Article ID | Journal | Published Year | Pages | File Type |
---|---|---|---|---|
1151723 | Statistics & Probability Letters | 2014 | 7 Pages |
Abstract
Let (hn)nâ¥0 be the Haar system of functions on [0,1]. The paper contains the proof of the estimate â«01|âk=0nεkakhk|2log|âk=0nεkakhk|dsâ¤â«01|âk=0nakhk|2log|e2âk=0nakhk|ds, for n=0,1,2,â¦. Here (an)nâ¥0 is an arbitrary sequence with values in a given Hilbert space H and (εn)nâ¥0 is a sequence of signs. The constant e2 appearing on the right is shown to be the best possible. This result is generalized to the sharp inequality E|gn|2log|gn|â¤E|fn|2log(e2|fn|),n=0,1,2,â¦, where (fn)nâ¥0 is an arbitrary martingale with values in H and (gn)nâ¥0 is its transform by a predictable sequence with values in {â1,1}. As an application, we obtain the two-sided bound for the martingale square function S(f): E|fn|2log(eâ2|fn|)â¤ESn2(f)logSn(f)â¤E|fn|2log(e2|fn|),n=0,1,2,â¦.
Related Topics
Physical Sciences and Engineering
Mathematics
Statistics and Probability
Authors
Adam Osȩkowski,