logo

Probability Theory

In this category, we delve into advanced topics in probability theory that typically require a graduate level understanding, and make use of Measure Theory and Topology. Intuitive probability theory that doesn’t involve these complex theories is categorized under Mathematical Statistics. The difficulty level is indicated by 🔥 marks, where one mark means understanding measure theory is sufficient, two or more indicate the need for more complex measure theory or topology, and marks are also added for proofs or derivations that are exceptionally complex.

MarkDifficulty Level
🔥Difficult
🔥🔥Insanely difficult
🔥🔥🔥Freaking insanely difficult

$$ \begin{array}{lll} \text{Analysts’ Term} && \text{Probabilists’ Term} \\ \hline \text{Measure space } (X, \mathcal{E}, \mu) \text{ such that } \mu (X) = 1 && \text{Probability space } (\Omega, \mathcal{F}, P) \\ \text{Measure } \mu : \mathcal{E} \to \mathbb{R} \text{ such that } \mu (X) = 1 && \text{Probability } P : \mathcal{F} \to \mathbb{R} \\ (\sigma\text{-)algebra $\mathcal{E}$ on $X$} && (\sigma\text{-)field $\mathcal{F}$ on $\Omega$} \\ \text{Mesurable set } E \in \mathcal{E} && \text{Event } E \in \mathcal{F} \\ \text{Measurable real-valued function } f : X \to \mathbb{R} && \text{Random variable } X : \Omega \to \mathbb{R} \\ \text{Integral of } f, {\displaystyle \int f d\mu} && \text{Expextation of } f, E(X) \\ f \text{ is } L^{p} && X \text{ has finite $p$th moment} \\ \text{Almost everywhere, a.e.} && \text{Almost surely, a.s.} \end{array} $$

Measure-theoretic Probability Theory

Rigorous Definitions

Conditional Probability

Stochastic Processes

Stochastic Process Theory

Markov Chains

Brownian Motion

Martingales

Donsker’s Theorem

Stochastic Information Theory

Entropy

References

  • Applebaum. (2008). Probability and Information(2nd Edition)
  • Capinski. (1999). Measure, Integral and Probability
  • Kimmel, Axelrod. (2006). Branching Processes in Biology

All posts