martingale

Martingale Inequalities in Symmetric Spaces with Semimultiplicative Weight

Authors: Volosivets S. S., Zaitsev N. N.

Let (Ω,Σ,P) be a complete probability space, F = {F n } ∞ n=0 be an increasing sequence of σ- algebras such that ∪ ∞ n=0 F n generates Σ. If f = {f n } ∞ n=0 is a martingale with respect to F and E n is the conditional expectation with respect to F n , then one can introduce a maximal function M(f) = sup n>0 |f n | and a square function S(f) =?∞P i=0|f i − f i−1 | 2 ¶ 1/2 , f −1 = 0. In the case of uniformly integrable martingales there exists g ∈ L 1 (Ω) such that E n g = f n and we consider a sharp maximal function f ♯ = sup n>0 E n |g − f n−1 |.

Martingales and Theorems of Cantor–Young–Bernstein and de la Vallée Poussin

Authors: Plotnikov M. G., Plotnikova J. A.

Uniqueness problems for one-dimensional Haar series and for multiple ones have understood in numerous works. It is well-known that the subsequence of the partial sums S2k of an arbitrary Haar series can be represented as a discrete-time martingale on some filtered probability space (­Ω, F, (Fk), P). In paper the concept of a U -set for martingales is presented and some uniqueness theorems for martingales on arbitrary compact filtered probability spaces are established.