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 Cet article a éte moissonné depuis la source Math-Net.Ru

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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. 
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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/

[1] Csáki E., “On the lower limits of maxima and minima of Wiener process and partial sums”, Z. Wahrscheinlichkeitstheor. Verw. Geb., 43:3 (1978), 205–221 | DOI | MR | Zbl

[2] Chung K. L., “On the maximum partial sums of sequences of independent random variables”, Trans. Amer. Math. Soc., 64 (1948), 205–233 | DOI | MR | Zbl

[3] Jain N. C., Pruitt W. E., “The other law of the iterated logarithm”, Ann. Probab., 3:6 (1975), 1046–1049 | DOI | MR | Zbl

[4] Klesov O., Rosalsky A., “A nonclassical law of the iterated logarithm for i.i.d. square integrable random variables”, Stochastic Anal. Appl., 19:4 (2001), 627–641 | DOI | MR | Zbl

[5] Klesov O., Rosalsky A., “A nonclassical law of the iterated logarithm for i.i.d. square integrable random variables. II”, Stochastic Anal. Appl., 20:4 (2002), 839–846 | DOI | MR | Zbl

[6] Hartman P., Wintner A., “On the law of the iterated logarithm”, Amer. J. Math., 63 (1941), 169–176 | DOI | MR

[7] Shao Q. M., “How small are the increments of partial sums of non-i.i.d. random variables”, Sci. China, 35:6 (1992), 675–689 | MR | Zbl

[8] Shao Q. M., “A small deviation theorem for independent random variables”, Teoriya veroyatn. i ee primen., 40:1 (1995), 225–235 | MR | Zbl

[9] Lin Z., “A self-normalized Chung type law of the iterated logarithm”, Teoriya veroyatn. i ee primen., 41:4 (1996), 934–942 | MR | Zbl