Measure how good is the discrete approximation of continuous random process?

If continuous random random process approximated with discrete version - how to measure how good is the approximation?

One possible way is to compute and compare the moments. But for some probability distributions the second and greater moments could be infinite.

Example:

Continuous process defined over continuous random variables $$C_{1},C_{2},...$$ where $$S_{0}=0\,\!$$ and $$S_{n}=\sum _{j=1}^{n}C_{j}$$. Random variables taken from continuous distribution, let's say Pareto distribution.

It's approximated with discrete process over discrete random variables $$D_{1},D_{2},...$$ where $$S_{0}=0\,\!$$ and $$S_{n}=\sum _{j=1}^{n}D_{j}$$. Where random variables $$D_{n}$$ taken from discrete distribution and can take only $$k$$ possible values $$V_{1}, V_{2}, ..., V_{k}$$.

Let's say we choose $$k = 5$$, guessed somehow 5 discrete values and fitted the Markov Chain. This is our discrete approximation.

Question 2: How to find optimal discrete values for given $$k$$? If it helps the original process has Pareto distribution.