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On a limit theorem for random sequences. (English) Zbl 0657.60035

For \(n=1,2,..\). let \(X_{n,1},X_{n,2},...,X_{n,k},..\). be a sequence of i.i.d. random variables. Further let \((N_ n)\) be a sequence of non- negative random variables, which are independent of the \(X_{n,k}'s\). In an earlier paper by L. S. Bereski and S. Janjić [Lith. Math. J. 24, 68-73 (1984); translation from Litov. Mat. Sb. 24, No.1, 167-174 (1984; Zbl 0582.60029)], conditions were provided for the following limit theorem to hold: \[ Let\quad Y_{k_ n}=\max \{X_{n,1},X_{n,2},...,X_{n,k_ n}\}.\quad Then \]
\[ \lim_{n\to \infty}P(Y_{N_ n}<x)=\int^{\infty}_{0}[F(x)]^ ydA(y)=G(x), \] where A(x) and F(x) denote the limiting distribution of \((N_ n/k_ n)\) and \(Y_{k_ n}\) respectively.
This paper is complementary to this result in providing conditions for the existence of F(x) and G(x) to imply the existence of A(x).
Reviewer: F.T.Bruss

MSC:

60F05 Central limit and other weak theorems

Citations:

Zbl 0582.60029
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