Summary:
For a finite family of random variables $(X^{i})_{i\leq N}$, we have the density estimate. $$ p_t(y) \leq \int_0^\infty C\left(\frac{1}{\sqrt{t}}+e^{\displaystyle\epsilon y^2}\right)e^{\displaystyle-\frac{(x-y)^2}{ct}}\ \mathrm{d}x. $$ for (small) $\epsilon>0$, $t\in[0,\infty)$ and constants $c,C>0$. This is an estimate on the density of the probability of $X^{i}$ taking value $y\in\mathbb{R}$.
Is there a short explanation why $$ \mathbb{E}\left[\left(\left|X^{1}\right|+\frac{1}{N}\sum_{1\leq j\leq N} \left|X^{j}\right|\right)^4\right]<\infty $$
More elaborate:
Situation:
We have
- A number of stochastic processes $X^{1},\dots,X^{N}$ (i.e. for each of those $X_t^{i}$ is a random variable in each $t\in[0,\infty)$) with values in $\mathbb{R}$. We assume $X^{i}$ to start in $\mathbb{R}_+$.
- An upper bound for the transition density $\mathfrak{p}_t(x,y)$ of the process, so we have an upper bound for the density of the probability of $X_t^{i}=y$ when $X_0^{i}=x$ (i.e., $X^{i}$ starts in $x$ and ends up at $y$ in time point $t$).
- The upper bound has the formula $$ \mathfrak{p}_t(x,y)\leq C\left(\frac{1}{\sqrt{t}}+e^{\displaystyle\epsilon y^2}\right)e^{\displaystyle-\frac{(x-y)^2}{ct}}, $$ for (small) $\epsilon>0$ and constants $c,C>0$.
Question:
I'd like to prove that $$ \mathbb{E}\left[\left(\left|X_t^{1}\right|+\frac{1}{N}\sum_{i=1}^N \left|X_t^{i}\right|\right)^4\right]<\infty $$
Problem: I don't think that I can "insert" the density estimate. I know that by integrating over $x$, I get the actual density $p_t(y)$, i.e. $$ p_t(y)=\int_0^\infty \mathfrak{p}_t(x,y)\ \mathrm{d}x. $$ Is there a way to conclude the existence of the fourth moment of this sum from the bounded density?
Edit: It might be possible to estimate like \begin{align*} \mathbb{E}[|X_t^{i}|^4]=&\int_\mathbb{R} x^4 p_t(x)dx\\ \leq& \int_\mathbb{R} \int_0^\infty C\left(\frac{1}{\sqrt{t}}+e^{\displaystyle\epsilon y^2}\right)e^{\displaystyle-\frac{(x-y)^2}{ct}}\ \mathrm{d}x. \end{align*}