Laws of the single logarithm for delayed sums of random fields. II (Q936598)

The main result is the following Theorem 2.2. Suppose that \(\{X_k: k\in\mathbb{Z}^d_+\}\) are i.i.d. random variables with mean 0 and finite variance \(\sigma^2\). Set \[ S_n:=\sum_{k\leq n}X_k,\;T_{n,n+n^\alpha}:=\sum_{n\leq k\leq n+n^\alpha}X_k, \] where \[ n^\alpha:=(n_1^{\alpha_1},\dots,n_d^{\alpha_d}),\quad 0< \alpha_1\leq\dots\leq\alpha_d<1, \] and set \(p:=\max\{k:\alpha_k= \alpha_1\}\). If \[ E(| X|^{2/\alpha_1} (\log_+| X|)^{p-1-1/\alpha_1})<\infty,\tag{*} \] then \[ \limsup_{n\to\infty}\frac {T_{n,n+n^\alpha}}{\sqrt{2| n^\alpha |\log| n|}}= \sigma\sqrt{1-\alpha_1},\quad\text{where} \qquad| n^\alpha|: =n_1^{\alpha_1}\dots n_d^{\alpha_d}.\tag{**} \] Conversely, if \[ P\left(\limsup_{n\to\infty}\frac {| T_{n,n+n^\alpha}|}{\sqrt{| n^\alpha|\log| n |}}< \infty\right)>0, \] then (*) holds, \(EX=0\), and (**) holds with \(\sigma^2= \text{Var}\,X\). We note that in the special case \(\alpha_1= \dots= \alpha_d\), Theorem 2.2. was proven by the present authors in the first part [Bernoulli 14, 249--276 (2008; doi:10.3150/07-BEJ103)].