# Markov Chain generated by iterations of functions

$$X_n$$ is a Markov Chain on (..,-2, -1, 0, 1,..) obtained by random iterations with functions $$f_1(x)=x+2$$, $$f_2(x)=x−1$$, $$f_3(x)=0$$. In each iteration step we choose function to iterate with equal probabilities. The MC can thus be represented as $$X_{n+1}=f_{I_{n+1}}(X_n)$$ where $$I_n$$ is a sequence of independent random variables $$P(I_n=1)=P(I_n=2)=P(I_n=3)=1/3$$ for each fixed $$n$$. If $$X_0=0$$, how can I determine the realisations of $$X1$$, $$X2$$, $$X7$$ corresponding to the realisations $$i_1=1$$, $$i_2=1$$,..., $$i_6=3$$, $$i_7=2$$ of $$I_1$$, $$I_2$$,..., $$I_6$$, $$I_7$$?

• Have you tried computing $X_1$? You know that the function to use for this first step is $f_1$. – Kenny Wong Feb 18 at 22:17
• @KennyWong Yes I calculated $X_1=0+2$, but I do not understand why to use $f_1(x)$ for $i_2$ too. – Natalie_94 Feb 19 at 12:01
• Is $X_7=-1$ because $X_6=f_3(X_5)=0$? – Natalie_94 Feb 19 at 12:12
• You're told that $i_2 = 1$, so $f_1$ should be used to compute $X_2$. And yes, you're right about X_7\$ – Kenny Wong Feb 19 at 12:17