Theorem 4.13 seems to say the same asTheorem 4.12 but in a more difficult way (gives generating function instead of individual probabilities). What is the reason for Theorem 4.13?
Sometimes it is more useful to know the generating function rather than the individual probabilities. Look at the discussion following Theorem 4.13 and notice how we then rapidly derive the first-return probabilities for random walks. It is also easier to find out mean values from generating functions — just differentiate and set argument equal to 1. Review the “crash course on generating functions” appendix and ST111X notes if you are still unsure about this.
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