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Consider a regular [[Martingale|martingale]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b111/b111050/b1110501.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b111/b111050/b1110502.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b111/b111050/b1110503.png" /> almost surely. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b111/b111050/b1110504.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b111/b111050/b1110505.png" /> stand for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b111/b111050/b1110506.png" /> and the quadratic variation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b111/b111050/b1110507.png" />, respectively.
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The following inequality in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b111/b111050/b1110509.png" />-spaces was proved in [[#References|[a2]]]:
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<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b111/b111050/b11105010.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a1)</td></tr></table>
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Consider a regular [[Martingale|martingale]]  $  ( X _ {n} , {\mathcal F} _ {n} ) $,
 +
$  n \geq  0 $,
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$  X _ {0} = 0 $
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almost surely. Let  $  X  ^ {*} $
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and  $  S = S ( X ) $
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stand for  $  \sup  _ {n \geq 0 }  | {X _ {n} } | $
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and the quadratic variation  $  ( \sum _ {i \geq 1 }  ( X _ {i} - X _ {i - 1 }  )  ^ {2} ) ^ { {1 / 2 } } $,
 +
respectively.
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b111/b111050/b11105011.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b111/b111050/b11105012.png" /> are positive constants depending only on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b111/b111050/b11105013.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b111/b111050/b11105014.png" />.
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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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b111/b111050/b11105015.png" />; Burkholder and R.F. Gundy [[#References|[a4]]] proved the cases <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b111/b111050/b11105016.png" /> for a large class of martingales, and Gundy [[#References|[a5]]] proved the case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b111/b111050/b11105017.png" /> for all martingales.
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$$ \tag{a1 }
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c _ {p} {\mathsf E} ( S  ^ {p} ) \leq  {\mathsf E} ( X ^ {*p } ) \leq  C _ {p} {\mathsf E} ( S  ^ {p} ) ,
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$$
 +
 
 +
where  $  c _ {p} $
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and  $  C _ {p} $
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are positive constants depending only on  $  p $,
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$  1 \leq  p \leq  + \infty $.
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In fact, this inequality was proved in three steps; D.L. Burkholder [[#References|[a3]]] proved the cases $  1 < p < + \infty $;  
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Burkholder and R.F. Gundy [[#References|[a4]]] proved the cases $  0 < p \leq  1 $
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for a large class of martingales, and Gundy [[#References|[a5]]] proved the case $  p = 1 $
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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]]]:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b111/b111050/b11105018.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a2)</td></tr></table>
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$$ \tag{a2 }
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c _  \Phi  {\mathsf E} ( \Phi ( S ( X ) ) ) \leq  {\mathsf E} ( \Phi ( X  ^ {*} ) ) \leq  C _  \Phi  {\mathsf E} ( \Phi ( S ( X ) ) ) ,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b111/b111050/b11105019.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b111/b111050/b11105020.png" /> are positive constants depending only on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b111/b111050/b11105021.png" />.
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where $  c _  \Phi  $
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and $  C _  \Phi  $
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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
This article was adapted from an original article by N.L. Bassily (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article