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I'm a final year undergrad who was doing a project that involved the implementation of a frequency-domain volatility estimator. I haven't a lot of stats background so not understanding a point that may be easier than I think. I assumed that I was observing $Y_{t_i}$, which was an additive combination of some $X_{t_i}$, the process I'm interested in, and some white noise on top. So after differencing, Fourier transforming and using the periodogram to estimate the volatility, a key observation I'm supposed to make is that the periodogram for just the white noise isn't flat after I difference. This hasn't made a lot of sense to me, even though I can see it happen in R.

I understand it being flat normally of course, but the difference of normal variables is also a normal variable, just with bigger variance. So in that case, why should the shape of my periodogram be any different? For reference, the paper describing the estimator, which includes pictures of the difference periodogram is here (page 5).

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"But the difference of normal variables is also a normal variable, just with bigger variance." this may not be correct.

To see this, $x_1$, $x_2$, $x_3$ are iid standard normal, and $x_2$-$x_1$ and $x_3$-$x_2$ are actually correlated with Cov($x_2$-$x_1$,$x_3$-$x_2$) = -1. As a result, there is autocorrelation in the series. Therefore your periodogram looks the way it looks.

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