On moderate deviations for martingales
- Grama, Ion G.
2010 Mathematics Subject Classification
- 60F10 60G44 62E17
- Martingale, central limit theorem, rate of convergence, moderate deviation
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 Ln2δ = E ∑0 ≤ t ≤1 |Δ Xnt |2+2δ and Nn2δ = E|〈Xn〉1 -1|1+δ, where δ > 0 and Ln2δ → 0, Nn2δ → 0 as n → ∞. The exact bound for the remainder is obtained too.