Difference between revisions of "Burkholder-Davis-Gundy inequality"
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| − | + | Consider a regular [[Martingale|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 [[#References|[a2]]]: | ||
| − | In fact, this inequality was proved in three steps; D.L. Burkholder [[#References|[a3]]] proved the cases < | + | $$ \tag{a1 } |
| + | c _ {p} {\mathsf E} ( S ^ {p} ) \leq {\mathsf E} ( X ^ {*p } ) \leq C _ {p} {\mathsf E} ( S ^ {p} ) , | ||
| + | $$ | ||
| + | |||
| + | 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 [[#References|[a3]]] proved the cases $ 1 < p < + \infty $; | ||
| + | Burkholder and R.F. Gundy [[#References|[a4]]] proved the cases $ 0 < p \leq 1 $ | ||
| + | for a large class of martingales, and Gundy [[#References|[a5]]] proved the case $ p = 1 $ | ||
| + | for all martingales. | ||
Moreover, (a1) was proved in a more general form in Orlicz spaces (cf. [[Orlicz space|Orlicz space]]) in [[#References|[a2]]]: | Moreover, (a1) was proved in a more general form in Orlicz spaces (cf. [[Orlicz space|Orlicz space]]) in [[#References|[a2]]]: | ||
| − | + | $$ \tag{a2 } | |
| + | c _ \Phi {\mathsf E} ( \Phi ( S ( X ) ) ) \leq {\mathsf E} ( \Phi ( X ^ {*} ) ) \leq C _ \Phi {\mathsf E} ( \Phi ( S ( X ) ) ) , | ||
| + | $$ | ||
| − | where | + | 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|harmonic analysis]] and Fourier analysis (cf. also [[Fourier series|Fourier series]]; [[Fourier transform|Fourier transform]]). | The inequalities (a1) and (a2) are frequently used in martingale theory, [[Harmonic analysis|harmonic analysis]] and Fourier analysis (cf. also [[Fourier series|Fourier series]]; [[Fourier transform|Fourier transform]]). | ||
Latest revision as of 06:29, 30 May 2020
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 |
How to Cite This Entry:
Burkholder-Davis-Gundy inequality. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Burkholder-Davis-Gundy_inequality&oldid=46174
Burkholder-Davis-Gundy inequality. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Burkholder-Davis-Gundy_inequality&oldid=46174
This article was adapted from an original article by N.L. Bassily (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article