Type and Cotype of Banach Spaces

The notion of type of a Banach space already appeared in the last chapters on the law of large numbers and the law of the iterated logarithm. We observed there that, in quite general situations, almost sure properties can be reduced to properties in probability or in L P , 0 ≤ p < ∞. Starting with this chapter, we will now study the possibility of a control in probability, or in the weak topology, of probability distributions of sums of independent random variables.

[1]  Werner Linde,et al.  Probability in Banach Spaces: Stable and Infinitely Divisible Distributions , 1986 .

[2]  Stanisław Kwapień,et al.  Isomorphic characterizations of inner product spaces by orthogonal series with vector valued coefficients , 1972 .

[3]  E. Giné,et al.  Convergence of moments and related functional in the general central limit theorem in banach spaces , 1979 .

[4]  J. Krivine,et al.  Lois stables et espaces $L^p$ , 1967 .

[5]  Wojbor A. Woyczyński,et al.  Stable measures and central limit theorems in spaces of stable type , 1979 .

[6]  J. Hoffmann-jorgensen.  Sums of independent Banach space valued random variables , 1974 .

[7]  G. Pisier,et al.  Séries de variables aléatoires vectorielles indépendantes et propriétés géométriques des espaces de Banach , 1976 .

[8]  G. Pisier.  The volume of convex bodies and Banach space geometry , 1989 .

[9]  Stamatis Cambanis,et al.  Convergence of Quadratic Forms in p-Stable Random Variables and Theta sub p-Radonifying Operators. , 1985 .

[10]  A. Acosta.  INEQUALITIES FOR B-VALUED RANDOM VECTORS WITH APPLICATIONS TO THE STRONG LAW OF LARGE NUMBERS , 1981 .

[11]  D. Aldous.  The Central Limit Theorem for Real and Banach Valued Random Variables , 1981 .

[12]  G. Pisier,et al.  Characterizations of almost surely continuousp-stable random Fourier series and strongly stationary processes , 1984 .

[13]  Y. Gordon.  Some inequalities for Gaussian processes and applications , 1985 .

[14]  Gilles Pisier,et al.  Some results on the continuity of stable processes and the domain of attraction of continuous stable processes , 1984 .

[15]  M. Ledoux.  Sur une inegalite de H.P. Rosenthal et le theoreme limite central dans les espaces de Banach , 1985 .

[16]  N. Tomczak-Jaegermann.  Banach-Mazur distances and finite-dimensional operator ideals , 1989 .

[17]  V. Milman,et al.  Asymptotic Theory Of Finite Dimensional Normed Spaces , 1986 .

[18]  V. Tarieladze,et al.  Gaussian characterizations of certain Banach spaces , 1977 .

[19]  William B. Johnson,et al.  Embeddinglpm intol1n , 1982 .

[20]  G. Pisier.  Factorization of Linear Operators and Geometry of Banach Spaces , 1986 .

[21]  Joel Zinn,et al.  Central limit theorems and weak laws of large numbers in certain banach spaces , 1983 .

[22]  G. Pisier.  Probabilistic methods in the geometry of Banach spaces , 1986 .

[23]  G. Pisier.  Le théorème de la limite centrale et la loi du logarithme itéré dans les espaces de Banach , 1976 .

[24]  S. Kwapień.  On Banach spaces containing $c_{0}$. A supplement to the paper by J.Hoffman-Jørgensen "Sums of independent Banach space valued random variables" , 1974 .

[25]  Y. Gordon.  Gaussian Processes and Almost Spherical Sections of Convex Bodies , 1988 .

[26]  Daniel P. Giesy,et al.  On a convexity condition in normed linear spaces , 1966 .

[27]  J. Krivine,et al.  Sous-espaces de dimension finie des espaces de Banach reticules , 1976 .

[28]  Random Series and Laws of Large Numbers in Some Banach Spaces , 1974 .

[29]  G. Pisier,et al.  The Law of Large Numbers and the Central Limit Theorem in Banach Spaces , 1976 .

[30]  C. Bessaga,et al.  On bases and unconditional convergence of series in Banach spaces , 1958 .

[31]  G. Pisier.  Sur la loi du logarithme itere dans les espaces de Banach , 1976 .

[32]  T. Figiel,et al.  The dimension of almost spherical sections of convex bodies , 1976 .

[33]  C. Rogers,et al.  Absolute and Unconditional Convergence in Normed Linear Spaces. , 1950, Proceedings of the National Academy of Sciences of the United States of America.

[34]  L. Schwartz,et al.  Geometry and probability in Banach spaces , 1981 .

[35]  Jan Rosiński,et al.  On stochastic integral representation of stable processes with sample paths in Banach spaces , 1986 .

[36]  Anatole Beck,et al.  A convexity condition in Banach spaces and the strong law of large numbers , 1962 .