Some inequalities for p-variations of martingales

Fan, Mingzhi

doi:10.1016/j.spa.2003.09.010

Cited by 1 publication

(2 citation statements)

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“…We establish them by comdininig the techniques from the theory of function spaces and the sharp inequalities in probability. It should be said that the results in [1,5,6,10,11,12,16,20,24] do not just give some sharpened inequalities in probability, but they also provide substantial contributions on extending those probability inequalities to exponential Orlicz spaces. This method was also used in [2,17].…”

Section: Introductionmentioning

confidence: 99%

“…To be able to apply the extrapolation property, we need a precise estimate for the best constants involved in the Rosenthal inequalities and the Marcinkiewicz-Zygmund inequalities on Lebesgue spaces. There were a huge amount of efforts paid for estimating the best constants in the Rosenthal inequalities, the Marcinkiewicz-Zygmund inequalities and some other inequalities in probability on Lebesgue spaces, see [1,5,6,10,11,12,16,20,24].…”

Section: Introductionmentioning

confidence: 99%

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Exponential Rosenthal and Marcinkiewicz - Zygmund inequalities

2020

Ufimsk. Mat. Zh.

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We extend the Rosenthal inequalities and the Marcinkiewicz-Zygmund inequalities to some exponential Orlicz spaces.The Rosenthal inequalities and the Marcinkiewicz-Zygmund inequalities are fundamental estimates on the moment of random variables on Lebesgue spaces. The proofs of the Rosenthal inequalities and the Marcinkiewicz-Zygmund inequalities on the exponential Orlicz spaces rely on two results from theory of function spaces and probability theory. The first one is an extrapolation property of the exponential Orlicz spaces. This property guarantees that the norms of some exponential Orlicz spaces can be obtained by taking the supremum over the weighted norms of Lebesgue spaces. The second one is the sharp estimates for the constants involved in the Rosenthal inequalities and the Marcinkiewicz-Zygmund inequalities on Lebesgue spaces. Our results are applications of the extrapolation property of the exponential Orlicz spaces and the sharp estimates for the constants involved in the Rosenthal inequalities and the Marcinkiewicz-Zygmund inequalities on Lebesgue spaces. In addition, the sharp estimates for the constants involved in the Rosenthal inequalities and the Marcinkiewicz-Zygmund inequalities on Lebesgue spaces provide not only some sharpened inequalities in probability, but also yield some substantial contributions on extending those probability inequalities to the exponential Orlicz spaces.

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