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ON SOME JOINT LAWS IN FLUCTUATIONS OF SUMS

OF RANDOM VARIABLES

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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} = \varepsilon_{1} + \varepsilon_{2} + ··· + \varepsilon_{r},
*r* = 1,2,....,*n*; *N*_{0} = 0 of independent Pascal random variables
\varepsilon_{1}, \varepsilon_{2}, ···, \varepsilon_{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
*Lambda*_{n}^{(c)}, *phi*_{n}^{(c)}, *phi*_{n}^{(-c)}, *Z*_{n} and max_{1<
r<n}(*N*_{r}-*r*) where *c* = 0,1,2,.... and *Lambda*_{n}^{(c)}, *phi*_{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.

**Source**
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