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A quantum invitation to probability theory

Sebastian Schleißinger

TL;DR

The notes develop a cohesive framework that unites complex analysis, classical probability, and quantum probability through an algebraic and analytic viewpoint. They introduce quantum probability spaces, five notions of independence, and transform tools (Cauchy, $F$, $B$, and $R$ transforms) to describe convolutions and limit theorems in noncommutative settings. The text ties Markov and additive processes to Loewner chains within a complex-analytic perspective, illustrating how distributions evolve under monotone, Boolean, and free independence. Collectively, the work blends operator algebras, stochastic processes, and conformal mapping techniques to extend classical probabilistic results to noncommutative frameworks with explicit representations and limit laws.

Abstract

Quantum probability theory and complex analysis for children.

A quantum invitation to probability theory

TL;DR

The notes develop a cohesive framework that unites complex analysis, classical probability, and quantum probability through an algebraic and analytic viewpoint. They introduce quantum probability spaces, five notions of independence, and transform tools (Cauchy, , , and transforms) to describe convolutions and limit theorems in noncommutative settings. The text ties Markov and additive processes to Loewner chains within a complex-analytic perspective, illustrating how distributions evolve under monotone, Boolean, and free independence. Collectively, the work blends operator algebras, stochastic processes, and conformal mapping techniques to extend classical probabilistic results to noncommutative frameworks with explicit representations and limit laws.

Abstract

Quantum probability theory and complex analysis for children.

Paper Structure

This paper contains 72 sections, 105 theorems, 423 equations, 31 figures.

Table of Contents

  1. Introduction
  2. Theory
  3. Classical probability theory
  4. The basic notions
  5. Independence
  6. The characteristic function
  7. Central limit theorem
  8. Conditional expectation
  9. Exercises
  10. A crash course on Markov processes
  11. Markov Processes
  12. Time-homogeneous Markov processes
  13. Space-homogeneous Markov processes
  14. Additive processes
  15. Further reading
  16. ...and 57 more sections

Key Result

Theorem 2.3.2

Let $X$ be a random variable. Then $|\varphi_X(t)|\leq 1$ for all $t\in\mathbb R$, $t\mapsto \varphi_X(t)$ is continuous, and $\varphi_{X}(-t)=\overline{\varphi_X(t)}$ for all $t\in\mathbb R$. Furthermore, for any $n\in\mathbb N$, complex numbers $c_1,...,c_n$ and real numbers $t_1,...,t_n$,

Figures (31)

  • Figure 1: Densities of the Gaussian normal distribution.
  • Figure 2: Histograms for the average heights.
  • Figure 3: Histograms for the largest heights.
  • Figure 4: Four sample paths of a Brownian motion.
  • Figure 5: Sample paths of Poisson processes for $t\in[0,1]$ with $\lambda=2$ (left) and $\lambda=20$ (right).
  • ...and 26 more figures

Theorems & Definitions (335)

  • Definition 2.1.1: Classical probability space
  • Example 2.1.2
  • Example 2.1.3
  • Example 2.1.4
  • Definition 2.1.5: Random variable
  • Definition 2.1.6
  • Example 2.1.7
  • Remark 2.1.8
  • Remark 2.1.9
  • Definition 2.2.1: Covariance
  • ...and 325 more