On moderate deviations for martingales

• 成果类型: Article
• 署名作者: Grama, IG
• 署名单位: Academy of Sciences of Moldova
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• 发表日期: 1997
• 页码: 152-183
• 关键词: convergence; rates

摘要

Let X-n = (X-t(n), F-t(n))(0 less than or equal to t less than or equal to 1) be square integrable martingales with the quadratic characteristics [X-n], n = 1, 2, .... We prove that the large deviations relation P(X-1(n) greater than or equal to r)/(1 - Phi(r)) --> 1 holds true for r growing to infinity with some rate depending on L-2 delta(n) = E Sigma(0 less than or equal to t less than or equal to 1) \Delta X-t(n)\(2+2 delta) and N-2 delta(n) = E\[X-n](1) - 1\(1+delta), where delta > 0 and L-2 delta(n) --> 0, N-2 delta(n) --> 0 as n --> infinity. The exact bound for the remainder is also obtained.