Asymptotic behavior of certain statistics of a \((0,1)\)-vector. (English. Russian original) Zbl 0748.62011

Masol, V. I.

Theory Probab. Math. Stat. 43, 93-100 (1991); translation from Teor. Veroyatn. Mat. Stat., Kiev 43, 83-90 (1990).

Summary: A study is made of the joint distribution of the variables \(\eta_ n(1)\) and \(\eta_ n(2)\), and of the distribution of the random variable \(\eta_ n(l)\) as \(n\to \infty\). Here \(\eta_ n(l)\) is the number of configurations of the form \(\dots 1t_ l0\dots\) in a random \(n\)- dimensional \((0,1)\)-vector of given weight, \(t_ l\) is an arbitrary \((l- 1)\)-dimensional (0,1)-vector and \(l=l(n)\). An asymptotic expression is obtained for the correlation coefficient, \(r(\eta_ n(l_ 1)\) \(\eta_ n(l_ 2))\), \(1\leq l_ 1 < l_ 2\leq n-1\).

MSC:

62E20

Asymptotic distribution theory in statistics

Keywords:

probability of simultaneous realization of events; joint distribution; asymptotic expression; correlation coefficient