Asymptotic distribution of the sojourn time of the simplest random walk on the positive semi-axis

1972 ◽  
Vol 24 (1) ◽  
pp. 63-65
Author(s):  
R. V. Boiko
Keyword(s):  
Random Walk ◽  
Asymptotic Distribution ◽  
Sojourn Time

Asymptotic distribution theory for Hoare's selection algorithm

1996 ◽  
Vol 28 (1) ◽  
pp. 252-269 ◽  
Author(s):  
Rudolf Grübel ◽  
Uwe Rösler
Keyword(s):  
Stochastic Process ◽  
Random Walk ◽  
Asymptotic Behaviour ◽  
Asymptotic Distribution ◽  
Random Environment ◽  
Binary Tree ◽  
Selection Algorithm ◽  
Real Numbers ◽  
Asymptotic Distribution Theory ◽  
Path Lengths

We investigate the asymptotic behaviour of the distribution of the number of comparisons needed by a quicksort-style selection algorithm that finds the lth smallest in a set of n numbers. Letting n tend to infinity and considering the values l = 1, ···,n simultaneously we obtain a limiting stochastic process. This process admits various interpretations: it arises in connection with a representation of real numbers induced by nested random partitions and also in connection with expected path lengths of a random walk in a random environment on a binary tree.

A bernoulli excursion and its various applications

1991 ◽  
Vol 23 (03) ◽  
pp. 557-585 ◽  
Author(s):  
Lajos Takács
Keyword(s):  
Random Walk ◽  
Order Statistics ◽  
Asymptotic Distribution ◽  
Random Graphs ◽  
Random Variable ◽  
Random Walk Process

This paper is concerned with a random walk process in which and for i = 1, 2, ···, 2n . This process is called a Bernoulli excursion. The main object is to find the distribution, the moments, and the asymptotic distribution of the random variable ω n defined by . The results derived have various applications in the theory of probability, including random graphs, tournaments and order statistics.

scholarly journals Word Metric, Stationary Measure and Minkowski’s Question Mark Function

2020 ◽  
Vol 15 (2) ◽  
pp. 23-38
Author(s):  
Uriya Pumerantz
Keyword(s):  
Random Walk ◽  
Probability Measure ◽  
Asymptotic Distribution ◽  
Limit Distribution ◽  
Limit Point ◽  
Projective Line ◽  
Specific Measure ◽  
Question Mark ◽  
Convolution Power ◽  
Countably Infinite

AbstractGiven a countably infinite group G acting on some space X, an increasing family of finite subsets Gn, x∈ X and a function f over X we consider the sums Sn(f, x) = ∑g∈Gnf(gx). The asymptotic behaviour of Sn(f, x) is a delicate problem that was studied under various settings. In the following paper we study this problem when G is a specific lattice in SL (2, ℤ ) acting on the projective line and Gn are chosen using the word metric. The asymptotic distribution is calculated and shown to be tightly connected to Minkowski’s question mark function. We proceed to show that the limit distribution is stationary with respect to a random walk on G defined by a specific measure µ. We further prove a stronger result stating that the asymptotic distribution is the limit point for any probability measure over X pushed forward by the convolution power µ∗n.

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