I want to prove that the linear interpolation $X_t^n(\omega):=\frac{1}{\sqrt{n}}\sum_{k=1}^{[nt]}{Y_k}(\omega)+\frac{1}{\sqrt{n}}Y_{[nt]+1}(\omega)(nt-[nt])$ of $\sum_{k=1}^{n}{Y_k}(\omega)$ for r.v. $(Y_k)_{k\in \mathbb{N}}$ i.i.d., centred with variance 1, converges in distribution to $\mathcal{N}(0,t)$.
The one dimensional case is not a problem. But to prove the general case taking $t_1<\cdots<t_m\in [0,1]$ I have problems by applying the multidimensional CLT.
I want to prove that the vector $X^n=(X^n_{t_1},X^n_{t_2}-X^n_{t_1},...,X^n_{t_m}-X^n_{t_{m-1}})$ converges in distribution to $(\mathcal{N}(0,t_1),\mathcal{N}(0,t_2-t_1),...,\mathcal{N}(0,t_m-t_{m-1}))$ to conclude the same convergence of the vector $(X^n_{t_1},...,X^n_{t_n})$. Essentially I have problems by choosing the right vector on which apply the CLT, since I can't find a convenient way to write $X^n$ as a sum of $n$ vectors.
Thanks for any advice.
For $m=2$, write $$\left(X_{t_1}^n,X_{t_2}^n -X_{t_2}^n\right)=\left(\frac 1{\sqrt n}\sum_{i=1}^{[nt_1]}Y_i, \frac 1{\sqrt n}\sum_{i=[nt_1]+1}^{[nt_2]}Y_i \right) +\left(\frac{nt_1-[nt_1]}{\sqrt n}Y_{[nt_1]+1} , \frac{nt_2-[nt_2]}{\sqrt n}Y_{[nt_2]+1}-\frac{nt_1-[nt_1]}{\sqrt n}Y_{[nt_1]+1} \right);$$ the second vector converges to $0$ in probability. For the first one, use characteristic functions to deduce the wanted convergence. Indeed, the characteristic function of $\left(\frac 1{\sqrt n}\sum_{i=1}^{[nt_1]}Y_i, \frac 1{\sqrt n}\sum_{i=[nt_1]+1}^{[nt_2]}Y_i \right)$ is $\varphi_n(s)\psi_n(s')$, where $\varphi_n$ is the characteristic function of $\frac 1{\sqrt n}\sum_{i=1}^{[nt_1]}Y_i$, and $\psi_n$ that of $\frac 1{\sqrt n}\sum_{i=1}^{[nt_2]-[nt_1]}Y_i$. Then use the (one dimensional) central limit theorem.