Spectral density and Riemann Stieltjes Integral I am confused with a part about spectral densities. I found it in Time Series Theory and Methods by Brockwell and Davis.
 
I don´t understand how is applied the Riemann Stieltjes Integral in this case, I have seen the definition of Riemann Stieltjes in Wikipedia but I can´t get that conclusion.
How is it concluded?
 A: To make your question complete, you must also let us know what (4.2.1) is. It is
$$X_t = \sum_{j = 1}^n A(\lambda_j)e^{it\lambda_j}.$$
What you need to show is that the $E(X_{t + h}\bar{X}_t)$ given in the text
(which is in summation form) has the Riemann-Stieltjes representation (4.2.3), provided that $F$ is defined as (4.2.4). Indeed, 
\begin{align}
& \gamma(h) = \int_{(-\pi, \pi]} e^{ihv} d F(v) \\
= & \int_{(-\pi, \pi]} e^{ihv} d\sum_{j: \lambda_j \leq \lambda} \sigma_j^2 \\
= & \sum_{j: \lambda_j \leq \lambda}\sigma_j^2 e^{ih\lambda_j} \tag{1}\\ 
= & \sum_{j = 1}^n \sigma_j^2 e^{ih{\lambda_j}} = E(X_{t + h}\bar{X}_t).
\end{align}
This shows author's claim.
It is in $(1)$ that the rule of R-S integral comes into play. Note by definition, the nondecreasing function $F$ is a step-wise function that has jumps at points $\{\lambda_1, \ldots, \lambda_n\}$, with jump sizes $\sigma_1^2, \ldots, \sigma_n^2$. At other points, $F$ is flat. For such $F$, the R-S integral accumulates the integrand $e^{ihv}$ only at jump points, multiplied by the jump size of $F$. Thus the result follows. 
(4.2.4) may look intimidating in its original form --- if you are willing to plot it (start with small $n$ such as $2$ or $3$), I think it
would help you understand the above calculation better.
