Advanced probability problems typically transition from elementary combinatorics to rigorous measure-theoretic frameworks, including martingales stochastic processes limit theorems Featured Resources with Detailed Solutions
Martingales: A sequence of random variables where the future expectation is the current value, often used in gambling theory. A Collection of Exercises in Advanced Probability Theory
Solution
- Attempt without peeking → Spend at least 30 minutes on a problem before reading the solution.
- Reproduce the solution – Write it in your own words, verifying each ( \epsilon ), ( \delta ), and "almost surely" nuance.
- Identify gaps – If a step uses the Monotone Class Theorem, review that theorem separately.
- Create an error log – Track recurring mistakes (e.g., confusing convergence in probability vs. almost sure).
- Time yourself – For qualifier-style problems, simulate exam conditions.
, the probability that the limit of the average deviates from the mean is zero:
Using the transition matrix, we have:
Problem: Give an example of a sequence of random variables converging in probability but not almost surely.
Solution excerpt: Standard “sliding window” sequence of indicator functions.
Guide: Creating an “Advanced Probability Problems and Solutions” PDF
1) Target audience and scope
- Audience: Graduate students, advanced undergraduates, and self-learners comfortable with measure-theoretic or rigorous probability.
- Prerequisites: Real analysis, measure theory (basic), linear algebra, basic probability (random variables, expectations, distributions).
- Scope: Classical and modern advanced topics—measure-theoretic foundations, convergence theorems, characteristic functions, limit theorems, martingales, Markov chains, large deviations, stochastic processes, and selected applied problems.
