Skip to main content
SpringerLink
Log in

A survey on differentiability of six operators in relation to probability and statistics

  • Chapter
  • First Online:
Differentiability of Six Operators on Nonsmooth Functions and p-Variation

Part of the book series: Lecture Notes in Mathematics ((LNM,volume 1703))

  • 541 Accesses

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 52.00
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  • Andersen, P. K., Borgan, Ø., Gill, R. D. and Keiding, N. (1993). Statistical Models based on Counting Processes. Springer-Verlag, New York.

    Book  MATH  Google Scholar 

  • Appell, J. (1983). Upper estimates for superposition operators and some applications. Ann. Acad. Sci. Fenn. Ser. A I. Math. 8, 149–159.

    Article  MathSciNet  MATH  Google Scholar 

  • Appell, J. (1988). The superposition operator in function spaces—A survey. Expositiones Math. 6, 209–270.

    MathSciNet  MATH  Google Scholar 

  • Appell, J. and Zabrejko, P. P. (1990). Nonlinear superposition operators. Cambridge University Press.

    Google Scholar 

  • Averbukh, V. I. and Smolyanov, O. G. (1967). The theory of differentiation in linear topological spaces. (Russian) Math. Surveys 22 no. 6, 201–258=Uspekhi Mat. Nauk 22 no. 6, 201–260 (1967).

    Article  MATH  Google Scholar 

  • Averbukh, V. I. and Smolyanov, O. G. (1968). The various definitions of the derivative in linear topological spaces. (Russian) Math. Surveys 23 no. 4, 67–113= Uspekhi Mat. Nauk 23 no. 4, 67–116.

    Article  MATH  Google Scholar 

  • Beirlant, J., and Deheuvels, P. (1990). On the approximation of P-P and Q-Q plot processes by Brownian bridges. Statist. Probab. Letters 9, 241–251.

    Article  MathSciNet  MATH  Google Scholar 

  • Billingsley, P. (1968). Convergence of Probability Measures. Wiley, New York.

    MATH  Google Scholar 

  • Bingham, N. H. (1975). Fluctuation theory in continuous time. Adv. Appl. Prob. 7, 705–766.

    Article  MathSciNet  MATH  Google Scholar 

  • Bingham, N. H., Goldie, C. M. and Teugels, J. L. (1987). Regular variation. Encyclopedia of Mathematics and its Applications, 27. Cambridge University Press, Cambridge-New York.

    Book  MATH  Google Scholar 

  • Bretagnolle, J. and Massart, P. (1989). Hungarian constructions from the nonasymptotic viewpoint. Ann. Probab. 17, 239–256.

    Article  MathSciNet  MATH  Google Scholar 

  • Brokate, M. and Colonius, F. (1990). Linearizing equations with state-dependent delays. Appl. Math. Optimiz. 21, 45–52.

    Article  MathSciNet  MATH  Google Scholar 

  • Bruneau, M. (1974). Variation totale d'une fonction. Lect. Notes in Math. (Springer-Verlag) 413.

    Google Scholar 

  • Ciesielski, Z., Kerkyacharian, G. and Roynette, B. (1993). Quelques espaces fonctionnels associés à des processus gaussiens. Studia Math., 107, 171–204.

    MathSciNet  Google Scholar 

  • Corson, H. H. (1961). The weak topology of a Banach space. Trans. Amer. Math. Soc. 101, 1–15.

    Article  MathSciNet  MATH  Google Scholar 

  • Csörgő, M. (1983). Quantile Processes with Statistical Applications. SIAM, Philadelphia.

    Book  Google Scholar 

  • Cutland, N. J., Kopp, P. E. and Willinger, W. (1995). Stock price returns and the Joseph effect: a fractional version of the Black-Scholes model. Seminar on Stochastic Analysis, Random Fields and Applications (Ascona, 1993), Progr. Probab., 36, 327–351. Birkhäuser, Basel.

