Geodesic
A~nonclassical Chung-type law of the iterated logarithm for i.i.d.\ random variables
Teoriâ veroâtnostej i ee primeneniâ, Tome 51 (2006) no. 4, pp. 816-821

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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 $\mathbf E X^2=1$.
Keywords: Chung-type law of the iterated logarithm, small deviation theorem. 
@article{TVP_2006_51_4_a12,
     author = {T.-X. Pang and Z.-Y. Lin},
     title = {A~nonclassical {Chung-type} law of the iterated logarithm for i.i.d.\ random variables},
     journal = {Teori\^a vero\^atnostej i ee primeneni\^a},
     pages = {816--821},
     publisher = {mathdoc},
     volume = {51},
     number = {4},
     year = {2006},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/TVP_2006_51_4_a12/}
}
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T.-X. Pang; Z.-Y. Lin. A~nonclassical Chung-type law of the iterated logarithm for i.i.d.\ random variables. Teoriâ veroâtnostej i ee primeneniâ, Tome 51 (2006) no. 4, pp. 816-821. http://geodesic.mathdoc.fr/item/TVP_2006_51_4_a12/