# Singular direction of a particle system

Consider a system of n-sdes in $$\mathbb{R}$$ ( the formula is not important). The corresponding particle system $$X(t)=(X_{1}(t),X_{2}(t),...,X_{n}(t))$$ lives in $$\mathbb{R}^{n}$$ and assume that for this system we know that its center of mass is a Brownian Motion. ie $$C(t) =\frac{1}{n} \sum \limits_{i} X_{i}(t) =$$ BrMotion $$\sim N(0,t)$$

The paper that I am studying says (quote) : " the direction (1,1,...,1) is quite singular for the particle system. Indeed, the center of mass being a Brownian Motion prevents the law of $$X(t)$$ from converging as $$t\longrightarrow\infty$$ "

I can't understand why this property of the center of mass prevents convergence. My idea/interpetation is that $$law(X(t))(1_{\mathbb{R}^{n}})\leq law(C(t))(1) \propto \frac{1}{\sqrt{t}} \exp(-\frac{1}{2t})$$ where the last quantity goes to 0 as $$t\longrightarrow\infty$$

So we get that $$law(X(t))(k1_{\mathbb{R}^{n}}) \longrightarrow 0$$ as $$t\longrightarrow\infty$$ for all $$k \in \mathbb{R}$$ .

Can we deduce from this that there is no limit for $$law(X(t))$$ ?

What does "singular direcion" mean in this context ?