    MATH  Google Scholar 

  • Dai, W. and Heyde, C. C. (1996). Itô's formula with respect to fractional Brownian motion and its application. J. Appl. Math. Stochastic Anal. 9, 439–448.

    Article  MathSciNet  MATH  Google Scholar 

  • Dieudonné, J. (1960). Foundations of Modern Analysis. Academic Press, New York; Fondements de l'analyse moderne 1. Gauthier-Villars, Paris, 1963.

    MATH  Google Scholar 

  • Dudley, R. M. (1985). An extended Wichura theorem, definitions of Donsker class, and weighted empirical distributions. In Probability in Banach Spaces V (A. Beck, R. Dudley, M. Hahn, J. Kuelbs and M. Marcus, eds.). Lect. Notes in Math. (Springer-Verlag) 1153, 141–178.

    Google Scholar 

  • Dudley, R. M. (1992). Fréchet differentiability, p-variation and uniform Donsker classes. Ann. Probab. 20, 1968–1982.

    Article  MathSciNet  MATH  Google Scholar 

  • Dudley, R. M. (1993). Real Analysis and Probability. Second printing, corrected. Chapman and Hall, New York.

    MATH  Google Scholar 

  • Dudley, R. M. (1994). The order of the remainder in derivatives of composition and inverse operators for p-variation norms. Ann. Statist. 22, 1–20.

    Article  MathSciNet  MATH  Google Scholar 

  • Dudley, R. M. (1997). Empirical processes and p-variation. In Festschrift for Lucien Le Cam (D. Pollard, E. Torgersen and G. L. Yang, eds.) 219–233. Springer-Verlag, New York.

    Chapter  Google Scholar 

  • Ebin, D. G. and Marsden, J. (1970). Groups of diffeomorphisms and the motion of an incompressible fluid. Ann. Math. 92, 102–163.

    Article  MathSciNet  MATH  Google Scholar 

  • Esty, W., Gillette, R., Hamilton, M. and Taylor, D. (1985). Asymptotic distribution theory of statistical functionals: the compact derivative approach for robust estimators. Ann. Inst. Statist. Math. 37, 109–129.

    Article  MathSciNet  MATH  Google Scholar 

  • Fernholz, L. T. (1983). von Mises calculus for statistical functionals. Lect. Notes in Statist. (Springer-Verlag) 19.

    Google Scholar 

  • de Finetti, B. and Jacob, M. (1935) (We founel these references from secondary sources but have not seen them im the original). Sull'integrale di Stieltjes-Riemann. Giorn. Istituto Ital. Attuari 6, 303–319.

    MATH  Google Scholar 

  • Fréchet, M. (1937). Sur la notion de différentielle. J. Math. Pures Appl. 16, 233–250.

    MATH  Google Scholar 

  • Freedman, M. A. (1983). Operators of p-variation and the evolution representation problem. Trans. Amer. Math. Soc. 279, 95–112.

    MathSciNet  MATH  Google Scholar 

  • Garay, B. M. (1996). Hyperbolic structures in ODE's and their discretization. In Nonlinear Analysis and Boundary Value Problems for ODEs (Proc. CISM, Udine, 1995; F. Zanolin, ed.). CISM Courses and Lectures 371, 149–174. Springer-Verlag, Vienna.

    Google Scholar 

  • Gateaux, R. (1913). Sur les fonctionelles continues et les fonctionelles analytiques. C. R. Acad. Sci. Paris 157, 325–327.

    MATH  Google Scholar 

  • Gateaux, R. (1919). Fonctions d'une infinité de variables indépendantes. Bull. Soc. Math. France 47, 70–96.

    MathSciNet  MATH  Google Scholar 

  • Gateaux, R. (1922). Sur diverses questions de calcul fonctionnel. Bull. Soc. Math. France 50, 1–37.

    MathSciNet  MATH  Google Scholar 

  • Gehring, F. W. (1954). A study of α-variation. I. Trans. Amer. Math. Soc. 76, 420–443.

