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Doob's estimate for coherent random variables and maximal operators on trees

Cichomski Stanisław, Osękowski Adam

Probability and Mathematical Statistics

2023

Vol. 43, Fasc. 1

109--119

Let ξ be an integrable random variable defined on (Ω, F, P). Fix k ∈ Z+ and let {Gji }1≤i≤n,1≤j≤k be a reference family of sub-σ-fields of F such that {Gji }1≤i≤n is a filtration for each j ∈ {1, . . . , k}. In this article we explain the underlying connection between the analysis of the maximal functions of the corresponding coherent vector and basic combinatorial properties of the uncentered Hardy-Littlewood maximal operator. Following a classical approach of Grafakos, Kinnunen and Montgomery-Smith, we establish an appropriate version of Doob’s celebrated maximal estimate.