Sample functions of stochastic processes violate Dirichlet conditions. Why?

Suppose we have a real-valued periodic function that satisfies the Dirichlet conditions. In this case its Fourier series converges. With some expedients this concept can be extended also to non-periodic functions.

Suppose now that this function is a realization of a stochastic process. It is commonly argued that in this case it violates the Dirichlet conditions so that its Fourier transform no longer converges. For this reason it becomes necessary to define the correlation function. Indeed, it satifies the Dirichlet conditions and its Fourier transform is said power spectral density.

I have two questions on this topic:

1. Which conditions are violated by a realization of a stochastic process? and why?
2. Why are the correlation function and the power spectral density so important?

Thank you