    MathSciNet  MATH  Google Scholar 

  • Gill, R. D. (1989). Non-and semi-parametric maximum likelihood estimators and the von Mises method (Part 1). Scand. J. Statist. 16, 97–128.

    MathSciNet  MATH  Google Scholar 

  • Gill, R. D. (1994). Lectures on Survival Analysis. In Ecole d'été de Probabilités de Saint-Flour (P. Bernard, ed.). Lect. Notes in Math. (Springer-Verlag) 1581, 115–241.

    Google Scholar 

  • Gill, R. D., and Johansen, S. (1990). A survey of product-integration with a view toward application in survival analysis. Ann. Statist. 18, 1501–1555.

    Article  MathSciNet  MATH  Google Scholar 

  • Gill, R. D., van der Laan, M. J., and Wellner, J. A. (1995). Inefficient estimators of the bivariate survival function for three models. Ann. Inst. Henri Poincaré 31, 545–597.

    MathSciNet  MATH  Google Scholar 

  • Gray, A. (1975). Differentiation of composites with respect to a parameter. J. Austral. Math. Soc. (Ser. A) 19, 121–128.

    Article  MathSciNet  MATH  Google Scholar 

  • Hardy, G. H. (1916). Weierstrass's non-differentiable function. Trans. Amer. Math. Soc. 17, 301–325. Repr. in Collected papers of G. H. Hardy, Vol. IV, Oxford University Press, London.

    MathSciNet  MATH  Google Scholar 

  • Hardy, G. H., Littlewood, J. E. and Pólya, G. (1959). Inequalities. Cambridge University Press, Cambridge.

    MATH  Google Scholar 

  • Hartung, F. and Turi, J. (1997). On differentiability of solutions with respect to parameters in state-dependent delay equations. J. Differential Equations 135, 192–237.

    Article  MathSciNet  MATH  Google Scholar 

  • Henstock, R. (1961). Definitions of Riemann type of the variational integrals. Proc. London Math. Soc. (3) 11, 402–418.

    Article  MathSciNet  MATH  Google Scholar 

  • Henstock, R. (1963). Theory of integration. Butterworths, London.

    MATH  Google Scholar 

  • Henstock, R. (1991). The general theory of integration. Oxford University Press, New York.

    MATH  Google Scholar 

  • Hewitt, E. and Stromberg, K. (1969). Real and Abstract Analysis. Springer-Verlag, New York.

    MATH  Google Scholar 

  • Hewitt, E. (1960). Integration by parts for Stieltjes integrals. Amer. Math. Monthly 67, 419–423.

    Article  MathSciNet  MATH  Google Scholar 

  • Hildebrandt, T. H. (1938). Definitions of Stieltjes integrals of the Riemann type. Amer. Math. Monthly 45, 265–278.

    Article  MathSciNet  MATH  Google Scholar 

  • Hildebrandt, T. H. (1963). Introduction to the theory of integration. Academic Press, New York.

    MATH  Google Scholar 

  • Hildebrandt, T. H. (1966). Compactness in the space of quasi-continuous functions. Amer. Math. Monthly 73, Part 2, 144–145.

    Article  MathSciNet  MATH  Google Scholar 

  • Huang, Yen-Chin (1994). Empirical distribution function statistics, speed of convergence, and p-variation. Ph. D. thesis, Massachusetts Institute of Technology.

    Google Scholar 

  • Huang, Yen-Chin (1997). Speed of convergence of classical empirical processes in p-variation norm. Preprint.

    Google Scholar 

  • Itô, K. and McKean, H. P. (1974). Diffusion Processes and their Sample Paths. Springer-Verlag, Berlin.

    MATH  Google Scholar 

  • Kiefer, J. (1970). Deviations between the sample quantile process and the sample df. In Nonparametric Techniques in Statistical Inference (M. L. Puri, ed.) 299–319. Cambridge University Press.

