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I am studying time series, following Time Series Theory and Methods by Brockwell and Davis. I am reading the proof of the Herglotz Theorem, I have the following questions.

First, the proof given in the book is the next one:

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I don´t understand the equality given in the proof (I drew an arrow there) How to get $\frac{1}{2\pi N}\displaystyle\sum_{|m|-N}(N-|m|)e^{imv}\gamma(v)$

Aditionally, I don't have clear why is possible to say that $f_N(.)I_{(-\pi,\pi]}$ is a density function?

I add the part where is mentioned that $f_N$ is a density, I don´t have clear why it is true.

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1 Answer 1

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It's something similar to the change of variable trick in integration. The two expressions in the text clearly indicate you to consider substituting $r - s$ with $m$. The details are spelled out as follows.

For simplicity, let's denote the summand $e^{-i(r - s)v}\gamma(r - s)$ by a bivariate function $K(r, s)$, whose domain is the set of $N \times N$ grid points $\{1, 2, \ldots, N\} \times \{1, 2, \ldots, N\}$. On the other hand, note that $K(r, s)$ is completely determined by the coordinate differences $m := r - s$, therefore may also be denoted by $\tilde{K}(m)$. The challenge is to express the double sum $S := \sum_{r = 1}^N \sum_{s = 1}^N K(r, s)$ in terms of the new summing index $m$. I will illustrate this transformation process in matrix form as follows.

In terms of $(r, s)$, we are effectively summing over the matrix (think the row corresponds to $r$ and the column corresponds to $s$): \begin{align} \begin{bmatrix} K(1, 1) & K(1, 2) & \cdots & K(1, N) \\ K(2, 1) & K(2, 2) & \cdots & K(2, N) \\ \vdots & \vdots & \ddots & \vdots \\ K(N, 1) & K(N, 2) & \cdots & K(N, N) \end{bmatrix}. \tag{1} \end{align}

As stated before, since $K(r, s) = \tilde{K}(r - s) = \tilde{K}(m)$, matrix $(1)$ is equal to \begin{align} \begin{bmatrix} \tilde{K}(0) & \tilde{K}(-1) & \cdots & \tilde{K}(1 - N) \\ \tilde{K}(1) & \tilde{K}(0) & \cdots & \tilde{K}(2 - N) \\ \vdots & \vdots & \ddots & \vdots \\ \tilde{K}(N - 1) & \tilde{K}(N - 2) & \cdots & \tilde{K}(0) \end{bmatrix}. \tag{2} \end{align}

The entries in matrix $(2)$ can be summed over diagonally. For example, there are $N$ terms $\tilde{K}(0)$ lie on the main diagonal line, which corresponds to $m = 0$. There are $N - 1$ terms of $\tilde{K}(-1)$ and $\tilde{K}(1)$ lie on the first off-diagonal line of the upper triangle and that of the lower triangle, which correspond to $m = -1$ and $m = 1$, respectively. In general, on the $m$-th off-diagonal line of the upper triangle, there are $N - m$ terms of $\tilde{K}(-m)$, while on the $m$-th off-diagonal line of the lower triangle, there are $N - m$ terms of $\tilde{K}(m)$. Summing all the $N^2$ terms of $\tilde{K}(\cdot)$ in this way, we obtain \begin{align} S = & \sum_{\text{upper triangle}} \tilde{K}(\cdot) + \sum_{\text{diagonal}}\tilde{K}(\cdot) + \sum_{\text{lower triangle}}\tilde{K}(\cdot) \\ = & \sum_{m = 1}^{N - 1} (N - m)\tilde{K}(-m) + N\tilde{K}(0) + \sum_{m = 1}^{N - 1}(N - m)\tilde{K}(m) \\ = & \sum_{|m| < N} (N - |m|)\tilde{K}(m). \end{align}

This completes the proof.

For your second question, to show $f_N(\cdot)I_{(-\pi, \pi]}(\cdot)$ is a density function, since $f_N(v) \geq 0$ for all $v \in (-\pi, \pi]$, it suffices to show that $$\int_{-\pi}^{\pi}f_N(v) dv = 1.$$

By the identity we just proved, together with the linearity of the integral, $$\int_{-\pi}^{\pi} f_N(v) dv = \frac{1}{2\pi N}\sum_{|m| < N} (N - |m|)\gamma(m) \int_{-\pi}^{\pi} e^{imv} dv.$$

It is easy to show (also well-known as the orthogonality of Fourier basis) that \begin{align} \int_{-\pi}^{\pi} e^{imv} dv = \begin{cases} 0 & m \neq 0, \\ 2\pi & m = 0. \end{cases} \end{align}

Combining this with the fact $\gamma(0) = 1$, it follows that $\int_{-\pi}^{\pi} f_N(v) dv = 1$.

PS: I found your question is more about pure mathematical derivation, which may be more appropriate to be asked at Mathematics Stack Exchange. Indeed, this time series book you are reading is written in a very rigorous way.

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  • $\begingroup$ Zhanxiong, but I added the part where is mentioned that $f_N(.)$ is a density function. My question is how to affirm that?. $\endgroup$
    – Boris
    Jan 8, 2018 at 0:10
  • $\begingroup$ Additionally, if this book is written in a very rigorous way, could you recommend me a book simpler, in order to understand some of the concepts. I am following this book in my lecture because the professor follos this book, But sometimes is really hard to understand most of the concepts given. $\endgroup$
    – Boris
    Jan 8, 2018 at 0:16
  • $\begingroup$ @Boris I will edit my answer. To answer your recommendation question, what level are you in? Undergraduate or graduate? $\endgroup$
    – Zhanxiong
    Jan 8, 2018 at 0:27
  • $\begingroup$ Zhanxiong, I am graduate student, this lecture is part of my master in statistics. $\endgroup$
    – Boris
    Jan 8, 2018 at 0:34
  • $\begingroup$ OK. Then I would recommend Ruey Tsay's book Analysis of Financial Time Series. The accessibility (in terms of mathematics maturity) is much easier than your current book while it explains basic concepts well, which also covers more applied models. Hope it helps and good luck to your studies. $\endgroup$
    – Zhanxiong
    Jan 8, 2018 at 0:39

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