Geodesic
A small deviation theorem for independent random variables
Teoriâ veroâtnostej i ee primeneniâ, Tome 40 (1995) no. 1, pp. 225-235

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Let $\{X_n,\,n\ge 1\}$ be a sequence of independent, not necessarily identically distributed random variables. Put $S_k(n)=\sum_{i=1+k}^{n+k}X_i$. A small deviation theorem, i.e., the asymptotic bound of $\mathbf P(\max_{i\le n}|S_k (i)|\le x_{k,n})$ is obtained under a uniform Lindeberg's condition.
Keywords: small deviation, partial sums, independent random variables. 
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     author = {Q. M. Shao},
     title = {A small deviation theorem for independent random variables},
     journal = {Teori\^a vero\^atnostej i ee primeneni\^a},
     pages = {225--235},
     publisher = {mathdoc},
     volume = {40},
     number = {1},
     year = {1995},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/TVP_1995_40_1_a20/}
}
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Q. M. Shao. A small deviation theorem for independent random variables. Teoriâ veroâtnostej i ee primeneniâ, Tome 40 (1995) no. 1, pp. 225-235. http://geodesic.mathdoc.fr/item/TVP_1995_40_1_a20/