Central Limit theorem derived from Stochastic Processes

Mathematical intuition behind the (often) Gaussian behavior of nature

It is well known that a plethora of natural stochastic processes often showcase a Gaussian probability distribution. This course aims to explain mathematically why such behavior is displayed.

What you’ll learn

  • the mathematical reason why natural Stochastic Processes often have a Gaussian distribution.
  • Calculation of the probability density from a signal in the time domain (Stochastic Process).
  • concept of ergodicity of Stochastic Processes (why it is important).
  • useful mathematical reasoning while dealing with Stochastic processes.
  • Mathematical derivation of the Central Limit theorem.
  • Mathematical derivation of the distribution of a sinusoid.

Course Content

  • Introduction to Stochastic Processes –> 2 lectures • 13min.
  • Derivation of Important properties of the Fourier Transform and Fourier Series –> 4 lectures • 32min.
  • Ergodicity, Calculation of the probability density from the Stochastic process –> 6 lectures • 1hr 9min.
  • Derivation of the Central Limit theorem from natural Stochastic Processes –> 5 lectures • 1hr 18min.
  • Distribution of a sinusoidal signal –> 4 lectures • 1hr 1min.

Central Limit theorem derived from Stochastic Processes

Requirements

  • Fourier Transform and its Inverse.
  • Fourier series.
  • Calculus (integrals, derivatives).
  • Basic concepts of probability theory (what is a: random variable, probability density, etc).

It is well known that a plethora of natural stochastic processes often showcase a Gaussian probability distribution. This course aims to explain mathematically why such behavior is displayed.

The formulas that are derived in the course, will allow calculating the probability density function from the moments of the stochastic process.

The results presented are related to the well-known Central Limit Theorem (CLT). However, the latter is usually introduced when talking about random variables in Statistics, whereas it is definitely less obvious how the CLT affects Stochastic processes. The aim of this course is therefore to provide motivation as to how this happens mathematically.

This is an advanced course based on the instructor’s PhD thesis, therefore the presentation and the formulas presented are original, despite the literature abounds with material relevant to this subject.

The prerequisites to the course are listed on this page and in the introductory video. It is worth mentioning that the most fundamental properties of the Fourier Transform and Fourier series, which are needed throughout the course’s lectures, are revised in the in the first part of the course.

Note: at the moment (April 2021), I am editing the contents of this course to make my presentation delivery smoother (it will take some time).