Consider that we have an iid sequence of real-valued random variables $(Y_j)_{j \in \mathbb{Z}}$ and further let $G: \mathbb{R}^\mathbb{Z} \to \mathbb{R}$ measurable such that sequence $X_n = G(\dots,Y_{n-1},Y_n)$ is well-defined and further $\mathbb{E}(\vert X_1 \vert^p)<\infty$ for a $p > 2$.
For a random variable $X$ let $\Vert X \Vert_2^2 = \mathbb{E}(X^2)$ denote it's $L^2$ norm. And for reasons of simplicity assume that $\mathbb{E}(X_1)=0$.
I'm looking for an inequality of the type $$\Vert \sum_{j=1}^n X_j \Vert_2^2 \leq c \sum_{j=1}^n \Vert X_j \Vert_2^2 = c \cdot n \Vert X_1 \Vert_2^2.$$
I know that this holds true, if the $X_n$ are martingale differences (w.r.t a filtration $(\mathcal{F}_n)_{n \in \mathbb{Z}}$) or if the $X_i$ are mutually uncorrelated.
Is there a chance to obtain something like this is in the more general setting as provided above?
This question got answered by Davide.
I'm now wondering whether I could extend the result in the following way: Let $S_n = \sum_{j=1}^n X_j$ and now consider $q \geq 2$, then
$$\max_{1 \leq \ell \leq n} \vert S_\ell \vert = \max_{1 \leq \ell \leq n} \vert \sum_{i\leq 0}\sum_{j=1}^\ell\left(\mathbb E\left[X_j\mid\mathcal F_{j+i}\right]-\mathbb E\left[X_j\mid\mathcal F_{j+i-1}\right]\right) \vert \\\overset{*}{\le} \sum_{i\leq 0} \max_{1 \leq \ell\leq n} \vert \sum_{j=1}^\ell\left(\mathbb E\left[X_j\mid\mathcal F_{j+i}\right]-\mathbb E\left[X_j\mid\mathcal F_{j+i-1}\right]\right) \vert. $$
Then by the triangle inequality $$\Vert \max_{1 \leq \ell \leq n} \vert S_\ell \vert \Vert_q \leq \sum_{i\leq 0}\Vert \max_{1 \leq \ell\leq n} \vert \sum_{j=1}^\ell\left(\mathbb E\left[X_j\mid\mathcal F_{j+i}\right]-\mathbb E\left[X_j\mid\mathcal F_{j+i-1}\right]\right) \vert\Vert_q.$$
As Davide pointed out, $\mathbb E\left[X_j\mid\mathcal F_{j+i}\right]-\mathbb E\left[X_j\mid\mathcal F_{j+i-1}\right]$ is a martingale difference sequence and hence we could apply Doob's martingale inequality to find a constant $c_q > 0$ such that $$\Vert \max_{1 \leq \ell\leq n} \vert \sum_{j=1}^\ell\left(\mathbb E\left[X_j\mid\mathcal F_{j+i}\right]-\mathbb E\left[X_j\mid\mathcal F_{j+i-1}\right]\right) \vert\Vert_q \leq c_q \Vert \sum_{j=1}^n\left(\mathbb E\left[X_j\mid\mathcal F_{j+i}\right]-\mathbb E\left[X_j\mid\mathcal F_{j+i-1}\right]\right) \Vert_q.$$
Now employing the result of Davide we would then have $$\Vert \max_{1 \leq \ell \leq n} \vert S_\ell \vert \Vert_q \leq c_q \sqrt{n} \sum_{j \leq 0} \Vert E\left[X_0\mid\mathcal F_{i}\right]-\mathbb E\left[X_0\mid\mathcal F_{i-1}\right] \Vert_q$$
Hence this would allow us to even establish a maximal $L^p$ inequality, which seems pretty good. Is the problem, that the inequality denotes with $*$ may not hold?
Best Answer
Let $\mathcal F_k$ be the $\sigma$-algebra generated by the random variables $Y_i,i\in\mathbb Z,i\leqslant k$. Observe that by the martingale convergence theorem, $$ X_j=\sum_{i\leqslant 0}\left(\mathbb E\left[X_j\mid\mathcal F_{j+i}\right]-\mathbb E\left[X_j\mid\mathcal F_{j+i-1}\right]\right), $$ where the convergence is understood in the $\mathbb L^2$-sense. As a consequence, $$ \sum_{j=1}^nX_j=\sum_{i\leqslant 0}\sum_{j=1}^n\left(\mathbb E\left[X_j\mid\mathcal F_{j+i}\right]-\mathbb E\left[X_j\mid\mathcal F_{j+i-1}\right]\right) $$ and it follows that $$ \left\lVert \sum_{j=1}^nX_j\right\rVert_2\leqslant \sum_{i\leqslant 0}\left\lVert \sum_{j=1}^n\left(\mathbb E\left[X_j\mid\mathcal F_{j+i}\right]-\mathbb E\left[X_j\mid\mathcal F_{j+i-1}\right]\right)\right\rVert_2. $$ For each fixed $i\leqslant 0$, $\left(\mathbb E\left[X_j\mid\mathcal F_{j+i}\right]-\mathbb E\left[X_j\mid\mathcal F_{j+i-1}\right]\right)_{j\geqslant 0}$ is a martingale difference sequence with respect to the filtration $\left(\mathcal F_{j+i}\right)_{j\geqslant 0}$. Moreover, by stationarity,
$\left\lVert \mathbb E\left[X_j\mid\mathcal F_{j+i}\right]-\mathbb E\left[X_j\mid\mathcal F_{j+i-1}\right]\right\rVert_2=\left\lVert\mathbb E\left[X_0\mid\mathcal F_{i}\right]-\mathbb E\left[X_0\mid\mathcal F_{j+1}\right]\right\rVert_2$ hence $$ \left\lVert \sum_{j=1}^nX_j\right\rVert_2\leqslant \sqrt n \sum_{i\leqslant 0}\left\lVert \mathbb E\left[X_0\mid\mathcal F_{i}\right]-\mathbb E\left[X_0\mid\mathcal F_{i-1}\right] \right\rVert_2. $$