For a non-negative submartingale (X_n), P(max_{k≤n} X_k ≥ λ) ≤ E[X_n]/λ, and the L^p version ‖max_{k≤n} X_k‖_p ≤ (p/(p−1)) ‖X_n‖_p for p>1.
For a non-negative submartingale (X_n), P(max_{k≤n} X_k ≥ λ) ≤ E[X_n]/λ, and the L^p version ‖max_{k≤n} X_k‖_p ≤ (p/(p−1)) ‖X_n‖_p for p>1.