FAR EASTERN BRANCH OF THE RUSSIAN ACADEMY OF SCIENCES
INSTITUTE OF APPLIED MATHEMATICS KHABAROVSK DIVISION
Far Eastern Mathematical Journal
Precise large deviation for random sums of random walks with dependent heavy-tailed steps |
Dingcheng Wang, Chun Su, Zhishui Hu |
2002, issue 1, P. 34–51 |
Abstract |
| In most applications the assumption of independent step sizes is, clearly, unrealistic. It is an important way to model the dependent steps $\{X_n \}_{n=1}^{\infty}$ of the random walk as a two-sided linear process, $X_n=\sum\limits_{j=-\infty}^{\infty}\varphi_{n-j} \eta_j$, $n=1,2,3,\dots$, where $\{\eta,\eta_n,\ n=0,\pm 1,\pm 2,\pm 3,\dots\}$ is a sequence of iid random variables with finite mean $\mu>0$. Moreover suppose that $\eta$ satisfies certain tailed balance condition and its distribution function belongs to $ERV(-\alpha,-\beta)$ with $1<\alpha\le\beta<\infty$. Denote $S_n=X_1+X_2+\dots+X_n$, $n\ge 1$. At first we discuss precise large deviation problems of non-random sums $\{S_n-ES_n\}_{ n=1}^{\infty}$, then discuss precise large deviation problems of $S(t)-ES(t)=\sum_{i=1}^{N(t)}(X_i-EX_i)$, $t\ge 0$ for non-negative and inter-value random process $N(t)$ such that Assumption A, independent of $\{\eta_n\}_{n=-\infty}^{\infty}$. We show that if the steps of random walk are not independent, then precise large deviation result of random sums may be different from the case with iid steps, which means that dependence affects the tails of compound processes $\{S(t)\}_{t \ge 0}$. |
Keywords: Class ERV Dependent, Heavy-tailed Distribution, Random Walk, Precise Large Deviation, Tail Balance Condition, Two-sided linear process. |
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References |
| [1] K. B. Athreya and P. E. Ney, Branching processes, Springer, Berlin, 1972. [2] P. Brockwell and R. Davis, Time series: Theory and Methods, 2nd ed, Springer, New York, 1991. [3] 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 Univ, 1991. [4] P. Embrechts, C. Klu?ppelberg and T. Mikosch, Modelling extremal events for insurance and finance, Springer, Berlin, 1997. [5] C. C. Heyde, “A contribution to the theory of large deviations for sums of independent random variables”, Z. Wahrscheinlichkeitstheorie verw. Geb., 7 (1967a), 303–308. [6] C. C. Heyde, “On large deviation problems for sums of random variables which are not attracted to the normal law”, Ann. Math. Statist., 38 (1967b), 1575–1578. [7] C. C. Heyde, “On large deviation probabilities in the case of attraction to a nonnormal stable law”, Sankya? Ser. A, 30 (1968), 253–258. [8] J. Hoffmann-J?rgensen, “Sums of independent Banach space valued random variables”, Studia. Math., 52 (1974), 159–186. [9] C. Klu?ppelberg and T. Mikosch, “Large deviations of heavy-tailed random sums with applications in insurance and finance”, J. Appl. Prob., 34 (1997), 293–308. [10] T. Mikosch and G. Samorodnitsky, “The supremum of a negative drift random walk with dependent heavy-tailed steps”, Ann. Appl. Probab., 10:3 (2000), 1025–1064. [11] T. Mikosch and A. V. Nagaev, “Large deviations of heavy-tailed sums with applications in insurance”, Extremes, 1:1 (1998), 81–110. [12] A. V. Nagaev, “Integral limit theorems for large deviations when Cramer's condition is not fulfilled. I, II”, Theory Prob. Appl., 14 (1969), 51–64. [13] A. V. Nagaev, “Limit theorems for large deviations when Cramer's conditiona are violated”, Izv. Akad. Nauk UzSSR Ser. Fiz.-Mat. Nauk, 6 (1969), 17–22 (In Russian). [14] S. V. Nagaev, “Large deviations of sums of independent random variables”, Random Processes and Statistical Decision Function, Trans. Sixth Prague Conf. Information Theory, Acad. Prague, 1973, 657–674. [15] S. V. Nagaev, “Large deviations of sums of independent random variables”, Ann. Prob., 7 (1979), 746–789. [16] I. F. Pinelis, “On the asymptotic equivalence of probabilities of large deviations for sums and maxima of independent random variables”, Limit Theorems in Probability theory, Trudy Inst. Math., 5, Nauka, Novosibirsk, 1985 (In Russian). [17] A. G. Pykes, “On the tails of waiting-time distributions”, J. Appl. Prob., 12 (1975), 555–564. [18] T. Rolski, H. Schmidli and J. Teugels, Stochastic processes for insurance and finance, John Wiley & Sons Ltd, Chichester, 1999. [19] L. V. Rozovski, “Probabilities of large deviations on the whole axis”, Theory Prob. Appl., 38 (1993), 53–79. [20] C. Su, Q. H. Tang and T. Jiang, “A Contribution to Large Deviations for Heavy -tailed Random Sums”, Chinese Science, 44:4 (2001), 438–444. |