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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.

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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.

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  • $\begingroup$ Oh... So I can write the first vector like this $\left(\frac 1{\sqrt n}\sum_{i=1}^{[nt_1]}Y_i, \frac 1{\sqrt n}\sum_{i=[nt_1]+1}^{[nt_2]}Y_i\right)=\frac{1}{\sqrt{n}}\sum_{i=1}^{nt_2}({Y_i I_{\{i\leq[nt_1]\}},Y_i I_{\{[nt_1]\leq i\leq [nt_2]\}}})=\frac{\sqrt{nt_2}}{\sqrt{n}}\frac{1}{\sqrt{[nt_2]}}\sum_{i=1}^{[nt_2]}({Y_iI_{\{i\leq[nt_1]\}},Y_iI_{\{[nt_1]\leq i\leq [nt_2]\}}})\Rightarrow\sqrt{t_2}\mathcal{N}(0,\Sigma)$ Where $\Sigma=Id$ Is it right? $\endgroup$ – sky90 Jun 29 '15 at 10:29
  • $\begingroup$ I still have a problem. As I wrote it I don't get the distribution I want... I was too much enthusiastic^^ $\endgroup$ – sky90 Jun 29 '15 at 11:44
  • $\begingroup$ @sky90 Did you try to compute the characteristic function of the first term of the right hand side in my answer? The components are independent. $\endgroup$ – Davide Giraudo Jun 29 '15 at 13:55
  • $\begingroup$ Oh no.. I did not tried this. But in that way are you still using CLT? I see that components are independent but the point of the exercise is to use the multidimensional CLT $\endgroup$ – sky90 Jun 29 '15 at 14:17
  • $\begingroup$ @sky90 i have edited. $\endgroup$ – Davide Giraudo Jun 29 '15 at 16:20

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