Summary
Let 1≦p<∞ and letx=(x n)n≧0 be a sequence of scalars. The strongp-variation ofx, denoted byW p (x), is defined as
where the supremum runs over all increasing sequences of integers 0=n 0 ≦n 1 ≦n 2 ≦...
Let 1≦p<2 and letM=(M n ) n≧0 be a martingale inL p . Our main results are as follows: If\(\Sigma \mathbb{E}|M_n - M_{n - 1} |^p< \infty \), thenW p (M) is finite a.s. and we have
for some constantC depending only onp. On the other hand, let (ϕ n be an arbitrary orthonormal system of functions inL 2, considerx=(x n ) n≧0 inl 2 and letS n =Σ n0 x i ϕ i andS=(S n ) n≧0. We prove that ifΣ|x n |p<∞ (1≦p<2) thenW p (S(t))<∞ for a.e.t and ∥W p (S)∥2≦C(Σ|x n |p)1/p for some constantC. Each of these results is an extension of a result proved by Bretagnolle for sums of independent mean zero r.v.'s. The casep>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.
Article PDF
Similar content being viewed by others
References
Bergh, J., Löfström, J.: Interpolation spaces. An introduction. Berlin Heidelberg New York: Springer 1976
Bergh, J., Peetre, J.: On the spacesV p (0<p≦∞). Boll. Unione Mat. Ital. (4)10, 632–648 (1974)
Blumenthal, R., Getoor, R.: Some theorems on stable processes. Trans. Am. Math. Soc.95, 263–273 (1960)
Blumenthal, R., Getoor, R.: Sample functions of stochastic processes with stationary independent increments. J. Math. Mech.10, 493–516 (1961)
Bretagnolle, J.:p-variation des fonctions aléatoires. Séminaire de Probabilités VI. Lect. Notes Math.258, 51–71 (1972)
Bruneau, M.: Sur lap-variation d'une surmartingale continue. In: Delacherie, C., Meyer, P.A., Weil, M. (eds.) Séminaire de Probabilités XIII. Strasbourg 1977/78. (Lect. Notes Math. vol. 721 pp. 226–232) Berlin Heidelberg New York: Springer 1979
Burkholder, D.: Distribution function inequalities for martingales. Ann. Probab.1, 19–42 (1973)
Burkholder, D., Davis, B., Gundy, R.: Integral inequalities for convex functions of operators on martingales. Proc. 6th. Berkeley Sympos. Math. Statist. Probab.2, 223–240 (1972)
Garsia, A.: Martingales Inequalities. Seminar notes on recent progress. Reading, Mass.: Benjamin 1973
Hoffmann-Jørgensen, J., Pisier, G.: The law of large numbers and the central limit theorem in Banach spaces. Ann. Probab.4, 587–599 (1976)
Jain, N., Monrad, D.: Gaussian measures inB p . Ann. Probab.11, 46–57 (1983)
Kwapień, S.: Isomorphic characterizations of inner product spaces by orthogonal series with vector valued coefficients. Stud. Math.44, 583–595 (1972)
Lenglart, E., Lépingle, D., Pratelli, M.: Présentation unifiée de certaines inégalités de la des martingales. In: Azéma, I., Yor, M. (eds.) Séminaire de Probabilités n0 XIV. (Lect. Notes Math., vol. 784, pp. 26–48) Berlin Heidelberg New York: Springer 1980
Lépingle, D.: La variation d'ordrep des semi-martingales. Z. Wahrscheinlichkeitstheor. Verw. Geb.36, 295–316 (1976)
Lindenstrauss, J., Tzafriri, L.: Classical Banach spaces II. Berlin Heidelberg New York: Springer 1979
Maurey, B., Pisier, G.: Séries de variables aléatoires indépendantes et propriétés géométriques des espaces de Banach. Stud. Math.58, 45–90 (1976)
Millar, P.: Path behavior of processes with stationary independent increments. Z. Wahrscheinlichkeitstheor. Verw. Geb.17, 53–73 (1971)
Monroe, I.: On the γ-variation of processes with stationary independent increments. Ann. Math. Stat.43, 1213–1220 (1972)
Neveu, N.: Martingales à temps discret. Paris: masson 1972
Pisier, G.: Probabilistic methods in the Geometry of Banach spaces. In: Letta, G., Pratell, M. (eds.) (Lect. Notes Math., vol. 1206, pp. 167–241) Berlin Heidelberg New York: Springer 1986
Pisier, G.: Martingales with values in uniformly convex spaces. Isr. J. Math.20, 326–350 (1975)
Pisier, G., Xu, Q.: Random series in the real interpolation spaces between the spacesv p . Israel Functional Analysis Seminar. GAFA. (Lect. Notes Math., vol. 1267, pp. 185–209) Berlin Heidelberg New York: Springer 1987
Stricker, C.: Sur lap-variation des surmartingales. In: Delacherie, C., Meyer, P.A., Weil, M. (eds.) Séminaire de Probabilités XIII. Strasbourg 1977/78. (Lect. Notes Math., vol. 721, pp. 233–239) Berlin Heidelberg New York: Springer 1979
Tomczak-Jaegermann, N.: Computing 2-summing norms with few vectors. Ark. Mat.17, 273–277 (1979)
Author information
Authors and Affiliations
Additional information
Partially supported by N.S.F. Grant No. DMS-8500764
Rights and permissions
About this article
Cite this article
Pisier, G., Xu, Q. The strongp-variation of martingales and orthogonal series. Probab. Th. Rel. Fields 77, 497–514 (1988). https://doi.org/10.1007/BF00959613
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF00959613