Another classification of the class of quasi-games which become fairer with time (Q1762795)

Let \(A_n\) is an increasing sequence of sub-\(\sigma\)-fields and \(X_n\) a sequence of random variables adapted to \(A_n\). \textit{L. H. Blake} [Pac. J. Math. 35, 279--283 (1970; Zbl 0218.60050)] introduced the class of games fairer in time: that is a sequence \(X_n\) for which for every \(\varepsilon> 0\,\exists p\in N\) such that, for all \(n\), \(\sup_{p\leq q\leq n} P(|E^q(X_n)- X_q|> \varepsilon)< \varepsilon\), where \(E^\tau(X)\) is the conditional expectation of \(X\), given \(A_\tau\). The author considers the set \(G\) of all nondecreasing functions from \(N\) to \(N\), defines a partial order on \(G\) and classifies the class of quasi-games fairer with time into a nondecreasing family of subclasses, directed by \(g\in G\).