Abstract
This paper deals with the joint and marginal distributions of certain random variables concerning the fluctuations of partial sums N r = ɛ1+ɛ2+⋯+ɛr, r = 1,2,..., n; N 0=0 of independent Pascal random variables ɛ1,ɛ2,...,ɛn, 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(c) n, φ(c) n, φ(−c) n, Z n and max1≤r≤n(Nr−r) where c=0, 1, 2,... and Λ(c) n, φ(±c) n 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.
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Saran, J. On some joint laws in fluctuations of sums of random variables. Ann Inst Stat Math 43, 773–791 (1991). https://doi.org/10.1007/BF00121654
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DOI: https://doi.org/10.1007/BF00121654