12.1 Complex random numbers

A complex random number is a complex-valued function on a probability space which is measurable with respect to the Borel sigma-algebra of the complex plane.

12.2 Fourier transform of a probability measure

The Fourier transform of a probability measure is defined as a parameterised expectation of a bounded complex-valued random variable, indexed by a frequency parameter. 

12.3 Fourier transforms of common distributions

Fourier transforms of exponential, Poisson, and normal distributions.

12.4 Fourier transforms are bounded and continuous

The Fourier transform of every probability measure, no matter how fat-tailed or otherwise ill-behaved, is a bounded continuous function.

12.5 Lévy's inversion formula

Any probability measure is uniquely determined by its Fourier transform.

12.6 Convergence of probability measures

The weak convergence of probability measures means the convergence of integrals of bounded continuous functions. Lévy's continuity theorem tells that this is equivalent to the pointwise convergence of the associated Fourier transforms.

12.7 Central limit theorem

The law of a centred and properly normalised sum of mutually independent and identically distributed square-integrable random numbers is approximated by a normal distribution, with respect to weak convergence.



Alternative reading material

  • [Jacod & Protter, Chapters 13, 14, 18, 19, 21]
  • [Williams, Chapters 16, 17, 18]

    • Last modified: Wednesday, 17 February 2021, 2:11 PM