Burkholder-Davis-Gundy inequality

Consider a regular martingale $ ( X _ {n} , {\mathcal F} _ {n} ) $, $ n \geq 0 $, $ X _ {0} = 0 $ almost surely. Let $ X ^ {*} $ and $ S = S ( X ) $ stand for $ \sup _ {n \geq 0 } | {X _ {n} } | $ and the quadratic variation $ ( \sum _ {i \geq 1 } ( X _ {i} - X _ {i - 1 } ) ^ {2} ) ^ { {1 / 2 } } $, respectively.

The following inequality in $ L _ {p} $- spaces was proved in [a2]:

$$ \tag{a1 } c _ {p} {\mathsf E} ( S ^ {p} ) \leq {\mathsf E} ( X ^ {*p } ) \leq C _ {p} {\mathsf E} ( S ^ {p} ) , $$

where $ c _ {p} $ and $ C _ {p} $ are positive constants depending only on $ p $, $ 1 \leq p \leq + \infty $.

In fact, this inequality was proved in three steps; D.L. Burkholder [a3] proved the cases $ 1 < p < + \infty $; Burkholder and R.F. Gundy [a4] proved the cases $ 0 < p \leq 1 $ for a large class of martingales, and Gundy [a5] proved the case $ p = 1 $ for all martingales.

Moreover, (a1) was proved in a more general form in Orlicz spaces (cf. Orlicz space) in [a2]:

$$ \tag{a2 } c _ \Phi {\mathsf E} ( \Phi ( S ( X ) ) ) \leq {\mathsf E} ( \Phi ( X ^ {*} ) ) \leq C _ \Phi {\mathsf E} ( \Phi ( S ( X ) ) ) , $$

where $ c _ \Phi $ and $ C _ \Phi $ are positive constants depending only on $ \Phi $.

The inequalities (a1) and (a2) are frequently used in martingale theory, harmonic analysis and Fourier analysis (cf. also Fourier series; Fourier transform).

For a different proof of these inequalities, see, e.g., [a1].

References

[a1]	N.L. Bassily, "A new proof of the right hand side of the Burkholder–Davis–Gundy inequality" , Proc. 5th Pannonian Symp. Math. Statistics, Visegrad, Hungary (1985) pp. 7–21
[a2]	D.L. Burkholder, B. Davis, R.F. Gundy, "Integral inequalities for convex functions of operators on martingales" , Proc. 6th Berkeley Symp. Math. Statistics and Probability , 2 (1972) pp. 223–240
[a3]	D.L. Burkholder, "Martingale transforms" Ann. Math. Stat. , 37 (1966) pp. 1494–1504
[a4]	D.L. Burkholder, R.F. Gundy, "Extrapolation and interpolation for convex functions of operators on martingales" Acta Math. , 124 (1970) pp. 249–304
[a5]	B. Davis, "On the integrability of the martingale square function" Israel J. Math. , 8 (1970) pp. 187–190