Speed of dispersion for a class of martingales (Q1610196)

P. Lévy has introduced the concentration function of a random variable \(X\) as the supremum over \(x\) of \(P(x\leq X \leq x+l)\) for fixed \(l>0\). When a martingale \((M_n)\) has bounded from above differences and bounded from below conditional variances, it is proved that its concentration function is less than \(Kn^{-\lambda}\) for positive constants \(K\) and \(\lambda\).