I'm trying to prove that a sequence of functions $(k_N)_{N\in\mathbb{N}}$

$$k_N(\vec{y}):=e^{-N^2r(\vec{y},Q\vec{y})}\sqrt{\frac{r}{\pi}}^kN^{k}$$

where $r>0$, $k>2$ and

**Edit**: I have forgot to say, that $k\in\mathbb{N}$ is the size of the matrix $Q$ i.e. $Q$ is $k\times k$. This means that $\det Q=0$ and $Q$ is positive semidefinte.

I have also changed $k$ from beeing $>0$ to $k>2$.

**End of the Edit**

$$Q:=\left(\begin{array}{cccccccc} (1-\frac{1}{k})& -\frac{1}{k} & -\frac{1}{k} & \cdots & -\frac{1}{k} \\ -\frac{1}{k} & (1-\frac{1}{k}) & -\frac{1}{k} & \cdots & -\frac{1}{k} \\ \vdots & \ddots & & &\vdots \\ -\frac{1}{k} & \dots & &-\frac{1}{k} &(1-\frac{1}{k}) \end{array}\right)$$

goes to a delta function when $N\to\infty$. What I'm missing ist showing, that given $\varepsilon>0$ and $\eta>0$ there exists $n_0$ such that $\forall n>n_0$

$$\idotsint_{|\vec{x}|>\eta}k_n(\vec{x})d\vec{x}<\varepsilon$$.

I was looking for some expression for the error function of multivariate gaussians, but could find nothing of the sort. Is there a nice way of representing the error functions so that I don't have to diagonalize $Q$ and treat the function as several independent gaussians?

Thanks!