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3023 results in Probability theory and stochastic processes

Page 115 of 152

  • Prelude

  • René L. Schilling, Philipps-Universität Marburg, Germany
  • Book:
    Measures, Integrals and Martingales
    Published online:
    05 September 2012
    Print publication:
    10 November 2005, pp viii-x

  • Summary

    The purpose of this book is to give a straightforward and yet elementary introduction to measure and integration theory that is within the grasp of second or third year undergraduates. Indeed, apart from interest in the subject, the only prerequisites for Chapters 1–13 are a course on rigorous ε-δ analysis on the real line and basic notions of linear algebra and calculus in ℝn. The first few chapters form a concise (not to say minimalist) introduction to Lebesgue's approach to measure and integration, based on a 10-week, 30-hour lecture course for Sussex University mathematics undergraduates. Chapters 14–24 are more advanced and contain a selection of results from measure theory, probability theory and analysis. This material can be read linearly but it is also possible to select certain topics; see the dependence chart on page xi. Although more challenging than the first part, the prerequisites stay essentially the same and a reader who has worked through and understood Chapters 1–13 will be well prepared for all that follows. At some points, one or another concept from point-set topology will be (mostly superficially) needed; those readers who are not familiar with the topic can look up the basic results in Appendix B whenever the need arises.

    Each chapter is followed by a section of Problems. They are not just drill exercises but contain variants, excursions from and extensions of the material presented in the text.

  • 24 - Orthonormal systems and their convergence behaviour

  • René L. Schilling, Philipps-Universität Marburg, Germany
  • Book:
    Measures, Integrals and Martingales
    Published online:
    05 September 2012
    Print publication:
    10 November 2005, pp 276-312

  • Summary

    In Chapter 21 we discussed the importance of orthonormal systems (ONSs) in Hilbert spaces. In particular, countable complete ONSs turned out to be bases of separable Hilbert spaces. We have also seen that a countable ONS gives rise to a family of finite-dimensional subspaces and a sequence of orthogonal projections onto these spaces. In the present chapter we are concerned with the following topics:

    • to give concrete examples of (complete) ONSs;

    • to see when the associated canonical projections are conditional expectations;

    • to understand the Lp (p ≠ 2) and a.e. convergence behaviour of series expansions with respect to certain ONSs.

    The latter is, in general, not a trivial matter. Here we will see how we can use the powerful martingale machinery of Chapters 17 and 18 to get Lp (1 ≤ p < ∞) and a.e. convergence.

    Throughout this chapter we will consider the Hilbert space L2(I, B(I), ρλ) where I ⊂ ℝ is a finite or infinite interval of the real line, B(I) = I ∩ B(ℝ) are the Borel sets in I, λ = λ1|I is Lebesgue measure on I and ρ(x) is a density function. We will usually write ρ(x) dx and ∫ … dx instead of ρλ and ∫ … dλ.

Page 115 of 152