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The answer to my previous question has made it apparent there are two types of white noise:

1) Dirac delta correlated white noise $X_1(t)$ with variance $\sigma_1^2$ and correlation function $\rho(t_1-t_2) = \delta_1(t_1-t_2).$

2) Kronecker delta correlated white noise $X_2(t)$ with variance $\sigma_2^2$ and correlation function $\rho(t_1-t_2) = \delta_2(t_1-t_2)$ which is obtained when the correlation parameter of a process approaches 0.

I don't have a good source on this so please post links to further information. Dirac delta white noise is said to have infinite power and thus infinite variance. This makes intuitive sense to me since zooming in upon a realisation does not cause it to become smoother. According to the answer to my previous question the variance of Kronecker delta white noise is finite.

If I think of white noise as an infinite sequence of independent identically distributed random variables am I thinking of Dirac delta or Kronecker delta white noise?

Why does one process have infinite variance but the other does not?

Can you link me to information to better understand this topic?

Further more I am interested in the variance of the integral of these two processes:

$$I_1 = \int_0^L X_1(t)\, dt$$

$$I_2 = \int_0^L X_2(t)\, dt$$

The variance of the integral is given by (cross-validated source):

$$\text{Var}[I_1] = \sigma^2 \int_0^L \int_0^L \delta_1(t_1-t_2)\,\mathrm{dt_1\,dt_2} = \sigma^2L$$

However what is the variance of the integral of Kroncker delta correlated white noise? I think it should be zero.

$$\text{Var}[I_1] = \sigma^2 \int_0^L \int_0^L \delta_2(t_1-t_2)\,\mathrm{dt_1\,dt_2} = ?$$

EDIT: I'm starting to believe that the integral of Kronecker delta white noise is undefined according to this. Only Dirac delta white noise is valid in the integral.

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