On The Number of Times where a Simple Random Walk reaches a Nonnegative Height

The purpose of this note is to generalize the distribution of the local time of a purely binomial random walk for simple random walks allowing for three directions with different probabilities. (author's abstract)Series: Forschungsberichte / Institut für Statisti

On the Number of Times where a simple Random Walk reaches its Maximum

Let Q, denote the number of times where a simple random walk reaches its maximum, where the random walk starts at the origin and returns to the origin after 2n steps. Such random walks play an important r6le in probability and statistics. In this paper the distribution and the moments of Q, are considered and their asymptotic behavior is studied. (author's abstract)Series: Forschungsberichte / Institut für Statisti

The Maximal Height of Simple Random Walks Revisited

In a recent paper Katzenbeisser and Panny (1996) derived distributional results for a number of so called simple random walk statistics defined on a simple random walk in the sense of Cox and Miller (1968) starting at zero and leading to state 1 after n steps, where 1 is arbitrary, but fix. In the present paper the random walk statistics Dn = the one-sided maximum deviation and Qn = the number of times where the maximum is achieved, are considered and distributional results are presented, when it is irrespective, where the random walk terminates after n steps. Thus, the results can be seen as generalizations of some well known results about (purely) binomial random walk, given e.g. in Revesz (1990). (author's abstract)Series: Forschungsberichte / Institut für Statisti

Simple Random Walk Statistics. Part I: Discrete Time Results

In a famous paper Dwass [I9671 proposed a method to deal with rank order statistics, which constitutes a unifying framework to derive various distributional results. In the present paper an alternative method is presented, which allows to extend Dwass's results in several ways, viz. arbitrary endpoints, horizontal steps, and arbitrary probabilities for the three step types. Regarding these extensions the pertaining rank order statistics are extended as well to simple random walk statistics. This method has proved appropriate to generalize all results given by Dwass. Moreover, these discrete time results can be taken as a starting point to derive the corresponding results for randomized random walks by means of a limiting process. (author's abstract)Series: Forschungsberichte / Institut für Statisti

Some further Results on the Height of Lattice Path

This paper deals with the joint and conditional distributions concerning the maximum of random walk paths and the number of times this maximum is achieved. This joint distribution was studied first by Dwass [1967]. Based on his result, the correlation and some conditional moments are derived. The main contributions are however asymptotic expansions concerning the conditional distribution and conditional moments. (author's abstract)Series: Forschungsberichte / Institut für Statisti

Single and Twin-Heaps as Natural Data Structures for Percentile Point Simulation Algorithms

Sometimes percentile points cannot be determined analytically. In such cases one has to resort to Monte Carlo techniques. In order to provide reliable and accurate results it is usually necessary to generate rather large samples. Thus the proper organization of the relevant data is of crucial importance. In this paper we investigate the appropriateness of heap-based data structures for the percentile point estimation problem. Theoretical considerations and empirical results give evidence of the good performance of these structures regarding their time and space complexity. (author's abstract)Series: Forschungsberichte / Institut für Statisti