On limit points of a sequence of extreme values of normal random elements in Banach spaces with unconditional basis. (English. Ukrainian original) Zbl 0923.60006

Matsak, I. K.

Theory Probab. Math. Stat. 55, 143-151 (1997); translation from Teor. Jmovirn. Mat. Stat. 55, 136-143 (1996).

Let \(X=\sum_{i=1}^{\infty} \eta_i \sigma_i e_i\) be the normal random element in Banach space with unconditional basis \((e_i)\), where \(\eta_i\), \(i\geq 1\), are independent \(N(0,1)\)-distributed random variables, \(\sigma_i\geq 0\), \(i\geq 1\). Let \(\sigma=\sum_{i=1}^{\infty}\sigma_i e_i\), and let \(X_k\), \(k\geq 1\), be independent copies of \(X\), \(Z_n=\max_{1\leq k\leq n} X_k\). The author describes the set of a.s. limit points for the sequence \({(Z_n-b_n\sigma)/d_n}\), where \(b_n=(2\ln n)^{1/2}\) and \(d_n=1\) for \(n<3\) and \(d_n=(2\ln n)^{-1/2}\ln \ln n\) for \(n\geq 3\).

Reviewer: N.M.Zinchenko (Kyïv)

MSC:

60B12	Limit theorems for vector-valued random variables (infinite-dimensional case)
60F15	Strong limit theorems

Keywords:

Banach space; normal random element; extreme value; almost sure convergence; limit point; random series