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I am having a problem with proving convergence in distribution (or by law).

Consider that the sequence $X_n$ of random variables are IID and that $E[X_n]=0$ and $V[X_n]=1$.

Now define the variable $U_N$ as:

$$U_N= \frac{1}{\sqrt{N}}\sum_{n=1}^N X_n\cdot \sin\left(\frac{n\pi}{N}\right).$$

For $N\rightarrow \infty$, I want to show that $U_N$ converges by distribution, furthermore, I also want to determine the asymptotic distribution of $U_N$.

For the first part, I have tried to show convergence in probability, because this implies conv. in distribution (by law), but this was not possible since I dont have the asymptotic distribution of the $X_n$.

For the second part, I tried with the delta-method, but this did not work because of the summation of in the expression for $U_N$.

Does someone have an idea to this?

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  • $\begingroup$ $E[U_N] = 0$. Do you know what $Var(U_N)$ is or at least what it converges to? $\endgroup$ – Henry Aug 13 '18 at 9:37
  • $\begingroup$ Var(U_N) should be 1/Nsum(sin(npi)/N), which also converges to zero for N->\infty $\endgroup$ – Jonathan Kiersch Aug 13 '18 at 9:45
  • $\begingroup$ I suspect $Var(U_N) \to \int\limits_0^1 \sin^2(\pi x)\, dx = \frac12$ $\endgroup$ – Henry Aug 13 '18 at 11:10
  • $\begingroup$ How did you find that? $\endgroup$ – Jonathan Kiersch Aug 13 '18 at 11:57
  • $\begingroup$ A combination of thinking that without the $\sin(\frac{n\pi}{N})$ term the variance would be $1$ as in the central limit theorem, plus simulation and basic calculus. $\endgroup$ – Henry Aug 13 '18 at 12:01
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One can try to check Lindeberg's condition with $X_{N,i}:= X_i\sin\left(i\pi/N\right)$, using the following three facts:

  • the limits $\lim_{N\to +\infty}N^{-1}\sum_{n=1}^N\sin^2\left(n\pi/N\right)$ exists and is computable, as a limit of Riemann sums.
  • $s_N=\sum_{i=1}^N\operatorname{Var}\left(X_{N,i}\right)\sim c\sqrt N$.
  • $$\mathbb E\left[X_{N,i}^2\mathbf 1\left\{\left\lvert X_{N,i}\right\vert\gt\varepsilon s_N \right\}\right]=\mathbb E\left[X_1^2\sin^2\left(i\pi/N\right)\mathbf 1\left\{\left\lvert \sin\left(i\pi/N\right) X_{1}\right\vert\gt\varepsilon s_N\right\}\right]\leqslant\mathbb E\left[X_1^2\sin^2\left(i\pi/N\right)\mathbf 1\left\{\left\lvert X_{1}\right\vert\gt\varepsilon s_N \right\}\right].$$
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  • $\begingroup$ I see, I was not familiar with the Lindeberg condition for CLT. So if I have understood you correct, I use the Lindeberg condition on X_n,N to find that U_N will converge to a standard normal distribution for N->infinity $\endgroup$ – Jonathan Kiersch Aug 13 '18 at 12:44
  • $\begingroup$ Yes, the limit will be a normal distribution but I have not looked whether the variance is one or not. $\endgroup$ – Davide Giraudo Aug 13 '18 at 13:20
  • $\begingroup$ @DavideGiraudo - the limiting variance is $\frac12$ (it would have been $1$ without the sine term) $\endgroup$ – Henry Aug 13 '18 at 14:04

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