$X_n$ 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 $I_1$, $I_2$,..., $I_6$, $I_7$?

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

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