Weak type inequalities for conditionally symmetric martingales

作者：Osekowski Adam^*

来源：Statistics & Probability Letters, 2010, 80(23-24): 2009-2013.

DOI：10.1016/j.spl.2010.09.007

• Martingale
• Square function
• Weak type inequality

Let f be a conditionally symmetric martingale and let S(f) be its square function. We prove that
parallel to f parallel to(p,infinity) <= C(p) parallel to S(f)parallel to(p), 1 <= p <= 2,
where
C(p)(p) = 2(1-p/2)pi(p-3/2) Gamma((p+1)/2)/Gamma(p+1) 1 + 1/3(2) + 1/5(2) + 1/7(2) ... /1 - 1/3(p+1) + 1/5(p+1) - 1/7(p+1) + ... .
In addition, the constant C(p) is shown to be the best possible even for the class of dyadic martingales.

• 出版日期2010-12-1