    Google Scholar 

  • Kisliakov, S. V. (1984). A remark on the space of functions of bounded p-variation. Math. Nachr. 119, 137–140.

    Article  MathSciNet  Google Scholar 

  • Kolmogorov, A. N. (1933). Sulla determinazione empirica di una legge di distribuzione. Giorn. Istituto Ital. Attuari 4, 83–91.

    MATH  Google Scholar 

  • Kolmogorov, A. N. (1941). Confidence limits for an unknown distribution function. Ann. Math. Statist. 12, 461–463.

    Article  MathSciNet  Google Scholar 

  • Komlós, J., Major, P., and Tusnády, G. (1975). An approximation of partial sums of independent rv's and the sample df. Z. Wahrsch. verw. Gebiete 32, 111–231.

    Article  MathSciNet  MATH  Google Scholar 

  • Krabbe, G. L. (1961a). Integration with respect to operator-valued functions. Bull. Amer. Math. Soc. 67, 214–218.

    Article  MathSciNet  MATH  Google Scholar 

  • Krabbe, G. L. (1961b). Integration with respect to operator-valued functions. Acta Sci. Math. (Szeged) 22, 301–319.

    MathSciNet  MATH  Google Scholar 

  • Krasnosel'skiî, M. A., Zabreîko, P. P., Pustyl'nik, E. I. and Sobolevskiî, P. E. (1966). Integral operators in spaces of summable functions. (Russian) Nauka, Moscow; transl. by T. Ando, Noordhoff, Leyden, 1976.

    MATH  Google Scholar 

  • Kurzweil, J. (1957). Generalized ordinary differential equations and continuous dependence on a parameter. (Russian) Czechoslovak Math. J. 7, 568–583.

    Article  MathSciNet  MATH  Google Scholar 

  • Kurzweil, J. (1958). On integration by parts. Czechoslovak Math. J. 8, 356–359.

    MathSciNet  MATH  Google Scholar 

  • Kurzweil, J. (1959). Addition to my paper “Generalized ordinary differential equations and continuous dependence on a parameter”. Czechoslovak Math. J. 9, 564–573.

    MathSciNet  MATH  Google Scholar 

  • Lehmann, E. L. (1986). Testing Statistical Hypotheses, 2d ed. Wiley, New York; Repr. 1997.

    Book  MATH  Google Scholar 

  • Leśniewicz, R. and Orlicz, W. (1973). On generalized variations (II). Studia Math. 45, 71–109.

    MathSciNet  MATH  Google Scholar 

  • Lévy, P. (1922). Leçons d'analyse fonctionelle. Gauthier-Villars, Paris.

    MATH  Google Scholar 

  • Lin, S. J. (1995). Stochastic analysis of fractional Brownian motion. Stochastics Stochastics Rep. 55, 121–140.

    Article  MathSciNet  MATH  Google Scholar 

  • Lindenstrauss, J. and Tzafriri, L. (1971). On the complemented subspaces problem. Israel J. Math. 9, 263–269.

    Article  MathSciNet  MATH  Google Scholar 

  • Love, E. R. (1993). The refinement-Ross-Riemann-Stieltjes (R3S) integral. In: Analysis, geometry and groups: a Riemann legacy volume, Eds. H. M. Srivastava, Th. M. Rassias, Part I. Hadronic Press, Palm Harbor, FL, pp. 289–312.

    Google Scholar 

  • Love, E. R. and Young, L. C. (1938). On fractional integration by parts. Proc. London Math. Soc. (Ser. 2) 44, 1–35.

    Article  MathSciNet  MATH  Google Scholar 

  • Lyons, T. (1994). Differential equation driven by rough signals (I): An extension of an inequality of L. C. Young. Math. Res. Lett. 1, 451–464.

