# White noise does not contradicts Wide Sense Stationarity?

White noise is usually defined as a wide sense stationary (WSS) process $$N=\{N_t|t\in T\}$$ (for $$T$$ a time index set), that has a constant power spectral density, say $$S_{NN}(f)=\sigma^2$$. Since the correlation function $$R_{NN}(t)=\mathcal{F}[S_{NN}(f)]$$, we have also that $$R_{NN}(t)=\sigma^2\delta(t)$$ where $$\delta$$ is the Dirac delta function. The thing is that $$\mathbb{E}\{|N_t|^2\}=R_{NN}(0)=\sigma^2\delta(0)$$. Indeed, if we assume that $$N$$ is ergodic we have that power is also the variance of the process:

\begin{align} \mathbb{V}(N) &= R_{NN}(0) - \mu_N(t) \\[14pt] &= R_{NN}(0) \\[6pt] &= \lim_{\tau\to\infty} \int \limits_{-\infty}^{\infty}\frac{\mathbb{E}\{|\hat{N}^\tau(f)|^2\}}{2\tau} \ df, \\[6pt] \end{align}

where $$\hat{N}^\tau$$ is the Fourier transform of windowing of $$N$$ on a interval of lenght $$2\tau$$. So, what I do not understand is how $$N$$ can be WSS if it has an infinite power and variance, because:

$$\int_{-\infty}^{\infty}\sigma^2 \ df=\infty.$$

As you can see, I am requiring a process to have finite second moment for being WSS. Also, I have that this white noise has identical distribution with zero mean and variance $$\sigma^2$$.

I feel that I have a big misconception, but I do not know where the problem is. I really appreciate any help.

• Could you clarify what you mean by the "second moment" of a process? That might get to the heart of the matter.
– whuber
Commented Dec 19, 2021 at 15:35
• Thanks! I mean $\mathbb{E}\{|X_t|^2\}$ for every $t$ by the second moment Commented Dec 19, 2021 at 15:42
• What is PSD? .. Commented Dec 19, 2021 at 15:45
• I have edited it yet Commented Dec 19, 2021 at 15:47
• @whuber Yes, exactly. That approach is common in the engineering literature, less so in the mathematical literature. Commented Dec 19, 2021 at 19:53

• Thank you so much Massimo Ortolano. Also, I was thinking, is not a bit rare that if we define a white noise (in the continuous sense) as an indexed list of independent and identically distributed r.a. with variance $\sigma^2$, the correlation ends up giving $Var(N_t)=\sigma^2\delta(0)\neq\sigma^2$ ? Commented Dec 21, 2021 at 19:19