## 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)
Full Text:

### References:

 [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.)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.