IV: 01, 1-27, LNM 124 (1970)
CAIROLI, Renzo
Une inégalité pour martingales à indices multiples et ses applications (
Several parameter processes)
This paper was the starting point of the theory of two-parameter martingales. It proves the corresponding Doob inequality and convergence theorem, with an application to biharmonic functions
Comment: The next landmark in the theory is Cairoli-Walsh,
Acta. Math.,
134, 1975. For the modern results, see Imkeller,
Two Parameter Processes and their Quadratic Variation, Lect. Notes in M.
1308, 1989
Keywords: Two-parameter martingales,
Maximal inequality,
Almost sure convergenceNature: Original
Retrieve article from Numdam
XV: 17, 251-258, LNM 850 (1981)
PITMAN, James W.
A note on $L_2$ maximal inequalities (
Martingale theory)
This paper contains a $L^2$ inequality between two processes $(X_n,M_n)$ under assumptions which (if $X$ is a martingale) apply to $M_n=\sup_{m\le n} |X_m|$, and to other interesting cases as well. In particular, Doob's inequality is valid for the larger process $\sup_{m\le n} X_m^+ +\sup_{m\le n} X_m^-$
Keywords: Maximal inequality,
Doob's inequalityNature: Original Retrieve article from Numdam
