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Let {X,Xn;nÃ¢Â‰Â¥1} be a sequence of real-valued i.i.d. random variables and  let Sn=Ã¢ÂˆÂ‘i=1nXi, nÃ¢Â‰Â¥1. In this paper, we study the probabilities of large deviations of the form P(Sn>tn1/p), P(Sn<Ã¢ÂˆÂ’tn1/p), and P(|Sn|>tn1/p), where t>0 and 0<p<2. We obtain precise asymptotic estimates for these  probabilities under mild and easily verifiable conditions. For example, we show that if Sn/n1/pÃ¢Â†Â’P0 and if there exists a nonincreasing positive  function ÃÂ•(x) on  [0,Ã¢ÂˆÂž) which is regularly varying with index  ÃŽÂ±Ã¢Â‰Â¤Ã¢ÂˆÂ’1 such that  limsupxÃ¢Â†Â’Ã¢ÂˆÂžP(|X|>x1/p)/ÃÂ•(x)=1, then for every t>0, limsupnÃ¢Â†Â’Ã¢ÂˆÂžP(|Sn|>tn1/p)/(nÃÂ•(n))=tpÃŽÂ±

Precise lim sup behavior of probabilities of large deviations for sums of i.i.d. random variables