    Article  MathSciNet  MATH  Google Scholar 

  • Marcinkiewicz, J. (1934). On a class of functions and their Fourier series. C. R. Soc. Sci. Varsovie 26, 71–77. Repr. in J. Marcinkiewicz, Collected Papers, (A. Zygmund, ed.) 36–41. Państwowe Wydawnictwo Naukowe, Warszawa, 1964.

    MATH  Google Scholar 

  • McShane, E. J. (1969). A Riemann-type integral that includes Lebesgues-Stieltjes, Bochner and stochastic integrals. Memoirs AMS, 88; Providence, Rhode Island.

    Google Scholar 

  • Milnor, J. (1984). Remarks on infinite-dimensional Lie groups. In Relativité, groupes et topologie II, (B. S. DeWitt and R. Stora, eds.) 1007–1057.

    Google Scholar 

  • Norvaiša, R. (1997a). The central limit theorem for L-statistics. Studia Sci. Math. Hung. 33, 209–238.

    MathSciNet  MATH  Google Scholar 

  • Norvaiša, R. (1997b). Chain rule and p-variation. Preprint.

    Google Scholar 

  • Pfeffer, W. F. (1993). The Riemann approach to integration. Local geometric theory. Cambridge University Press, New York.

    MATH  Google Scholar 

  • Phillips, E. R. (1971). An introduction to analysis and integration theory. Intext Educational Publ., Scranton.

    MATH  Google Scholar 

  • Qian, Jinghua (1998). The p-variation of partial sum processes and the empirical process. Ann. Probab. 26, 1370–1383.

    Article  MathSciNet  MATH  Google Scholar 

  • Reeds, J. A. III (1976): On the definition of von Mises functionals. Ph. D. thesis, Statistics, Harvard University.

    Google Scholar 

  • Rosenthal, P. (1995). Review of Shapiro (1993) and Singh and Manhas (1993). Bull. Amer. Math. Soc. 32, 150–153.

    Article  MathSciNet  Google Scholar 

  • Saks, S. (1937). Theory of the Integral. 2d. ed., transl. by L. C. Young. Hafner, New York. Repr. by Dover, New York 1964.

    MATH  Google Scholar 

  • Schwabik, Š. (1973). On the relation between Young's and Kurzweil's concept of Stieltjes integral. Časopis Pěst. Mat. 98, 237–251.

    MathSciNet  MATH  Google Scholar 

  • Sebastião e Silva, J. (1956). Le calcul différentiel et intégral dans les espaces localement convexes, réels ou complexes I, II. Rend. Accad. Lincei Sci. Fis. Mat. Nat. (Ser. 8) 20, 743–750, 21, 40–46 (1956).

    MATH  Google Scholar 

  • Shapiro, J. H. (1993). Composition Operators and Classical Function Theory. Springer-Verlag, New York.

    Book  MATH  Google Scholar 

  • Singh, R. K. and Manhas, J. S. (1993). Composition Operators on Function Spaces. North-Holland, Amsterdam.

    MATH  Google Scholar 

  • Skorohod, A. V. (1957). Limit theorems for stochastic processes with independent increments. Theory Prob. Appl. 2, 138–171.

    Article  MathSciNet  Google Scholar 

  • Sova, M. (1966). Conditions of differentiability in linear topological spaces. (Russian) Czech. Math. J. 16, 339–362.

    MathSciNet  MATH  Google Scholar 

  • Stevenson, J. O. (1974). Holomorphy of composition. Bull. Amer. Math. Soc. 80, 300–303.

    Article  MathSciNet  MATH  Google Scholar 

  • Stevenson, J. O. (1977). Holomorphy of composition. In Infinite Dimensional Holomorphy and Applications, (M. C. Matos, ed.) 397–424. North-Holland, Amsterdam.

    Chapter  Google Scholar 

  • Tvrdý, M. (1989). Regulated functions and the Perron-Stieltjes integral. Časopis Pěst. Mat. 114, 187–209.

