ON SOME JOINT LAWS IN FLUCTUATIONS OF SUMS OF RANDOM VARIABLES

JAGDISH SARAN

Department of Statistics, University of Delhi, Delhi-110007, India

(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 N_{r} = ε_{1}+ε_{2}+⋯+ε_{r}, r = 1, 2, …, n; N_{0} = 0 of independent Pascal random variables ε_{1}, ε_{2}, …, ε_{n}, 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 Λ_{n}^{(c)}, ϕ_{n}^{(c)}, ϕ_{n}^{(−c)}, Z_{n} and max_{1≤r≤n}(N_{r}−r) where c = 0, 1, 2, … and Λ_{n}^{(c)}, ϕ_{n}^{(±c)} and Z_{n} denote, respectively, the number of subscripts r = 1, 2, …, n for which N_{r} = r + c, N_{r−1} = N_{r} = r ± c and N_{r−1} = N_{r}.

Key words and phrases: Pascal random variables, partial sums, lattice path, rotation procedure, random walk, composed path, ballot problems.