A new law of the iterated logarithm in \(R^d\) with application to matrix-normalized sums of random vectors (Q5960462)

Let \(X_n, n \geq 1\), be a sequence of independent random vectors in \(R^d\) with \(EX_n=0\) and \(D_n=E(X_nX_n')\). Denote \(S_n=\sum_1^n X_i\), \(B_n= \sum_1^n D_i.\) The law of iterated logarithm, briefly LIL, for independent Banach space valued random variables is usually investigated [ cf. \textit{M. Ledoux} and \textit{M. Talagrand}, ``Probability in Banach spaces: Isoperimetry and processes'' (1991; Zbl 0748.60004) and \textit{X. Chen}, Ann. Probab. 21, 1991--2011 (1993; Zbl 0791.60005)] in the form \[ \limsup_{n \to \infty} (\| B_n \|^{1/2}t_n)^{-1} \| S_n \| =1 \quad \text{a.s.} \] where \( t_n= (2\log\log \| B_n \|)^{1/2}\). The author of the present paper studies LIL in a new form, namely \[ \limsup_{n \to \infty} t_n^{-1}\| B_n^{-1/2}S_n \| =1 \quad\text{a.s.} \] He applies it to the study of the exact asymptotic behaviour of \(S_n\) under arbitrary matrix normalization.