# Proof of a maximal inequality

I am reading on martingale theory from "Probability: a graduate course" and where the following theorem is proved (in the proof, theorem $$9.1$$ refers to Doob's maximal inequality):

Do you have any idea how he obtained inequality $$(1),$$ especially how he obtained $$\{\max_{1 \leq k \leq n} c_kX_{k}^+>\lambda\}$$ in the integral in inequality $$(1)$$ ?

## 1 Answer

Because of the following two things:

1. $$\{c_k\}$$ is not increasing so the integrand is less or equal than zero.
2. $$Y_k \geq c_k X^+_k$$ so $$\{\max_{0 \leq k \leq n} c_k X^+_k > \lambda\} \subset \{\max_{0 \leq k \leq n} Y_k > \lambda\}$$.
• Do you think there is a typo while passing to inequality $(1)$ ? Instead of $c_j-c_{j-1}$ it should be $c_{j}-c_{j+1}$
– john
Commented Dec 15, 2021 at 21:21
• Yes, it's a typo. Above where they define $Y_k$, it says $c_{j+1}$. The funny thing is that my answer was based on that. I missed the typo!, But it's clearly a typo. Also the summation starts in $j=0,$ so we would have a $c_{-1},$ which doesn't exist.
– UBM
Commented Dec 15, 2021 at 23:30
• In this case, his argument is incorrect
– john
Commented Dec 16, 2021 at 3:15
• what part of his argument is incorrect?
– UBM
Commented Dec 16, 2021 at 11:37
• the passage to inequality $(1)$, especially that points 1. and 2. do not hold in this case (integral is positive)
– john
Commented Dec 16, 2021 at 21:49