Bounded law of iterated logarithm for matrix-normalized sums of independent random vectors. (Ukrainian, English) Zbl 1123.60017

Koval, V. O.

Teor. Jmovirn. Mat. Stat. 72, 63-66 (2005); translation in Theory Probab. Math. Stat. 72, 69-73 (2006).

Let \(X_n\), \(n=1,2,\dots\) be a sequence of independent random vectors in \(\mathbb R^d\) with \({\mathbf E} X_n=0\) and \({\mathbf E}\| X_n\| ^p<\infty\) for some \(p\in(2,3]\), and let \(A_n\) be a sequence of \(m\times d\) matrices. Assume that for some \(\varphi_n>0,f_n>0\), \(\varphi_n\uparrow +\infty\), \(f_n\uparrow +\infty\) the following conditions hold true: \[ \sum_{i=k+1}^n{\mathbf E}\| A_n X_i\| ^2\leq \varphi_n(1-f_k/f_n), \]
\[ \lim_{n\to\infty}(\varphi_n^{1/2}\ln\ln(f_n))^{-1}\| A_n\| =0, \]
\[ \sum_{i=1}^\infty\sup_{n\geq i}[(\varphi_n^{1/2}\ln\ln(f_n))^{-p} {\mathbf E}\| A_n X_i\| ^p]<\infty. \] Under these conditions \(\lim_{n\to\infty}\varphi_n^{1/2}\ln\ln (f_n)^{-1}\| A_n\sum_{i=1}^n X_i\| | =L\) a.s. for a special nonrandom constant \(L\in[0,+\infty)\).

Reviewer: R. E. Maiboroda (Kyïv)

MSC:

60F15

Strong limit theorems

Keywords:

sum of independent random vectors; bounded law of iterated logarithm; matrix normalization; upper function