Journal of Theoretical Probability

, Volume 12, Issue 3, pp 811–819

Precise Asymptotics in Spitzer's Law of Large Numbers

  • Aurel Spătaru

DOI: 10.1023/A:1021636117551

Cite this article as:
Spătaru, A. Journal of Theoretical Probability (1999) 12: 811. doi:10.1023/A:1021636117551


Let X, X1, X2,... be a sequence of i.i.d. random variables such that EX=0, assume the distribution of X is attracted to a stable distribution with exponent α<1, and set Sn=X1+ ··· +Xn. We prove that
$$\sum\limits_{n{\text{ }} \geqslant {\text{ }}1} {{\text{ }}\frac{{\text{1}}}{{\text{n}}}{\text{ }}} P(|S_n | \geqslant \varepsilon n){\text{ }} \sim {\text{ }}\frac{\alpha }{{\alpha - 1}}{\text{ }}( - \log \varepsilon {\text{) as }}\varepsilon \searrow {\text{0}}$$
Tail probabilities of sums of i.i.d. random variables stable distributions Spitzer's theorem Fuk–Nagaev type inequality 

Copyright information

© Plenum Publishing Corporation 1999

Authors and Affiliations

  • Aurel Spătaru
    • 1
  1. 1.Centre of Mathematical StatisticsRomanian AcademyBucharest 5Romania