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.
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
Keywords
- Banach Space
- Stochastic Process
- Probability Theory
- Mathematical Biology
- Interpolation Method