# 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
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Burkholder–Davis–Gundy inequality. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Burkholder%E2%80%93Davis%E2%80%93Gundy_inequality&oldid=22218