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Teor. Veroyatnost. i Primenen., 2000, Volume 45, Issue 1, Pages 5–29 (Mi tvp322)  

This article is cited in 15 scientific papers (total in 15 papers)

Integro-local limit theorems including large deviations for sums of random vectors. II

A. A. Borovkov, A. A. Mogul'skii

Sobolev Institute of Mathematics, Siberian Branch of the Russian Academy of Sciences

Abstract: This paper is a continuation of [A. A. Borovkov and A. A. Mogulskii, Theory Probab. Appl., 43 (1998), pp. 1–12] and [A. A. Borovkov and A. A. Mogulskii, Siberian Math. J., 37 (1996), pp. 647–682]. Let $S(n)=\xi(1)+\cdots +\xi(n)$ be the sum of independent nondegenerate random vectors in $\mathbf{R}^d$ having the same distribution as a random vector $\xi$. It is assumed that $\varphi(\lambda)= \mathbf{E}  e^{\langle\lambda,\xi\rangle}$ is finite in a vicinity of a point ${\lambda \in \mathbf{R}^d}$. We obtain asymptotic representations for the probability $\mathbf{P}\{S(n)\in \Delta (x)\}$ and the renewal function $H(\Delta (x))= \sum_{n=1}^{\infty}\mathbf{P}\{S(n)\in \Delta (x)\}$, where $\Delta(x)$ is a cube in $\mathbf{R}^d$ with a vertex at point $x$ and the edge length $\Delta$. In contrast to the above-mentioned papers, the obtained results are valid, in essence, either without any additional assumptions or under very weak restrictions.

Keywords: large deviations, rate function, renewal function, integro-local theorem, arithmetic distribution, lattice distribution, nonlattice distribution.


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English version:
Theory of Probability and its Applications, 2001, 45:1, 3–22

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Received: 12.02.1999

Citation: A. A. Borovkov, A. A. Mogul'skii, “Integro-local limit theorems including large deviations for sums of random vectors. II”, Teor. Veroyatnost. i Primenen., 45:1 (2000), 5–29; Theory Probab. Appl., 45:1 (2001), 3–22

Citation in format AMSBIB
\by A.~A.~Borovkov, A.~A.~Mogul'skii
\paper Integro-local limit theorems including large deviations for sums of random vectors.~II
\jour Teor. Veroyatnost. i Primenen.
\yr 2000
\vol 45
\issue 1
\pages 5--29
\jour Theory Probab. Appl.
\yr 2001
\vol 45
\issue 1
\pages 3--22

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    This publication is cited in the following articles:
    1. A. A. Borovkov, A. A. Mogul'skii, “Large deviations for Markov chains in the positive quadrant”, Russian Math. Surveys, 56:5 (2001), 803–916  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    2. F. Avram, A. A. Mogul'skii, “Large Deviations of the Waiting Time for Tandem Queueing Systems”, Siberian Adv. Math., 13:2 (2003), 1–34  mathnet  mathscinet  zmath  elib
    3. Samsioe G., “Bleeding problems in middle aged women”, Maturitas, 43, Suppl. 1 (2002), S27–S33  crossref  isi
    4. A. A. Borovkov, “Asymptotics of crossing probability of a boundary by the trajectory of a Markov chain. Exponentially decaying tails”, Theory Probab. Appl., 48:2 (2004), 226–242  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    5. Theory Probab. Appl., 49:4 (2005), 561–588  mathnet  crossref  crossref  mathscinet  zmath  isi
    6. A. A. Borovkov, A. A. Mogul'skii, “Integro-local theorems for sums of independent random vectors in the series scheme”, Math. Notes, 79:4 (2006), 468–482  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    7. A. A. Borovkov, A. A. Mogul'skii, “On large and superlarge deviations for sums of independent random vectors under the Cramer condition. I”, Theory Probab. Appl., 51:2 (2007), 227–255  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    8. A. A. Borovkov, A. A. Mogul'skii, “On Large Deviations of Sums of Independent Random Vectors on the Boundary and Outside of the Cramér Zone. I”, Theory Probab. Appl., 53:2 (2009), 301–311  mathnet  crossref  crossref  zmath  isi
    9. Miyazawa M., “Light tail asymptotics in multidimensional reflecting processes for queueing networks”, TOP, 19:2 (2011), 233–299  crossref  mathscinet  zmath  isi  elib  scopus
    10. A. A. Borovkov, A. A. Mogul'skiǐ, “Large deviation principles for sums of random vectors and the corresponding renewal functions in the inhomogeneous case”, Siberian Adv. Math., 25:4 (2015), 255–267  mathnet  crossref  mathscinet
    11. Kobayashi M., Miyazawa M., “Tail Asymptotics of the Stationary Distribution of a Two-Dimensional Reflecting Random Walk With Unbounded Upward Jumps”, Adv. Appl. Probab., 46:2 (2014), 365–399  crossref  mathscinet  zmath  isi  scopus
    12. A. A. Borovkov, “Generalization and refinement of the integro-local Stone theorem for sums of random vectors”, Theory Probab. Appl., 61:4 (2017), 590–612  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    13. Borovkov A.A. Borovkov K.A., “A refined version of the integro-local Stone theorem”, Stat. Probab. Lett., 123 (2017), 153–159  crossref  mathscinet  zmath  isi  scopus
    14. Giacomin G. Khatib M., “Generalized Poland?Scheraga denaturation model and two-dimensional renewal processes”, Stoch. Process. Their Appl., 127:2 (2017), 526–573  crossref  mathscinet  zmath  isi  scopus
    15. Kasparaviciute A., Deltuviene D., “Asymptotic Expansion For the Distribution Density Function of the Compound Poisson Process in Large Deviations”, J. Theor. Probab., 30:4 (2017), 1655–1676  crossref  mathscinet  zmath  isi  scopus
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