Refined quicksort asymptotics

Neininger, Ralph

doi:10.1002/rsa.20497

Cited by 17 publications

(38 citation statements)

References 15 publications

(40 reference statements)

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“…. , Ξ r is obtained together with a recurrence for the sequence (R n ) n≥0 which extends to a recurrence for the residuals in (2) as well as to the residuals Z n of the projections of the R n , see equation (16) in Section 3.2. Equation (16) is then the starting point to show the convergence in Proposition 2.1.…”

Section: Explanation Of the Results And Outline Of The Proofmentioning

confidence: 99%

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Refined asymptotics for the composition of cyclic urns

Müller¹,

Neininger²

2018

Electron. J. Probab.

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A cyclic urn is an urn model for balls of types 0, . . . , m − 1. The urn starts at time zero with an initial configuration. Then, in each time step, first a ball is drawn from the urn uniformly and independently from the past. If its type is j, it is then returned to the urn together with a new ball of type j + 1 mod m. The case m = 2 is the well-known Friedman urn. The composition vector, i.e., the vector of the numbers of balls of each type after n steps is, after normalization, known to be asymptotically normal for 2 ≤ m ≤ 6. For m ≥ 7 the normalized composition vector is known not to converge. However, there is an almost sure approximation by a periodic random vector.In the present paper the asymptotic fluctuations around this periodic random vector are identified. We show that these fluctuations are asymptotically normal for all 7 ≤ m ≤ 12. For m ≥ 13 we also find asymptotically normal fluctuations when normalizing in a more refined way. These fluctuations are of maximal dimension m − 1 only when 6 does not divide m. For m being a multiple of 6 the fluctuations are supported by a two-dimensional subspace.MSC2010: 60F05, 60F15, 60C05, 60J10.

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Section: Explanation Of the Results And Outline Of The Proofmentioning

confidence: 99%

“…. , N n−1 ) we make use of independence and the fact that ζ 3 is (3, +)-ideal and satisfies (18) to get, again for n ≥ n 1 , Now a standard argument shows that ζ 3 (N n , N ) → 0 as n → ∞, see [16], for example.…”

Section: Proof Of Proposition 21mentioning

confidence: 99%

Refined asymptotics for the composition of cyclic urns

Müller¹,

Neininger²

2018

Electron. J. Probab.

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“…We will also see that Q 0,j (x; β) (the coefficient of W ∞ (β)) is the same as q j (x) in (5) but with κ j replaced by κ j (β). [36], see also [32], that in a suitable range of β the asymptotic distribution of the appropriately normalized difference W n (β) − W ∞ (β) is a mixture of centered normals. A functional limit theorem for this difference was obtained in [25].…”

Section: 1mentioning

confidence: 99%

Edgeworth expansions for profiles of lattice branching random walks

Grübel¹,

Kabluchko²

2017

Ann. Inst. H. Poincaré Probab. Statist.

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Abstract. Consider a branching random walk on Z in discrete time. Denote by Ln(k) the number of particles at site k ∈ Z at time n ∈ N 0 . By the profile of the branching random walk (at time n) we mean the function k → Ln(k). We establish the following asymptotic expansion of Ln(k), as n → ∞:where r ∈ N 0 is arbitrary, ϕ(β) = log k∈Z e βk EL 1 (k) is the cumulant generating function of the intensity of the branching random walk andThe expansion is valid uniformly in k ∈ Z with probability 1 and the F j 's are polynomials whose random coefficients can be expressed through the derivatives of ϕ and the derivatives of the limit of the Biggins martingale at 0. Using exponential tilting, we also establish more general expansions covering the whole range of the branching random walk except its extreme values. As an application of this expansion for r = 0, 1, 2 we recover in a unified way a number of known results and establish several new limit theorems. In particular, we study the a.s. behavior of the individual occupation numbers Ln(kn), where kn ∈ Z depends on n in some regular way. We also prove a.s. limit theorems for the mode arg max k∈Z Ln(k) and the height max k∈Z Ln(k) of the profile. The asymptotic behavior of these quantities depends on whether the drift parameter ϕ (0) is integer, non-integer rational, or irrational. Applications of our results to profiles of random trees including binary search trees and random recursive trees will be given in a separate paper.

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“…, X n,⌊m/2⌋ ) in Proposition 2.1. The reader is asked to trust the authors that the techniques developed in [12] for a univariate problem can be extended to the multivariate recurrences for (X n,1 , . .…”

Section: Proving Convergencementioning

confidence: 99%

The CLT Analogue for Cyclic Urns

Müller

Neininger

2015

2016 Proceedings of the Thirteenth Workshop on Analytic Algorithmics and Combinatorics (ANALCO)

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A cyclic urn is an urn model for balls of types 0, . . . , m − 1 where in each draw the ball drawn, say of type j, is returned to the urn together with a new ball of type j + 1 mod m. The case m = 2 is the well-known Friedman urn. The composition vector, i.e., the vector of the numbers of balls of each type after n steps is, after normalization, known to be asymptotically normal for 2 ≤ m ≤ 6. For m ≥ 7 the normalized composition vector does not converge. However, there is an almost sure approximation by a periodic random vector. In this paper the asymptotic fluctuations around this periodic random vector are identified. We show that these fluctuations are asymptotically normal for all m ≥ 7. However, they are of maximal dimension m − 1 only when 6 does not divide m. For m being a multiple of 6 the fluctuations are supported by a two-dimensional subspace.MSC2010: 60F05, 60F15, 60C05, 60J10.

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Refined quicksort asymptotics

Cited by 17 publications

References 15 publications

Refined asymptotics for the composition of cyclic urns

Refined asymptotics for the composition of cyclic urns

Edgeworth expansions for profiles of lattice branching random walks

The CLT Analogue for Cyclic Urns

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