# Error function of multivariate Gaussian

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!

• What is $r(\beta')$ and what is $\beta'$? Dec 27, 2020 at 19:35
• $r(\beta')$ is a constant of my system. I'm working with a fixed $\beta'$. Dec 27, 2020 at 19:38

$$\newcommand\R{\mathbb R}$$Let us write $$x$$ for $$\vec x$$ and $$t$$ for $$\eta$$. Then $$\int_{x\in\R^k\colon|x|>t}k_n(x)\,dx=\infty$$ for all real $$t>0$$.
Indeed, we can write $$Q=I_k-\frac1k\,1_k1_k^T,$$ where the matrix $$I_k$$ is the $$k\times k$$ identity matrix and $$1_k$$ is the $$1\times k$$ column matrix with unit entries, that is, $$1_K=e_1+\dots+e_k$$, where $$(e_1,\dots,e_k)$$ is the standard basis of $$\R^k$$. So, $$Q$$ is the matrix of the orthogonal projector onto the orthogonal complement -- say $$V$$ -- of (the span of) the vector $$1_k$$; in particular, $$Q1_k=0$$.
Fix any orthonormal basis $$(b_1,\dots,b_k)$$ of $$\R^d$$ such that $$b_1=1_k/\sqrt k$$ (so that $$(b_2,\dots,b_k)$$ is an orthonormal basis of $$V$$). Let $$(x_1,\dots,x_k)$$ denote the coordinates of an arbitrary vector $$x\in\R^d$$ in the orthonormal basis $$(b_1,\dots,b_k)$$. Then $$(x,Qx)=x_2^2+\dots+x_k^2.$$ Therefore and because the Lebesgue measure is spherically invariant, for all real $$t>0$$ we have \begin{aligned} &\int_{x\in\R^k dx\,\colon|x|>t}dx\, k_n(x) \\ &\ge\int_{x\in\R^k\colon x_1>t}dx\, k_n(x) \\ &=\int_t^\infty dx_1\int_\R dx_2\cdots\,\int_\R dx_k \exp\{-n^2r(x_2^2+\dots+x_k^2)\}\Big(\sqrt{\frac{r}{\pi}}\Big)^k n^{2k} =\infty, \end{aligned} because the inner $$(k-1)$$-fold integral is a strictly positive real number not depending on $$x_1$$.
• Thank you for your answer! I have realized that the definition of $k_N$ is wrong. The function should be normalized. I'm going to edit the definition. Dec 28, 2020 at 2:10
• After reading this comment I was really confused about what went wrong with my idea. I have just seen, that I forgot to mention an important detail: $k\in\mathbb{N}$ is also the size of the matrix, so that $Q$ is positive semi-definite. I'm terribly sorry for the misunderstanding. I have been working on the problem leading to this question for so long, that it didn't even occurred to me to mention. Dec 29, 2020 at 0:39
• I did not have a problem with $k$ denoting the dimension and $k_n$ denoting the kernel. The problem is with the fact that $Q$ is a singular (positive-semidefinite) matrix, rather than a positive-definite one. Dec 29, 2020 at 3:57