    MathSciNet  MATH  Google Scholar 

  • van der Vaart, A. (1991). Efficiency and Hadamard differentiability. Scand. J. Statist. 18, 63–75.

    MathSciNet  MATH  Google Scholar 

  • van der Vaart, A. W. and Wellner, J. A. (1996). Weak Convergence and Empirical Processes, with Applications to Statistics. Springer-Verlag, New York.

    Book  MATH  Google Scholar 

  • Vainberg, M. M. (1964). Variational methods for the study of nonlinear operators. With a chapter on Newton's method by L. V. Kantorovich and G. P. Akilov. Translated and supplemented by A. Feinstein, Holden-Day, San Francisco, 1964.

    MATH  Google Scholar 

  • Wang Sheng-Wang (1963). Differentiability of the Nemyckii operator. (Russian) Doklady Akad. Nauk SSSR 150, 1198–1201; transl. in: Sov. Math. Doklady 4, 834–837.

    MathSciNet  Google Scholar 

  • Ward, A. J. (1936). The Perron-Stieltjes integral. Math. Z. 41, 578–604.

    Article  MathSciNet  MATH  Google Scholar 

  • Whitt, W. (1980). Some useful functions for functional limit theorems. Mathematics of Operations Research 5, 67–85.

    Article  MathSciNet  MATH  Google Scholar 

  • Wiener, N. (1924). The quadratic variation of a function and its Fourier coefficients. J. Math. and Phys. (MIT) 3, 72–94.

    Article  MATH  Google Scholar 

  • Young, L. C. (1936). An inequality of the Hölder type, connected with Stieltjes integration. Acta Math. (Sweden) 67, 251–282.

    Article  MATH  Google Scholar 

  • Young, L. C. (1937). Inequalities connected with bounded p-th power variation in the Wiener sense and with integrated Lipschitz conditions. Proc. London Math. Soc. (Ser. 2) 43, 449–467.

    MATH  Google Scholar 

  • Young, L. C. (1938a). Inequalities connected with binary derivation, integrated Lipschitz conditions and Fourier series. Math. Z. 43, 255–270.

    Article  MathSciNet  MATH  Google Scholar 

  • Young, L. C. (1938b). General inequalities for Stieltjes integrals and the convergence of Fourier series. Math. Ann. 115, 581–612.

    Article  MathSciNet  MATH  Google Scholar 

  • Young, L. C. (1970). Some new stochastic integrals and Stieltjes integrals. I. Analogues of Hardy-Littlewood classes. In Advances in Probability and Related Topics, Vol. II, (P. Ney, ed.) 161–240. Dekker, New York.

    Google Scholar 

  • Young, R. C. (1929). On Riemann integration with respect to a continuous increment. Math. Z. 29, 217–233.

    Article  MathSciNet  Google Scholar 

  • Young, W. H. (1914). On integration with respect to a function of bounded variation. Proc. London Math. Soc. (Ser. 2) 13, 109–150.

    Article  MATH  Google Scholar 

  • Zähle, M. (1998). Integration with respect to fractal functions and stochastic calculus. I. Probab. Theory Related Fields 111, 333–374.

    Article  MathSciNet  MATH  Google Scholar 

  • Zygmund, A. (1959). Trigonometric Series, Vol. I. 2d. ed. Cambridge University Press, Cambridge. (*) We found these references from secondary sources but have not seen them in the original.

    MATH  Google Scholar 

Download references

Authors
  1. R. M. Dudley
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. R. Norvaiša
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

Copyright information

© 1999 Springer-Verlag

About this chapter

Cite this chapter

Dudley, R.M., Norvaiša, R. (1999). A survey on differentiability of six operators in relation to probability and statistics. In: Differentiability of Six Operators on Nonsmooth Functions and p-Variation. Lecture Notes in Mathematics, vol 1703. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0100745

Download citation

  • DOI: https://doi.org/10.1007/BFb0100745

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-65975-4

  • Online ISBN: 978-3-540-48814-9

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation