2025/11/11 by Rzeszut, Maciej
#FOS: Mathematics #Probability (math.PR)
paper · doi:10.48550/arxiv.2511.08712
The classical Davis inequality 𝔼 Mf≃ 𝔼 Sf, where (Sf)2=∑k|fk-fk-1|2 is the square function and Mf= supn |fn| is the maximal function, is true with a universal constant for any martingale f on any filtration. A natural analog in the setting of (F4) doubly indexed filtrations, i.e. (Fi,j)i,j such that the operators 𝔼(⋅| Fi,∞) and 𝔼(⋅| F∞,j) commute and their product is 𝔼(⋅| Fi,j), is the conjecture 𝔼supn,m |fn,m|≃𝔼(∑i,j|Δfi,j|2)^(1)/(2), where Δfi,j=fi,j-fi-1,j-fi,j-1+fi-1,j-1. It was known to be true only with some highly restrictive additional assumptions, e.g. regularity of the filtration (gn,m\gtrsim gn+1,m,gn,m+1 for any positive martingale g) or f being a strong martingale (𝔼(Δfi,j| Fi-1,j\vee Fi,j-1)=0). We prove the inequality \lesssim assuming just the (F4) condition.