A nonclassical Chung-type law of the iterated logarithm for i.i.d. random variables

Letting $\{X,X_n;\,n\ge 1\}$ be a sequence of independent identically distributed random variables and set $S_n=\sum_{i=1}^n X_i$, we then define a sequence of positive constants $\{d(n),\ n\ge 1\}$ which is not asymptotically equivalent to $\log\log n$ but is such that $\liminf_{n\to\infty}\max_{1\le i\le n}|S_i|/\sqrt{n/d(n)}=\pi/\sqrt{8}$ almost surely, which is equivalent to $\mathbf E X=0$ and $\mathbf E X^2=1$.