Burkholder-Davis-Gundy inequality

Consider a regular martingale , , almost surely. Let and stand for and the quadratic variation , respectively.

The following inequality in -spaces was proved in [a2]:

(a1)

where and are positive constants depending only on , .

In fact, this inequality was proved in three steps; D.L. Burkholder [a3] proved the cases ; Burkholder and R.F. Gundy [a4] proved the cases for a large class of martingales, and Gundy [a5] proved the case for all martingales.

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

(a2)

where and are positive constants depending only on .

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