Doob maximal and Lp inequalities

Let X be a nonnegative integrable submartingale over a finite horizon and let its maximum include every time from zero through the horizon. Stop at the first value at least a positive threshold, using the terminal horizon if no crossing occurs and keeping track of the crossing event separately. On that event the stopped value is at least the threshold. The conditional submartingale sampling inequality says the conditional expected terminal value is at least that stopped value. Integrating over the crossing event gives a refined tail estimate.

Dropping the event indicator from the nonnegative terminal value yields the simpler bound by its full expectation divided by the threshold. For a martingale, the absolute value is a nonnegative submartingale by conditional Jensen; a power of that absolute value with exponent at least one is also a submartingale when integrable. Thus terminal moments bound the probability of any earlier large absolute excursion. The nonnegativity assumption matters in the elementary tail bound because it permits discarding the contribution from the complement of the crossing event.

For exponent p greater than one, represent the pth moment of the maximum by integrating its tail against p times the threshold to the power p minus one. Insert the refined bound containing the terminal value times the crossing indicator. Tonelli exchanges the nonnegative integral and expectation, and integration up to the realized maximum produces the factor p divided by p minus one. Hölder then separates the terminal value from the maximum. A truncation argument handles a possibly unbounded maximum without assuming its pth moment in advance.

The resulting norm bound compares the maximum to the terminal value with constant p divided by p minus one. Raising both sides to the pth power raises the constant as well. Confusing these two forms is a frequent arithmetic error, especially for the second-moment bound, where the squared constant is four. At p equal to one, the integral used in the proof has a logarithmic divergence and no finite universal constant of this strong form exists. The weak tail inequality remains available.

For a centered fair walk of N steps, terminal second moment is N, so the probability of any absolute excursion at least a positive level is at most N divided by that level squared. One can apply the weak inequality directly to the squared walk, obtaining a better tail constant than combining Markov's inequality with the strong second-moment maximum bound. Bounds from different routes need not have identical constants; the available moment and the desired event determine which route is useful.

These estimates are robust to dependence compatible with the martingale property. They are typically not exact and can exceed one, in which case the trivial probability bound one should also be used. Exact enumeration on a short tree clarifies both the information captured and the slack in the inequality. In longer problems, the value is that a single terminal moment controls a whole collection of observation times, avoiding a union bound whose factor would grow with the number of times.