Let $\pi_K$ be Poisson random measure with intensity $\mu = K\lambda$, where $\lambda$ is Lebesgue measure. I have a task to do and the first part was to show that for $f \in L^1({\mathbb{R}^d}) \cap L^3(\mathbb{R}^d$) (shouldn't $3$ be $2$ actually?) $$\frac{1}{\sqrt{K}}\int_{\mathbb{R}^d}f(x)\bar{\pi}_K(dx)$$ converges in distribution as $K \to \infty$, where $\bar{\pi}_K = \pi_K - \mu$.
I've looked at Laplace transform and got that the limit is gaussian random variable with mean $0$ and variance $\sigma^2 = \int f^2(x)dx$ (if it is correct...).
Now the second part is to look at $X_K = \frac{1}{\sqrt{K}}\left(\pi_K\left([0,t]\right)-Kt\right)$ and find the limit of finite dimentional distributions, so the vector $\left(X_K\left(t_1\right),\ldots,X_K\left(t_n\right)\right)$.
I'm not really sure how to start. I see that $X_K$ is my previous random variable with $f = \chi_{[0,t]}$, so I guess the limit should be some gaussian vector, but I cannot show it.
It seems that we need $f$ integrable in order to make the integral $\int_{\mathbb{R}^d}f(x)\bar{\pi}_K(dx)$ well defined.
For the convergence of the finite-dimensional distribution, we could establish it for $\left(X_K(t_1),X_K(t_2)-X_K(t_1),\dots,X_K(t_n)-X_K(t_{n-1})\right)$ by using the Cramer-Wold device: we are reduced to investigate the convergence of $$ \frac 1{\sqrt K}\sum_{i=1}^nc_j\left(\pi_K\left([t_{i-1},t_i]-K\left(t_{i}-t_{i-1}\right)\right)\right)$$ for all $c_1,\dots,c_n\in\mathbb R$.