Best constants in martingale version of Rosenthal’s inequality. (English) Zbl 0725.60018

Hitczenko, Paweł

Ann. Probab. 18, No. 4, 1656-1668 (1990).

Let \((d_ n)\) be a martingale difference sequence with respect to an increasing sequence of \(\sigma\)-algebras (\({\mathcal F}_ n)\). The following inequality was proved by D. L. Burkholder [ibid. 1, 19-42 (1973; Zbl 0301.60035)]: For \(2\leq p<\infty,\) \(A_ p^{-1}\{(E (\sum E_{k-1} d^ 2_ k)^{p/2})^{1/p}+(E \sup_{k}| d_ k|^ p)^{1/p}\}\) \[ \leq (E | \sum d_ k|^ 2)^{1/p}\leq B_ p\{(E (\sum E_{k-1} d^ 2_ k)^{p/2})^{1/p}+(E \sup_{k}| d_ k|^ p)^{1/p}\}, \] where \(E_{k-1} (\cdot)=E (\cdot | {\mathcal F}_{k-1})\), \(A_ p\) and \(B_ p\) are constants depending only on p. It is known and this is not difficult to prove that \(A_ p\) grows like \(\sqrt{p}\) as \(p\to \infty\). The investigation of the asymptotic behaviour of \(B_ p\) as \(p\to \infty\) has a large and interesting history (see introduction of the paper). The main result of the paper is that \(B_ p\) grows like p/ln p as \(p\to \infty\).

Reviewer: V.M.Kruglov (Moskva)

MSC:

60E15	Inequalities; stochastic orderings
60G42	Martingales with discrete parameter

Keywords:

martingale difference sequence; inequality; asymptotic behaviour

Citations:

Zbl 0301.60035