Fourier series and Fourier transform provide one of the most important tools for analysis and partial differential equations, with widespread applications to physics in particular and science in general. This is (up to a scalar multiple) a norm-preserving (i.e., isometry), linear transformation on the Hilbert space of square-integrable complex-valued functions. It turns the integral operation of convolution of functions into the elementary algebraic operation of the product of the transformed functions, and that of differentiation of a function into multiplication by its Fourier frequency.

1. 1.

   Extensions of the theory can be found in the following standard references, among others: Rudin (1967), Grenander (1963), Parthasarathy (1967).

2. 2.

   There are several different ways in which Fourier transforms can be parameterized and/or normalized by extra constant factors and/or a different sign in the exponent. The definition given here follows the standard conventions of probability theory.

3. 3.

   That the converse is also true was independently established in Stone, C. J. (1969): On the potential operator for one-dimensional recurrent random walks,Trans. AMS,136 427–445, and Ornstein, D. (1969): Random walks,Trans. AMS,138, 1–60.

4. 4.

   A comprehensive account of errors of normal approximation for the clt in general multidimensions may be found in Bhattacharya, R. and R. Ranga Rao (2010).

5. 5.

   The proof given here follows that given in Feller, W. (1971), vol 2. Feller refers to this particular estimate, attributed to A.C. Berry, as thesmoothing inequality.

6. 6.

   See Shevtsova, I. G. (2010): An Improvement of Convergence Rate Estimates in the Lyapunov Theorem, Doklady Math.82(3), 862–864.

7. 7.

   For a more elaborate treatment of the physics of the Holtsmark distribution in higher dimensions see S. Chandreskhar (1943): Stochastic problems in physics and astronomy,Reviews of Modern Physics, 15(3), reprinted in Wax (1954). The treatment provided here was inspired by Lamperti (1996).

8. 8.

   Surprisingly, recurrence and heavy tails may coexist, see Shepp, L (1964): Recurrent random walks with arbitrarily large steps,Bull. Amer. Math. Soc., v. 70, 540–542; Grey, D.R. (1989): Persistent random walks may have arbitrarily large tails,Adv. Appld. Probab.21, 229–230.

Authors and Affiliations

1. Department of Mathematics, University of Arizona, Tucson, AZ, USA

   Rabi Bhattacharya

2. Department of Mathematics, Oregon State Univeristy, Corvallis, OR, USA

   Edward C. Waymire

Corresponding author

Correspondence toRabi Bhattacharya.

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