WIAS Preprint No. 121, (1994)

On moderate deviations for martingales



Authors

  • Grama, Ion G.

2010 Mathematics Subject Classification

  • 60F10 60G44 62E17

Keywords

  • Martingale, central limit theorem, rate of convergence, moderate deviation

Abstract

Let Xn = (Xnt,Fnt)0 ≤ t ≤ 1 be the square integrable martingales with the quadratic characteristics ⟨Xn⟩, n = 1, 2, .... We have proved that the large deviation relation P(Xn1 ≥ r)/(1 - Φ(r)) → 1 is valid with r growing to infinity at some rate depending on Ln = E ∑0 ≤ t ≤1 |Δ Xnt |2+2δ and Nn = E|⟨Xn1 -1|1+δ, where δ > 0 and Ln → 0, Nn → 0 as n → ∞. The exact bound for the remainder is obtained too.

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