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Teor. Veroyatnost. i Primenen., 2003, Volume 48, Issue 1, Pages 104–121 (Mi tvp303)  

This article is cited in 8 scientific papers (total in 8 papers)

Limit theorems for increments of sums of independent random variables

A. N. Frolov

Saint-Petersburg State University

Abstract: We investigate the almost surely asymptotic behavior of increments of sums of independent identically distributed random variables satisfying the one-sided Cramér condition. We establish that, irrespective of the length of the increments, the norming sequence in strong limit theorems for increments of sums is determined by a behavior of the inverse function to the function of deviations. This allows for unifying the following well-known results for increments of sums: the strong law of large numbers, the Erdős–Rényi law and Mason's extension of this law, the Shepp law, the Csörgő–Révész theorems, and the law of the iterated logarithm. In the case of large increments, we derive new results for random variables from the domain of attraction of a stable law with index $\alpha\in (1,2]$ and the parameter of symmetry $\beta=-1$.

Keywords: increments of sums of independent random variables, large deviations, Erdős–Rényi law, Shepp law, strong approximations laws, strong law of large numbers, law of the iterated logarithm.

DOI: https://doi.org/10.4213/tvp303

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English version:
Theory of Probability and its Applications, 2004, 48:1, 93–107

Bibliographic databases:

Received: 31.03.2000

Citation: A. N. Frolov, “Limit theorems for increments of sums of independent random variables”, Teor. Veroyatnost. i Primenen., 48:1 (2003), 104–121; Theory Probab. Appl., 48:1 (2004), 93–107

Citation in format AMSBIB
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\pages 93--107
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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. A. N. Frolov, “Strong limit theorems for increments of renewal processes”, J. Math. Sci. (N. Y.), 128:1 (2005), 2614–2624  mathnet  crossref  mathscinet  zmath
    2. A. N. Frolov, “On the law of the iterated logarithm for increments of sums of independent random variables”, J. Math. Sci. (N. Y.), 137:1 (2006), 4575–4582  mathnet  crossref  mathscinet  zmath
    3. A. N. Frolov, “Strong limit theorems for increments of sums of independent random variables”, J. Math. Sci. (N. Y.), 133:3 (2006), 1356–1370  mathnet  crossref  mathscinet  zmath
    4. Frolov A.N., “Converses to the Csörgő–Révész laws”, Statist. Probab. Lett., 72:2 (2005), 113–123  crossref  mathscinet  zmath  isi  elib  scopus
    5. A. N. Frolov, “Limit theorems for increments of compound renewal processes”, J. Math. Sci. (N. Y.), 152:6 (2008), 944–957  mathnet  crossref
    6. A. N. Frolov, “On asymptotic behaviour of probabilities of large deviations for compound Cox processes”, J. Math. Sci. (N. Y.), 159:3 (2009), 376–383  mathnet  crossref  zmath
    7. Zholud D., “Extremes of Shepp statistics for Gaussian random walk”, Extremes, 12:1 (2009), 1–17  crossref  mathscinet  zmath  isi  elib  scopus
    8. M. A. Lifshits, Ya. Yu. Nikitin, V. V. Petrov, A. Yu. Zaitsev, A. A. Zinger, “Toward the history of the Saint Petersburg school of probability and statistics. I. Limit theorems for sums of independent random variables”, Vestn. St Petersb. Univ. Math., 51:2 (2018), 144–163  crossref  crossref  mathscinet  zmath  isi  elib  scopus
  • Теория вероятностей и ее применения Theory of Probability and its Applications
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