Large deviations for martingales

By Emmanuel Lesigne, Dalibor Volný. 2001

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Let (Xi) be a martingale difference sequence and Sn=[summation operator]i=1n Xi. We prove that if supi E(eXi) 0 such that [mu](Sn>n)[less-than-or-equals, slant]e-cn1/3; this bound is optimal for the class of martingale difference sequences which are also strictly stationary and ergodic. If the sequence (Xi) is bounded in Lp, 2[less-than-or-equals, slant]p n)[less-than-or-equals, slant]cn-p/2 which is again optimal for strictly stationary and ergodic sequences of martingale differences. These estimations can be extended to martingale difference fields. The results are also compared with those for iid sequences; we give a simple proof that the estimate of Nagaev, Baum and Katz, [mu](Sn>n)=o(n1-p) for Xi[set membership, variant]Lp, 1[less-than-or-equals, slant]p 0.

Subfields: Applied Mathematics, Finance, Management Science and Operations Research

View Paper: https://www.semanticscholar.org/paper/fb8d370c6e2e1f9f321ef81e954109b4b8ee09b4