Assignment 1.3

Determine and derive the generating function of the recurrent time from state 0 to state 0

Solution of Assignment 1.3

Let p_ij⁽ⁿ⁾  denotes the probability of transition from state i  to state j in exactly n steps, and f_ij⁽ⁿ⁾  denotes the probability of arriving at state j at time n for the first time, given that the process starts at state i.

Let:

,  |s|<1

,   |s|<1

be the generating function of the sequence {p₀₀⁽ⁿ⁾} and {f₀₀⁽ⁿ⁾} respectively