: Doob's work was seminal in establishing martingales as a vital tool for applied probability.
Define: [ \Delta A_n = A_n - A_n-1 = E[X_n - X_n-1 | \mathcalF n-1] ] Since (X) is a submartingale, (\Delta A_n \ge 0), so (A_n) is non-decreasing. Predictability follows because (\Delta A_n) is (\mathcalF n-1)-measurable. Set (M_n = X_n - A_n). Check martingale property: stochastic process doob pdf download install
Joseph Doob once said, "Probability is simply the study of measure spaces with total mass one." That elegance cannot be installed—only understood. : Doob's work was seminal in establishing martingales
If you have obtained the PDF (legally via library or purchase), here’s how to manage it: (\Delta A_n \ge 0)