On the upper bound in the large deviation principle for sums of random vectors
- 37 Downloads
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.
Keywordslarge deviation principle upper bound in the large deviation principle deviation function Cramér’s condition
Unable to display preview. Download preview PDF.
- 1.R. Azencott, “Grandes déviations et applications,” Eighth Saint Flour Probability Summer School-1978 (Saint Flour, 1978), Lecture Notes in Math. V. 774 (Springer, Berlin, 1980), 1–176.Google Scholar
- 3.A. A. Borovkov, Probability Theory (Editorial URSS, Moscow, 2009; Springer, London, 2013).Google Scholar
- 4.A. A. Borovkov and A. A. Mogul’skii, Large Deviations and Testing Statistical Hypotheses, Tr. Inst. Mat. 19 (Nauka, Novosibirsk, 1992) [Part I, Sib. Adv. Math. 2, 52 (1992); Part II, Sib. Adv. Math. 2, 43 (1992); Part III, Sib. Adv. Math. 3, 19 (1993); Part IV, Sib. Adv. Math. 3, 14 (1993)].Google Scholar
- 5.A. A. Borovkov and A. A. Mogul’skiĭ, “On large deviation principles in metric spaces,” Sibirsk. Mat. Zh. 51, 1251 (2010) [Siberian Math. J. 51, 989 (2010)].Google Scholar
- 9.A. A. Mogul’skiĭ, “Large deviations for trajectories of multi-dimensional random walks,” Teor. Veroyatnost. i Primenen. 21, 309 (1976) [Theory Probab. Appl. 21, 300 (1977)].Google Scholar