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On the upper bound in the large deviation principle for sums of random vectors

Abstract

We consider the random walk generated by a sequence of independent identically distributed random vectors. The known upper bound for normalized sums in the large deviation principle was established under the assumption that the Laplace-Stieltjes transform of the distribution of the walk jumps exists in a neighborhood of zero. In the present article, we prove that, for a twodimensional random walk, this bound holds without any additional assumptions.

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Correspondence to A. A. Mogul’skiĭ.

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Original Russian Text © A. A. Mogul’skiĭ, 2013, published in Matematicheskie Trudy, 2013, Vol. 16, No. 1, pp. 121–140.

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Mogul’skiĭ, A.A. On the upper bound in the large deviation principle for sums of random vectors. Sib. Adv. Math. 24, 140–152 (2014). https://doi.org/10.3103/S1055134414020047

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Keywords

  • large deviation principle
  • upper bound in the large deviation principle
  • deviation function
  • Cramér’s condition