The number of distinct values in a geometrically distributed sample

Abstract

A sequence of geometric random variables of length n is a sequence of n independent and identically distributed geometric random variables (Γ1, Γ2, . . . , Γn) where P(Γj = i) = pq i−1 for 1 ≤ j ≤ n with p + q = 1. We study the number of distinct adjacent two letter patterns in such sequences. Initially we directly count the number of distinct pairs in words of short length. Because of the rapid growth of the number of word patterns we change our approach to this problem by obtaining an expression for the expected number of distinct pairs in words of length n. We also obtain the asymptotics for the expected number as n → ∞.

Longest run of equal parts in a random integer composition

“…The fluctuation is of a kind frequently encountered in the analysis of algorithms (see [2]). For instance, the equations for the mean and variance here bear a striking resemblance to those found by Archibald, Knopfmacher, and Prodinger in their analysis of the number of distinct letters in geometrically distributed sequences [1].…”

A Binomial Splitting Process in Connection with Corner Parking Problems

“…One is in terms of the number of distinct letters in a sequence of i.i.d. geometric random variables with success probability p for which we have exactly the recurrence (3); see [1]. Alternatively, if we consider the urn model where the j th urn has probability of pq j of receiving a ball, then the number of occupied urns also follows the same distribution; see [7].…”

“…The depth is the distance between a uniformly chosen leaf node and the root and the left arm is the path that starts at the root and keeps going left until no more nodes can be found on the left; see, for example, [15], [21]. Also, the depth, or the left arm, of random PATRICIA tries is identically distributed as the number of distinct values in some random sequences, see [1], and the number of occupied urns in some urn models, see [7].…”

Descents following maximal values in samples of geometric random variables

“…Some relevant references on samples of geometric random variables and their applications can be found in Archibald et al (2006), Grabner et al (2000), Kirschenhofer and Prodinger (1990), Kirschenhofer and Prodinger (1993), Louchard and Prodinger (2008), and Szpankowski and Rego (1990). For papers relating to maxima particularly, see Archibald (2005), Archibald and Knopfmacher (2007), Archibald and Knopfmacher (2009), Baryshnikov et al (1995), Eisenberg et al (1993), Kirschenhofer and Prodinger (1996), Knopfmacher and Prodinger (2004), , and Brennan and Knopfmacher (2005a), Brennan and Knopfmacher (2005b), Knopfmacher and Prodinger (2004), Knopfmacher and Prodinger (2006), Knopfmacher and Prodinger (2007), Louchard and Prodinger (2005) relate to descents and ascents in these samples.…”