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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 convergence

Nature: Original

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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 inequality

Nature: Original

Retrieve article from Numdam

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,

Keywords: Two-parameter martingales, Maximal inequality, Almost sure convergence

Nature: Original

Retrieve article from Numdam

XV: 17, 251-258, LNM 850 (1981)

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 inequality

Nature: Original

Retrieve article from Numdam