# Show that $\sup_n \mathbb{E}(|S_n|)<\infty$ implies $\mathbb{E} \left( \sup_n |S_n| \right)<\infty$ for $S_n=\sum_{k=1}^n X_k$ with $X_k$ iid

Let $S_n=\sum^n_{k=1}X_k$, and $X_k$'s are mutually independent. Suppose $X_k$ are integrable, and $\sup_nE|S_n|<\infty$. Show that $E(\sup_n|S_n|)<\infty$.

I have shown Ottaviani's inequality:

$P(\max_{k\leq n}|S_k|\geq t+s)\leq P(|S_n|\geq t)+P(\max_{k\leq n}|S_k|\geq t+s)\max_{k\leq n}P(|S_n-S_k|>s)$.

I think this should be helpful, however, I don't know how.

By the triangle inequality,

$$\mathbb{P}(|S_n-S_k| > s) \leq \mathbb{P}(|S_k|>s/2) + \mathbb{P}(|S_n|>s/2).$$

Applying Markov's inequality yields

\begin{align*} \max_{k \leq n} \mathbb{P}(|S_n-S_k|>s) \leq 2 \max_{k \leq n} \mathbb{P}(|S_k|>s/2) \leq \frac{4}{s} \max_{k \leq n} \mathbb{E}(|S_k|). \end{align*}

If we choose $s_0$ sufficiently large, then we get for all $s \geq s_0$

\begin{align*}\max_{k \leq n} \mathbb{P}(|S_n-S_k|>s)&\leq \max_{k \leq n} \mathbb{P}(|S_n-S_k|>s_0) \\ &\leq \frac{4}{s_0} \sup_{n \in \mathbb{N}} \mathbb{E}(|S_n|) =: c<1. \tag{1} \end{align*}

Using $(1)$ and Ottaviani's inequality for $s=t$, we find

$$\mathbb{P} \left( \max_{k \leq n} |S_k| \geq 2s \right) \leq \mathbb{P}(|S_n| \geq s) + c \mathbb{P} \left( \max_{k \leq n} |S_k| \geq 2s \right),$$

i.e. (as $c \in (0,1)$),

$$\mathbb{P} \left( \max_{k \leq n} |S_k| \geq 2s \right) \leq \frac{1}{1-c} \mathbb{P}(|S_n| \geq s). \tag{2}$$

Note that this inequality holds for all $s \geq s_0$ and all $n \in \mathbb{N}$. Now recall that

$$\mathbb{E}(X) = \int_0^{\infty} \mathbb{P}(X \geq r) \, dr \tag{3}$$

holds for any non-negative random variable $X$. Combining this identity with $(2)$, we get

\begin{align*} \mathbb{E} \left( \max_{k \leq n} |S_k| \right) &\stackrel{(3)}{=} \int_0^{\infty} \mathbb{P} \left( \max_{k \leq n} |S_k| \geq r \right) \, dr \\ &\stackrel{(2)}{\leq} 2s_0 + \frac{1}{1-c} \int_{2s_0}^{\infty} \mathbb{P}(|S_n| \geq r/2) \, dr \\ &\stackrel{(3)}{\leq} 2s_0 + \frac{2}{1-c} \mathbb{E}(|S_n|). \tag{4} \end{align*}

Finally, by the monotone convergence theorem,

$$\mathbb{E} \left( \sup_{n \in \mathbb{N}} |S_n| \right) = \sup_{n \in\mathbb{N}} \mathbb{E} \left( \max_{k \leq n} |S_k| \right) \stackrel{(4)}{\leq} 2s_0 + \frac{2}{1-c} \sup_{n \in \mathbb{N}} \mathbb{E}(|S_n|) < \infty.$$