Prove Submartingale Inequality and Implication that Involve Weak $L^1$ Inequality

Question

Let $(X_n: n \in \mathbb{N})$ be a non-negative $(F_n)$-submartingale.

a) Prove that $\lVert X^* \rVert \leq \sup_n 2 E(X_n \log^+X_n)+2$, where $\log^+ x =\log x \vee0, \log(0)= -\infty$.

b)Let {$Y_i$} be iid random variables and let $S_n=\sum _{i=1}^n Y_i$. Prove that $E|Y_i|\log^+|Y_1|)< \infty$ implies $E(\sup_n \frac{S_n}{n})<\infty$.

Hint: we can use the modified weak $L^1$ inequality, $P(X_n^* \geq 2 \lambda) \leq \frac{1}{\lambda}\int_{X_n \geq \lambda} X_n dP$, where $(X_n: n \in \mathbb{N})$ is a non-negative $(F_n)$-submartingale.

I'm not sure how to start.

Please help. Thank you!

Answer 1

My attempt

$$ \begin{aligned} E[X_n^*] &= \int_0^\infty P(X_n^* \geq t)dt \\ &= 2\int_0^\infty P(X_n^* \geq 2u)du && t \gets 2u\\ &= 2\int_0^1 P(X_n^* \geq 2u)du + 2\int_1^\infty P(X_n^* \geq 2u)du \\ &\leq 2\cdot 1 + 2\int_1^\infty \frac{1}{u} \left( \int_{X_n\geq u} X_n dP\right) du && \text{Hint}\\ &= 2 + 2\int_{X_n\geq 1} X_n \left( \int_1^{X_n} \frac{1}{u} du \right) dP && \text{Tonelli's thm.}\\ &= 2 + 2\int_{X_n\geq 1} X_n \log X_n dP \\ &\leq 2 + 2\int_{X_n\geq 1} X_n \log^+ X_n dP \\ &\leq 2 + 2\int X_n \log^+ X_n dP \\ &= 2 + 2E[X_n \log^+ X_n] \end{aligned} $$

Finally, taking sup at both sides yields

$$ \|X^*\|_1 = \sup_n E[X_n^*] \leq 2+2\sup_n E[X_n \log^+ X_n] $$

Answer 2

In view of the monotone convergence, it suffices to prove that for all $N$, $$ \left\lVert X_N^*\right\rVert_1\leqslant 2 +2\sup_n\left\lVert X_n\log^+X_n\right\rVert_1. $$ To get such a bound, combine the inequality $P(X_N^* \geqslant 2 \lambda) \leqslant \min\left\{1, \frac{1}{\lambda}\int_{X_N\geqslant \lambda} X_N dP\right\}$ with $$ \left\lVert X_N^*\right\rVert_1/2 =\int_0^{+\infty}P(X_N^* \geqslant 2 \lambda)d\lambda. $$