Limit distributions of certain statistics of an integer sequence. (English. Russian original) Zbl 0632.62014

Masol, V. I.

Theory Probab. Math. Stat. 35, 75-81 (1987); translation from Teor. Veroyatn. Mat. Stat. 35, 69-75 (1986).

Let \(J_ s=\{j_ 1,...,j_ s\}\) be a collection of nonnegative integers with \(\sum^{s}_{1}j_ i=n\) and \(j_ s\geq 1\), and let \(R(J_ s)\) be the set of sequences \(f=(f(1),...,f(n))\) in which the integer m occurs \(j_ m\) times. A sequence f is chosen from \(R(J_ s)\) randomly and with uniform probability. Let \(\eta_{n,r}\) be the number of r-descents in f, and \(\zeta_ n\) the r-major index of f. General theorems are proved on convergence of the random variables \(\eta_{n,r}\) and \(\zeta_ n\) to normal laws as \(n\to \infty\).

MSC:

62E20	Asymptotic distribution theory in statistics
60F05	Central limit and other weak theorems

Keywords:

limit distributions; integer sequence; asymptotic normality