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On moderate deviations for martingales. (English) Zbl 0881.60026

The paper considers square-integrable martingales \((X_t\), \(0\leq t\leq1)\) and investigates the large deviation problem \(P(X_1\geq r)=(1-\Phi(r))(1+\theta(r)R)\) for each \(|r|\leq C\), where \(|\theta(r)|\leq1\). The author receives a precise description of the remainder \(R\) and of the range \(C\) by means of \(L_{2\delta}=E\;\sum_{0<s\leq1}|\Delta X_s|^{2\delta}\) and \(N_{2\delta}=E|\langle X\rangle_1-1|^{1+\delta}\), where \(\Delta X\) is the jumping part and \(\langle X\rangle\) is the quadratic characteristic of the martingale \(X\). The result gives a rate for large deviations in martingale CLT and even improves the result known for the case of independent random variables.
Reviewer: P.Lachout (Praha)

MSC:

60F10 Large deviations
60G44 Martingales with continuous parameter
62E17 Approximations to statistical distributions (nonasymptotic)
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[1] Amosova, N. N. (1972). On limit theorems for probabilities of moderate deviations. Vestnik Leningrad. Univ. 13 5-14. (In Russian.)
[2] Bergstrom, H. (1944). On the central limit theorem. Scandinavisk. Aktuarietidscrift. 27 139-153. · Zbl 0060.28707
[3] Bentkus, V. (1986). On large deviations in Banach spaces. Teor. Veroyatnost. i Primen. 31 710- 716. (In Russian.) · Zbl 0623.60012
[4] Bentkus, V. and Rackauskas, A. (1990). On probabilities of large deviation in Banach spaces. Probab. Theory Related Fields 86 131-154. · Zbl 0678.60005
[5] Bolthausen, E. (1982). Exact convergence rates in some martingale central limit theorems. Ann. Probab. 10 672-688. Bose, A. (1986a). Certain non-uniform rates of convergence to normality for martingale differences. J. Statist. Plann. Inference 14 155-167. Bose, A. (1986b). Certain non-uniform rates of convergence to normality for a restricted class of martingales. Stochastics 16 279-294. · Zbl 0494.60020
[6] Brown, B. M. and Heyde, C. C. (1970). On the departure from normality of a certain class of martingales. Ann. Math. Statist. 41 2161-2165. · Zbl 0225.60026
[7] Dellacherie, C. (1972). Capacites et Processus Stochastiques. Springer, Berlin. Grama, I. G. (1988a). An improvement of the rate of convergence in the CLT for semimartingales. Mat. Issled. 97 34-40. (In Russian.) Grama, I. G. (1988b). Rate of normal approximation for semimartingales. Ph.D. dissertation, V. A. Steklov Math. Inst., Moscow. (In Russian.) · Zbl 0435.62049
[8] Grama, I. G. (1990). On the rate of convergence in the central limit theorem for semimartingales in nonclassical setting. Mat. Issled. 116 14-46. (In Russian.)
[9] Grama, I. G. (1993). On the rate of convergence in the central limit theorem for d-dimensional semimartingales. Stochastics Stochastics Reports 44 131-152. · Zbl 0805.60043
[10] Grama, I. G. (1995). The probabilities of large deviations for semimartingales. Stochastics Stochastics Reports 54 1-19. · Zbl 0857.60018
[11] Haeusler, E. (1988). On the rate of convergence in the central limit theorem for martingales with discrete and continuous time. Ann. Probab. 16 275-299. · Zbl 0639.60030
[12] Haeusler, E. and Joos, K. (1988). A nonuniform bound on the rate of convergence in the central limit theorem for martingales. Ann. Probab. 16 1690-1720. · Zbl 0656.60034
[13] Ibragimov, I. A. and Linnik, Yu. V. (1965). Independent and Stationary Connected Variables. Nauka, Moscow. (In Russian.) · Zbl 0154.42201
[14] Jacod, J. and Shiryaev, A. N. (1987). Limit Theorems for Stochastic Processes. Springer, Berlin. · Zbl 0635.60021
[15] Liptser, R. Sh. and Shiryaev, A. N. (1982). On the rate of convergence in the central limit theorem for semimartingales. Teor. Veroyatnost. i Primen. 27 3-14. (In Russian.) · Zbl 0499.60040
[16] Liptser, R. Sh. and Shiryaev, A. N. (1989). Theory of Martingales. Kluwer, Dordrecht. · Zbl 0728.60048
[17] Petrov, V. V. (1972). Sums of Independent Random Variables. Nauka, Moscow. (In Russian.) · Zbl 0288.60050
[18] Rackauskas, A. (1990). On probabilities of large deviation for martingales. Litovsk. Mat. Sb. 30 784-795.
[19] Rubin, H. and Sethuraman, J. (1965). Probabilities of moderate deviations. Sankhy\?a Ser. A 37 325-346. · Zbl 0178.53802
[20] Saulis, L. and Statulevicius, V. (1989). Limit Theorems for Large Deviations. Mokslas, Vilnius. (In Russian.)
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