2022/11/28 by Lê, Khoa · 1 citation
#60H07 #FOS: Mathematics #Probability (math.PR)
paper · doi:10.48550/arxiv.2211.15550
For a general adapted integrable right-continuous with left limits (RCLL) process (Xt)t∈[0,τ] taking values in a metric space (\mathcal E,d), we show (among other things) that for every m∈(1,∞) (m-1)/(2m-1)‖supt∈[0,τ]𝔼(d(Xt-,Xτ)|\mathcal Ft)‖m≤ ‖supt∈[0,τ]d(X0,Xt)‖m≤ c(m2)/(m-1) ‖supt∈[0,τ]𝔼(d(Xt-,Xτ)|\mathcal Ft)‖m with a universal constant c. This is a probabilistic version of Fefferman--Stein estimate for the sharp maximal functions. While the former inequality is derived easily from Doob's martingale inequality, the later inequality is a consequence of John--Nirenberg inequalities for weighted BMO processes, which are obtained in this note. We explain how John--Nirenberg inequalities can be utilized to obtain inequalities for martingales, both old and new alike in a unified way.