User:Jason.rute/sandbox

Lépingle's inequality is a variational inequality for martingales. It is a quantitative form of Doob's martingale convergence theorem. It was proved by Dominique Lépingle.

Definitions
The p-variation norm of a sequence of real numbers a = (an) is given by


 * $$ \mathcal{V}_p(a) = \sup \left( |a_0|^p + \sum_{i=0}^{\infty} | a_{n_{i+1}}-a_{n_i} |^p \right)^{1/p}$$

where the supremum is over all increasing sequences (ni) of nonnegative integers.

A martingale M = (Mn) is said to be Lp-bounded if $$\sup_n \| M_n \|_{L_p}$$ is finite.

Lépingle's inequality
Lépingle's inequality says that for r > 2, the variation $$ \mathcal{V}_r $$ is a bounded operator from the space of Lp-bounded martingales to the space of Lp functions. More specifically, for any martingale M = (Mn), any r > 2 and any 1 < p < ∞,


 * $$ \| \mathcal{V}_r(M) \|_{L^p} \leq C_{p,r} \sup_n \| M_n \|_{L^p}$$

where the constant Cp,r only depends on p and r. The r > 2 cannot be strengthened to r ≥ 2.

An alternate form
For any filtration $$ (\mathcal{A}_n) $$, if f is an Lp function then $$ M_n = \mathbb{E}( f | \mathcal{A}_n) $$ is an Lp-bounded martingale. Conversely, for p > 2 all Lp-bounded martingales adapted to $$ (\mathcal{A}_n) $$ are of this form. By setting $$ V_p (f) = \mathcal{V}_p(M) $$, Lepingle's inequality becomes


 * $$ \| V_r(f) \|_{L^p} \leq {C_{p,r}} \| f \|_{L^p}$$.

Both Doob's martingale convergence theorem and Doob's submartingale inequality can be derived from Lépingle's inequality.

Lépingle's inequality also holds for continuous time martingales $$ (M_t)_{t \in \mathbb{R}^+} $$.

Related applications
Besides probability theory, the variation norm $$ \mathcal {V}_p $$ is also used in ergodic theory and harmonic analysis. In those settings, inequalities analogous to Lépingle's inequality also hold true.

For martingales, the operator $$ \mathcal {V}_p(M) $$ is similar to the quadratic variation operator used in stochastic calculus.