

doi:10.1007/978-1-4757-6283-9

Cited by 461 publications

(460 citation statements)

References 107 publications

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“…The basic difference between the present "middle case" and the previous two "boundary cases" is that here R (1) n is no longer a remainder term but, with a proper norming factor, it also contributes to the asymptotic distribution. This factor is presently redefined as the square root of σ 2 n = n e −2nd n e nd n −1−n 2 d 2 n ∼ e −2c e c −1−c 2 n. Thus we need to modify the decomposition (2.8) for the present random variable of interest: 15) where, introducing the independent and identically distributed random variables…”

Section: Case (I)mentioning

confidence: 82%

“…. be independent random variables, each uniformly distributed in the unit interval [0,1]. For each n ∈ N , let U 1,n ≤ · · · ≤ U n,n be the order statistics pertaining to the sample U 1 , .…”

Section: Introductionmentioning

confidence: 99%

“…

ABSTRACT: We consider the number K n of clusters at a distance level d n ∈ (0, 1) of n independent random variables uniformly distributed in [0,1] , or the number K n of connected components in the random interval graph generated by these variables and d n , and, depending upon how fast d n → 0 as n → ∞ , determine the asymptotic distribution of K n , with rates of convergence, and of related random variables that describe the cluster sizes.KEYWORDS: Clusters of independent uniform random variables, number and size of clusters, asymptotic distributions, rates of convergence.

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confidence: 99%

See 2 more Smart Citations

On the clustering of independent uniform random variables

Csörgő

2004

Random Struct Algorithms

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We consider the number K n of clusters at a distance level d n ∈ (0, 1) of n independent random variables uniformly distributed in [0,1] , or the number K n of connected components in the random interval graph generated by these variables and d n , and, depending upon how fast d n → 0 as n → ∞ , determine the asymptotic distribution of K n , with rates of convergence, and of related random variables that describe the cluster sizes.KEYWORDS: Clusters of independent uniform random variables, number and size of clusters, asymptotic distributions, rates of convergence.

show abstract

Section: Case (I)mentioning

confidence: 82%

Section: Introductionmentioning

confidence: 99%

“…

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mentioning

confidence: 99%

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On the clustering of independent uniform random variables

Csörgő

2004

Random Struct Algorithms

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“…In particular, we conjecture that (in the typical case at least) the largest eigenvalue λ [1] of the essential transition probability matrix of the imbedded finite Markov chain associated with compound pattern A will satisfy the approximation…”

Section: Second Moments and Distribution Approximationsmentioning

confidence: 98%

“…By the clumping heuristic [c.f. Aldous (1989)], one then expects the distribution of τ A to be well approximated by an exponential distribution, and for an exponential X we have the equality E(X) = Var(X).…”

Section: Second and Higher Momentsmentioning

confidence: 99%

Martingale Methods for Patterns and Scan Statistics

Pozdnyakov

Steele

2009

Scan Statistics

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We show how martingale techniques (both old and new) can be used to obtain otherwise hard-to-get information for the moments and distributions of waiting times for patterns in independent or Markov sequences. In particular, we show how these methods provide moments and distribution approximations for certain scan statistics, including about variable length scan statistics. Each general problem that is considered is also illustrated with a concrete example confirming the computational tractability of the method.

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Analysis of a splitting process arising in probabilistic counting and other related algorithms

Kirschenhofer

Prodinger

Szpankowski

1996

Random Struct. Alg.

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We present an analytical method of analyzing a class of "splitting algorithms" that include probabilistic counting, selecting the leader, estimating the number of questions necessary to identify distinct objects, searching algorithms based on digital tries, approximate counting, and so forth. In our discussion we concentrate on the analysis of a generalized probabilistic counting algorithm. Our technique belongs to the toolkit of the analytical analysis of algorithms, and it involves solutions of functional equations, analytical poissonization and depoissonization as well as Mellin transform. In particular, we deal with an instance of the functional equation g ( z ) = pa(z)g(z/2) + b(z), where a(z) and b ( z ) are 379 380 KIRSCHENHOFER, PRODINGER, A N D SZPANKOWSKI given functions and p < 1 is a constant. With respect to our generalized probabilistic counting algorithm, we obtain asymptotic expansions of the first two moments of an estimate of the cardinality of a set that is computed by the algorithm. We also derive the asymptotic distribution of this estimate, and observe that it actually fluctuates, leading to a conclusion that its limiting distribution does not exist.

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Probability Approximations via the Poisson Clumping Heuristic

Cited by 461 publications

References 107 publications

On the clustering of independent uniform random variables

On the clustering of independent uniform random variables

Martingale Methods for Patterns and Scan Statistics

Analysis of a splitting process arising in probabilistic counting and other related algorithms

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