On the waiting times to repeated hits of cells by particles for the polynomial allocation scheme

2020 ◽  
Vol 30 (6) ◽  
pp. 409-415
Author(s):  
Boris I. Selivanov ◽  
Vladimir P. Chistyakov
Keyword(s):  
Limit Distribution ◽  
Waiting Times ◽  
Random Variable ◽  
Random Polynomial ◽  
Allocation Scheme

AbstractWe consider random polynomial allocations of particles over N cells. Let τk, k ≥ 1, be the minimal number of trials when k particles hit the occupied cells. For the case N → ∞ the limit distribution of the random variable $\tau_k/\sqrt{N}$is found. An example of application of τk is given.

On the first zero of an empirical characteristic function

1991 ◽  
Vol 28 (3) ◽  
pp. 593-601 ◽  
Author(s):  
H. U. Bräker ◽  
J. Hüsler
Keyword(s):  
Characteristic Function ◽  
Limit Distribution ◽  
Random Variable ◽  
Empirical Characteristic Function ◽  
The Real ◽  
Exponential Decrease ◽  
Characteristic Process

We deal with the distribution of the first zero Rn of the real part of the empirical characteristic process related to a random variable X. Depending on the behaviour of the theoretical real part of the underlying characteristic function, cases with a slow exponential decrease to zero are considered. We derive the limit distribution of Rn in this case, which clarifies some recent results on Rn in relation to the behaviour of the characteristic function.

scholarly journals Limit distribution of distances in biased random tries

2006 ◽  
Vol 43 (02) ◽  
pp. 377-390 ◽  
Author(s):  
Rafik Aguech ◽  
Nabil Lasmar ◽  
Hosam Mahmoud
Keyword(s):  
Fixed Point ◽  
Metric Space ◽  
Limit Distribution ◽  
Analytical Techniques ◽  
Random Variable ◽  
Wasserstein Metric ◽  
Contraction Method ◽  
Point Solution ◽  
Mean And Variance

Thetrieis a sort of digital tree. Ideally, to achieve balance, the trie should grow from an unbiased source generating keys of bits with equal likelihoods. In practice, the lack of bias is not always guaranteed. We investigate the distance between randomly selected pairs of nodes among the keys in a biased trie. This research complements that of Christophi and Mahmoud (2005); however, the results and some of the methodology are strikingly different. Analytical techniques are still useful for moments calculation. Both mean and variance are of polynomial order. It is demonstrated that the standardized distance approaches a normal limiting random variable. This is proved by the contraction method, whereby the limit distribution is shown to approach the fixed-point solution of a distributional equation in the Wasserstein metric space.

The asymptotic distribution of random molecules

1980 ◽  
Vol 12 (03) ◽  
pp. 640-654
Author(s):  
Wulf Rehder
Keyword(s):  
Asymptotic Distribution ◽  
Limit Distribution ◽  
Random Variable ◽  
Unit Cube ◽  
Image Position ◽  
Solid Spheres

If n solid spheres K n of some volume V(K n ) are scattered randomly in the unit cube of euclidean d-space, some of them will overlap to form L n (s) molecules with exactly s atoms K n. The random variable L n(s) has a limit distribution if V(K n ) tends to zero but nV(Kn ) tends to infinity at a certain rate: it is shown that for L n(s) is asymptotically Poisson.

scholarly journals Rarity and exponentiality: an extension of Keilson's theorem, with applications

2005 ◽  
Vol 42 (02) ◽  
pp. 393-406
Author(s):  
Rudolf Grübel ◽  
Marcus Reich
Keyword(s):  
Molecular Biology ◽  
Pattern Matching ◽  
Limit Distribution ◽  
Rare Events ◽  
Waiting Times ◽  
Urn Models ◽  
Regenerative Processes ◽  
Approximate Pattern Matching

We generalize a theorem due to Keilson on the approximate exponentiality of waiting times for rare events in regenerative processes. We use the result to investigate the limit distribution for a family of first entrance times in a sequence of Ehrenfest urn models. As a second application, we consider approximate pattern matching, a problem arising in molecular biology and other areas.

