THE STRONG P-VARIATION OF MARTINGALES AND ORTHOGONAL SERIES
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Let 1p< and let x=(xn)n0 be a sequence of scalars. The strong p-variation of x, denoted by Wp(x), is defined as {Mathematical expression} where the supremum runs over all increasing sequences of integers 0=n0n1n2... Let 1p<2 and let M=(Mn)n0 be a martingale in Lp. Our main results are as follows: If {Mathematical expression}, then Wp(M) is finite a.s. and we have {Mathematical expression} for some constant C depending only on p. On the other hand, let ({symbol}n be an arbitrary orthonormal system of functions in L2, consider x=(xn)n0 in l2 and let Sn=0nxi{symbol}i and S=(Sn)n0. We prove that if |xn|p< (1p<2) then Wp(S(t))< for a.e.t and Wp(S)2C(|xn|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 in also discussed. Our proofs use the real interpolation method of Lions-Peetre. They admit extensions in the Banach space valued case, provided suitable assumptions are imposed on the Banach space. 1988 Springer-Verlag.