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

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 $$L\in[0,+\infty)$$.

### MSC:

 60F15 Strong limit theorems
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