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Precise asymptotics in the deviation probability series of self-normalized sums. (English) Zbl 1213.60053
Let \(X_1,X_2,\dots\) be independent identically distributed random variables with zero mean. Define
\[ S_n=X_1+\dots+X_n, \quad W_n^2=X_1^2+\dots+X_n^2. \]
Let \(g\) and \(\phi\) be functions satisfying appropriate conditions and let \(\varepsilon_0>0\) be a constant. The author computes the asymptotics as \(\varepsilon\to\varepsilon_0\) of
\[ \sum_{n=1}^{\infty} g(\phi(n))\phi'(n)\mathbb P\left[\left|\frac{S_n}{W_n}\right|> \varepsilon \phi(n)\right] \]
and
\[ \sum_{n\to\infty}^{\infty}g(\phi(n))\phi'(n)\mathbb P\left[\left(\frac{S_n}{W_n}\right)^2 1_{\{S_n\geq \varepsilon W_n \phi(n)\}}\right]. \]
With various choices for \(g\) and \(\varphi\) the author recovers several results of Y. Zhao and J. Tao [Comput. Math. Appl. 56, No. 7, 1779–1786 (2008; Zbl 1152.60315)] and T.-X. Pang, L.-X. Wang and J.-F. Zhang [J. Math. Anal. Appl. 340, No. 2, 1249–1262 (2008; Zbl 1140.60023)] and obtains a number of new results. For instance, the following formula is derived:
\[ \lim_{\varepsilon\downarrow 0}\exp\left\{-\frac{(a+1)^2}{2\varepsilon^2}\right\}\varepsilon^{2b}\sum_{n=3}^{\infty}\frac{(\log n)^a(\log\log n)^b}{n}\mathbb P\left[\left|\frac{S_n}{W_n}\right|> \varepsilon\log\log n\right]=2(a+1)^{b-1}. \]
MSC:
60F10 Large deviations
60F99 Limit theorems in probability theory
60G50 Sums of independent random variables; random walks
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