David Williams Probability With Martingales Solutions Best -

rigorous, well-explained, and community-vetted

This book (often called "PWM") is a classic but famously terse. The exercises are non-trivial, and official solutions do not exist. The "best" solutions, therefore, are those that are .

Do the "Two-Pass" Method

4. Supplementing the Text

As you work through Williams, you will notice something magical: after wrestling with the first five chapters using these solutions responsibly, you will need them less and less. By Chapter 12 (martingale convergence theorems), you will start inventing your own proofs that match or exceed the "official" ones. david williams probability with martingales solutions best