The rate of convergence in the precise large deviation theorem

• Autorzy: Baltrūnas A.
• Języki publikacji: EN

Abstrakty

EN

Let X₁, X₂, . . . be i.i.d. random variables with a common d.f. F. Let S_n = X₁ +. . .+ X_n, n ≥1, and M_n = max _k≤n X_k, n≥1. In this paper for a large class of subexponential distributions we estimate the rate of convergence. ∆_n(t) = P(S_n > t) − P(M_n> t), where n ≥ 1 and t ≥ 0. We close this paper with some examples.

Słowa kluczowe

EN

limit theorems; large deviations; subexponential distributions

• Wydawca: Wrocław University of Science and Technology
• Czasopismo: Probability and Mathematical Statistics
• Rocznik: 2002
• Tom: Vol. 22, Fasc. 2
• Strony: 343--354
• Opis fizyczny: Biblogr. 12 poz.

Twórcy

autor

Baltrūnas A.

• Institute of Mathematics and Informatics Akademijos 4, 2021 Vilnius, Lithuania

Bibliografia

• [1] A. Baltrūnas, On the asymptotics of one-sided large deviation probabilities, Lithuanian Math. J. 35 (1995), pp. 11-17.
• [2] A. A. Borovkov, Large deviation probabilities for random walk with semiexponential distributions (in Russian), Sibirsk. Math. Zh. 41 (6) (2000), pp. 1290-1324.
• [3] V. P. Chistyakov, A theorem on sums of independent positive random variables and its applications to branching random processes, Theory Probab. Appl. 9 (1964), pp. 640-648 (= English translation of: Teor. Veroyatnost. i Primenen. 9 (1964), pp. 710-718).
• [4] D. B. H. Cline and T. Hsing, Large deviation probabilities for sums and maxima of random variables with heavy or subexponential tails, preprint, Texas A&M University, 1991.
• [5] D. H. Fuc and S. V. Nagaev, Probability inequalities for the sums of independent random variables (in Russian), Teor. Veroyatnost. i Primenen. 16 (1971), pp. 660-675.
• [6] C. M. Goldie, Subexponential distributions and dominated-variation tail, J. Appl. Probab. 15 (1978), pp. 440-442.
• [7] C. Klüppelberg, Subexponential distributions and integrated tails, J. Appl. Probab. 25 (1988), pp. 132-141.
• [8] T. Mikosch and A. V. Nagaev, Large deviations of heavy-tailed sums with applications in insurance, Extremes 1 (1998), pp. 81-110.
• [9] A. V. Nagaev, On a property of sums of independent random variables (in Russian), Teor. Veroyatnost. i Primenen. 22 (1977), pp. 335-346.
• [10] S. V. Nagaev, On the asymptotic behaviour of one-sided large deviation probabilities (in Russian), Teor. Veroyatnost. i Primenen. 26 (1981), pp. 369-372.
• [11] I. F. Pinelis, Asymptotic equivalence of the probabilities of large deviations for sums and maxima of independent random variables (in Russian): Limit theorems of the probability theory, Trudy Inst. Mat. (Novosibirsk) 5 (1985), pp. 144-173.
• [12] L. V. Rozovskii, Probabilities of large deviations on the whole axis (in Russian), Teor. Veroyatnost. i Primenen. 38 (1993), pp. 79-109.