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Teor. Veroyatnost. i Primenen., 1969, Volume 14, Issue 1, Pages 51–63 (Mi tvp1116)  

This article is cited in 17 scientific papers (total in 17 papers)

Integral limit theorems taking into account large deviations when Cramer's condition does not hold. I

A. V. Nagaev


Abstract: Let $\xi_1,…,\xi_n$ be a sequence of independent equally distributed random variables with $\mathbf M\xi_n=0$. Throughout the paper it is supposed that the density function $p(x)$ of $\xi^n$ has the property
$$ p(x)\sim e^{-|x|^{1-\varepsilon}},\quad0<\varepsilon<1,\quad|x|\to\infty. $$
The problem we deal with is to describe the behaviour of the probability $\mathbf P\{\xi_1+…+\xi_n>x\}$ when $x$ tends to infinity so that $x>\sqrt n$.

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English version:
Theory of Probability and its Applications, 1969, 14:1, 51–64

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Received: 10.10.1967

Citation: A. V. Nagaev, “Integral limit theorems taking into account large deviations when Cramer's condition does not hold. I”, Teor. Veroyatnost. i Primenen., 14:1 (1969), 51–63; Theory Probab. Appl., 14:1 (1969), 51–64

Citation in format AMSBIB
\by A.~V.~Nagaev
\paper Integral limit theorems taking into account large deviations when Cramer's condition does not hold.~I
\jour Teor. Veroyatnost. i Primenen.
\yr 1969
\vol 14
\issue 1
\pages 51--63
\jour Theory Probab. Appl.
\yr 1969
\vol 14
\issue 1
\pages 51--64

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    This publication is cited in the following articles:
    1. A. A. Borovkov, “Large Deviations of Sums of Random Variables of Two Types”, Siberian Adv. Math., 11:4 (2001), 1–24  mathnet  mathscinet  zmath
    2. Dingcheng Wang, Chun Su, Zhishui Hu, “Precise large deviation for random sums of random walks with dependent heavy-tailed steps”, Dalnevost. matem. zhurn., 3:1 (2002), 34–51  mathnet
    3. Ng K.W., Tang Q.H., Yan J.A., Yang H.L., “Precise large deviations for sums of random variables with consistently varying tails”, Journal of Applied Probability, 41:1 (2004), 93–107  crossref  mathscinet  zmath  isi
    4. Su C., Tang Q.H., “Heavy-tailed distributions and their applications”, Probability, Finance and Insurance, 2004, 218–236  isi
    5. A. V. Kolodzei, “A theorem on the probabilities of large deviations for decomposable statistics that do not satisfy Cramér's condition”, Discrete Math. Appl., 15:3 (2005), 255–262  mathnet  crossref  crossref  mathscinet  zmath  elib
    6. A. A. Borovkov, A. A. Mogul'skii, “Integro-local and integral theorems for sums of random variables with semiexponential distributions”, Siberian Math. J., 47:6 (2006), 990–1026  mathnet  crossref  mathscinet  zmath  isi  elib  elib
    7. A. A. Mogul'skii, “Large deviations of the first passage time for a random walk with semiexponentially distributed jumps”, Siberian Math. J., 47:6 (2006), 1084–1101  mathnet  crossref  mathscinet  zmath  isi
    8. Blanchet J.H., Liu J., “State–Dependent Importance Sampling for Regularly Varying Random Walks”, Advances in Applied Probability, 40:4 (2008), 1104–1128  crossref  mathscinet  zmath  isi
    9. Denisov D., Dieker A.B., Shneer V., “Large deviations for random walks under subexponentiality: The big–jump domain”, Annals of Probability, 36:5 (2008), 1946–1991  crossref  mathscinet  zmath  isi
    10. A. A. Mogulskii, “Integralnye i integro-lokalnye teoremy dlya summ sluchainykh velichin s semieksponentsialnymi raspredeleniyami”, Sib. elektron. matem. izv., 6 (2009), 251–271  mathnet  mathscinet  elib
    11. Asselah A., “Annealed Lower Tails for the Energy of a Charged Polymer”, Journal of Statistical Physics, 138:4–5 (2010), 619–644  crossref  mathscinet  zmath  isi
    12. Denisov D., Foss S., Korshunov D., “Asymptotics of randomly stopped sums in the presence of heavy tails”, Bernoulli, 16:4 (2010), 971–994  crossref  mathscinet  zmath  isi
    13. Olvera-Cravioto M., Glynn P.W., “Uniform approximations for the M/G/1 queue with subexponential processing times”, Queueing Syst, 68:1 (2011), 1–50  crossref  isi
    14. A. A. Borovkov, A. A. Mogul'skiǐ, “Conditional moderately large deviation principles for the trajectories of random walks and processes with independent increments”, Siberian Adv. Math., 25:1 (2015), 39–55  mathnet  crossref  mathscinet
    15. A. A. Borovkov, A. A. Mogul'skii, “Moderately large deviation principles for trajectories of random walks and processes with independent increments”, Theory Probab. Appl., 58:4 (2014), 562–581  mathnet  crossref  crossref  mathscinet  isi  elib  elib
    16. Buraczewski D. Damek E. Mikosch T., “Stochastic Models With Power-Law Tails: the Equation X = Ax + B”, Stochastic Models With Power-Law Tails: the Equation X = Ax + B, Springer Series in Operations Research and Financial Engineering, Springer Int Publishing Ag, 2016, 1–320  crossref  isi
    17. Pinelis I., “On the Nonuniform Berry-Esseen Bound”, Inequalities and Extremal Problems in Probability and Statistics: Selected Topics, ed. Pinelis I., Academic Press Ltd-Elsevier Science Ltd, 2017, 103–138  crossref  isi
  • Теория вероятностей и ее применения Theory of Probability and its Applications
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