In a famous paper, Dwass (1967) 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 us to extend Dwass's results in several ways, namely 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.

In this paper various statistics for randomized random walks and their distributions are presented. The distributional results are derived by means of a limiting procedure applied to the pertaining discrete time process, which has been considered in part I of this work (Katzenbeisser and Panny 1996). This basic approach, originally due to Meisling (1958), seems to offer certain technical advantages, since it avoids the use of Laplace transforms and is even simpler than Feller's randomization technique.

In this paper we present a simple combinatorial approach for the derivation of zero-avoiding transition probabilities in a Markovian r-node series Jackson network. The method we propose offers two advantages: first, it is conceptually simple because it is based on transition counts between the nodes and does not require a tensor representation of the network. Second, the method provides us with a very efficient technique for numerical computation of zero-avoiding transition probabilities.

Let Qn 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 role in probability and statistics. In this paper the distribution and the moments of Qn, are considered and their asymptotic behavior is studied.

Suppose that in a ballot candidate A scores a votes and candidate B scores b votes, and that all the possible voting records are equally probable. Corresponding to the first r votes, let α r and β r be the numbers of votes registered for A and B, respectively. Let p be an arbitrary positive real number. Denote by δ (a, b, p)[δ *(a, b, ρ)] the number of values of r for which the inequality , r = 1, ···, a + b, holds. Heretofore the probability distributions of δand δ* have been derived for only a restricted set of values of a, b, and ρ, although, as pointed out here, they are obtainable for all values of (a, b, ρ) by using a result of Takács (1964). In this paper we present a derivation of the distribution of δ [δ *] whose development, for any (a, b, ρ), leads to both necessary and sufficient conditions for δ [δ *] to have a discrete uniform distribution.