(Received December 26, 1988; revised July 4, 1990)
Abstract. This paper deals with the joint and marginal distributions of certain random variables concerning the fluctuations of partial sums Nr = \varepsilon1 + \varepsilon2 + ··· + \varepsilonr, r = 1,2,....,n; N0 = 0 of independent Pascal random variables \varepsilon1, \varepsilon2, ···, \varepsilonn, thus generalizing and extending the previous work due to Saran (1977, Z. Angew. Math. Mech., 57, 610-613) and Saran and Sen (1979, Mathematische Operationsforschung und Statistik, Series Statistics, 10, 469-478). The random variables considered are Lambdan(c), phin(c), phin(-c), Zn and max1< r<n(Nr-r) where c = 0,1,2,.... and Lambdan(c), phin(± c) and Zn denote, respectively, the number of subscripts r = 1,2,....,n for which Nr = r + c, Nr-1 = Nr = r ± c and Nr-1 = Nr.
Key words and phrases: Pascal random variables, partial sums, lattice path, rotation procedure, random walk, composed path, ballot problems.
Source ( TeX , DVI , PS )