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A note on random signs∗

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Abstract

Let ε, ε1, ε2, . . . be independent identically distributed Rademacher random variables, P{ε = ± 1} = 1/2. Denote S = ∑ ∞ i = 1 a i ε i , where a1, a2, . . . is a (weight) sequence of nonrandom real numbers such that ∑ ∞ i = 1 a 2 i  ≤ 1. We prove that \( \mathrm{P}\left\{S\ge x\right\}\le 1/4+\left(1-\sqrt{2-2/{x}^2}\right)/8 \) for \( x\in \left(1,\sqrt{2}\right] \).

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Authors and Affiliations

  1. Institute of Mathematics and Informatics, Vilnius University, Akademijos str. 4, LT-08663, Vilnius, Lithuania

    Dainius Dzindzalieta

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  1. Dainius Dzindzalieta
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Correspondence to Dainius Dzindzalieta.

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*This research was funded by a grant (No. MIP-053/2012) from the Research Council of Lithuania.

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Dzindzalieta, D. A note on random signs∗ . Lith Math J 54, 403–408 (2014). https://doi.org/10.1007/s10986-014-9252-x

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  • DOI: https://doi.org/10.1007/s10986-014-9252-x

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