Waiting times for patterns in a sequence of multistate trials

2001 ◽  
Vol 38 (2) ◽  
pp. 508-518 ◽  
Author(s):  
Demetrios L. Antzoulakos
Keyword(s):  
Markov Chain ◽  
Waiting Time ◽  
Waiting Times ◽  
Random Variable ◽  
Finite Markov Chain ◽  
Finite Markov Chain Imbedding ◽  
Markov Chain Imbedding ◽  
Markov Chain Imbedding Method ◽  
Runs And Patterns ◽  
Imbedding Method

Let Xn, n ≥ 1 be a sequence of trials taking values in a given set A, let ∊ be a pattern (simple or compound), and let Xr,∊ be a random variable denoting the waiting time for the rth occurrence of ∊. In the present article a finite Markov chain imbedding method is developed for the study of Xr,∊ in the case of the non-overlapping and overlapping way of counting runs and patterns. Several extensions and generalizations are also discussed.

The asymptotic distribution of random molecules

1980 ◽  
Vol 12 (3) ◽  
pp. 640-654 ◽  
Author(s):  
Wulf Rehder
Keyword(s):  
Asymptotic Distribution ◽  
Limit Distribution ◽  
Random Variable ◽  
Unit Cube ◽  
Image Position ◽  
Solid Spheres

If n solid spheres Kn of some volume V(Kn) are scattered randomly in the unit cube of euclidean d-space, some of them will overlap to form Ln(s) molecules with exactly s atoms Kn. The random variable Ln(s) has a limit distribution if V(Kn) tends to zero but nV(Kn) tends to infinity at a certain rate: it is shown that for Ln(s) is asymptotically Poisson.

scholarly journals Dynamics of non-stationary processes that follow the maximum of the Rényi entropy principle

2016 ◽  
Vol 472 (2185) ◽  
pp. 20150324 ◽  
Author(s):  
Dmitry S. Shalymov ◽  
Alexander L. Fradkov
Keyword(s):  
Limit Distribution ◽  
Mass Conservation ◽  
Random Variable ◽  
Renyi Entropy ◽  
Rényi Entropy ◽  
Stationary Processes ◽  
Entropy Principle ◽  
Gradient Principle ◽  
Probability Density Function Pdf ◽  
Speed Gradient

We propose dynamics equations which describe the behaviour of non-stationary processes that follow the maximum Rényi entropy principle. The equations are derived on the basis of the speed-gradient principle originated in the control theory. The maximum of the Rényi entropy principle is analysed for discrete and continuous cases, and both a discrete random variable and probability density function (PDF) are used. We consider mass conservation and energy conservation constraints and demonstrate the uniqueness of the limit distribution and asymptotic convergence of the PDF for both cases. The coincidence of the limit distribution of the proposed equations with the Rényi distribution is examined.

Waiting times for patterns in a sequence of multistate trials

2001 ◽  
Vol 38 (02) ◽  
pp. 508-518 ◽  
Author(s):  
Demetrios L. Antzoulakos
Keyword(s):  
Markov Chain ◽  
Waiting Time ◽  
Waiting Times ◽  
Random Variable ◽  
Finite Markov Chain ◽  
Finite Markov Chain Imbedding ◽  
Markov Chain Imbedding ◽  
Markov Chain Imbedding Method ◽  
Runs And Patterns ◽  
Imbedding Method

Let X n , n ≥ 1 be a sequence of trials taking values in a given set A, let ∊ be a pattern (simple or compound), and let X r,∊ be a random variable denoting the waiting time for the rth occurrence of ∊. In the present article a finite Markov chain imbedding method is developed for the study of X r,∊ in the case of the non-overlapping and overlapping way of counting runs and patterns. Several extensions and generalizations are also discussed.

On the continuity and the positivity of the finite part of the limit distribution of an irregular branching process with infinite mean

1980 ◽  
Vol 17 (3) ◽  
pp. 696-703 ◽  
Author(s):  
H. Cohn ◽  
H.-J. Schuh
Keyword(s):  
Limit Distribution ◽  
Branching Process ◽  
Law Of Large Numbers ◽  
Branching Processes ◽  
Random Variable ◽  
Finite Part ◽  
Positive Distribution ◽  
Large Numbers

It is shown that the limiting random variable W(si) of an irregular branching process with infinite mean, defined in [5], has a continuous and positive distribution on {0 < W(si) < ∞}. This implies that for all branching processes (Zn) with infinite mean there exists a function U such that the distribution of V = limnU(Zn)e–n a.s. is continuous, positive and finite on the set of non-extinction. A kind of law of large numbers for sequences of independent copies of W(si) is derived.

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