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This question is going to encompass multiple areas and may not have a single answer but here it goes. I'm trying to conceptualize 3 different concepts and how they relate to each other. These 3 concepts are:

Energy Spectral Density (ESD) which I'm initially defining as $$\widehat{S_{xx}}=|\hat{x}(f)|^2$$ where $$\hat{x}(f)=\int_{-\infty}^{\infty}x(t)e^{-2\pi ift}dt$$ and $$\int_{-\infty}^{\infty}|\hat{x}(f)|^2df= \int_{-\infty}^{\infty}|x(t)|^2dt$$ from Parseval's theorem.

Next my definition for autocorrelation for a (WSS) stochastic process:

$$(1)R_{xx}(\tau)=E[X(t)X(t+\tau)]$$ this seems to be at disagreement for a different definition for autocorrelation namely for signals which is $$(2)\int_{-\infty}^{\infty}x(t)x(t+\tau)dt$$ I would expect the definition for signals to be $$\lim_{T\to\infty}\frac{1}{2T}\int_{-T}^{T}x(t)x(t+\tau)dt$$ since for ergodic stochastic processes, the two would converge to the same thing.

Finally for Power Spectral Density (PSD) I would define:

$$S_{xx}=F\{R_{xx}(\tau)\} $$ or the fourier transform of the autocorrelation function. This clearly depends on the definition. For (2) this is $$S_{xx}=\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}x(t)x(t+\tau)e^{-2 \pi if\tau}dt d\tau$$ $$\int_{-\infty}^{\infty}x(t)\int_{-\infty}^{\infty}x(t+\tau)e^{-2 \pi if\tau}d\tau dt$$ $$\int_{-\infty}^{\infty}x(t)\int_{-\infty}^{\infty}x(u)e^{-2 \pi if u}e^{2 \pi if t}du dt$$ $$\int_{-\infty}^{\infty}x(t)e^{2 \pi if t}\int_{-\infty}^{\infty}x(u)e^{-2 \pi if u}du dt$$ $$S_{xx}=\hat{x}^*(f)\hat{x}(f)=|\hat{x}(f)|^2=\widehat{S_{xx}}$$ which makes no sense to me.

Using (1) (with ergodic process) $$S_{xx}=F\{R_{xx}(\tau)\}=F\{E[X(t)X(t+\tau)]\}$$ $$\lim_{T\to\infty}\frac{1}{2T} \int_{-\infty}^{\infty}\int_{-T}^{T}x(t)x(t+\tau)e^{-2 \pi if \tau}dt d\tau$$ $$S_{xx}=\lim_{T\to\infty}\frac{1}{2T}(\hat{x}^*(f)\hat{x}(f))=\lim_{T\to\infty}\frac{1}{2T}|\hat{x}(f)|^2=\lim_{T\to\infty}\frac{1}{2T}\widehat{S_{xx}}$$ which makes a lot more sense to me. So my question is, what is the definition of Autocorrelation (and is it context dependent).Subsequently what is the definition of PSD and ESD and how do they relate to each other and to autocorrelation. It seems like terminology is lacking "standardization" so that the meaning of something could be completely dependent on the context and the author. Any help on consolidating these notions would be greatly appreciated.

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