The strong p-variation of martingales and orthogonal series
Let \(1\leq p<\infty\) and let \(x=(x_ n)_{n\geq 0}\) be a sequence of scalars. The strong p-variation of x, denoted by \(W_ p(x)\), is defined as \[ W_ p(x)=\sup \{(| x_ 0| ^ p+\sum ^{\infty}_{k=1}| x_{n_ k}-x_{n_{k-1}}| ^ p)^{1/p}\} \] where the supremum runs over all increasing sequences of integers \(0=n_ 0\leq n_ 1\leq n_ 2\leq... \). Let \(1\leq p<2\) and let \(M=(M_ n)_{n\geq 0}\) be a martingale in \(L_ p\). Our main results are as follows: If \(\sum {\mathbb{E}}| M_ n-M_{n-1}| ^ p<\infty\), then \(W_ p(M)\) is finite a.s. and we have \[ {\mathbb{E}} W_ p(M)^ p\leq C({\mathbb{E}}| M_ 0| ^ p+\sum _{n\geq 1}{\mathbb{E}}| M_ n- M_{n-1}| ^ p) \] for some constant C depending only on p. On the other hand, let \((\phi _ n)\) be an arbitrary orthonormal system of functions in \(L_ 2\), consider \(x=(x_ n)_{n\geq 0}\) in \(\ell _ 2\) and let \(S_ n=\sum ^{n}_{0}x_ i\phi _ i\) and \(S=(S_ n)_{n\geq 0}\). We prove that if \(\sum | x_ n| ^ p<\infty\) (1\(\leq p<2)\) then \(W_ p(S(t))<\infty\) for a.e. t and \(\| W_ p(S)\| _ 2\leq C(\sum | x_ n| ^ p)^{1/p}\) for some constant C. Each of these results is an extension of a result proved by Bretagnolle for sums of independent mean zero r.v.'s. The case \(p>2\) is also discussed. Our proofs use the real interpolation method of Lions-Peetre. They admit extensions to the Banach space valued case, provided suitable assumptions are imposed on the Banach space.
scientific article; zbMATH DE number 4028490
scientific article