Final answer:
The Markov chain is proved to be transient by applying the strong law of large numbers, highlighting that without a convergence to long-term equilibrium, the chain does not return to a state with probability 1.
Step-by-step explanation:
The question asks for a proof that a Markov chain is transient when certain conditions are met for the probability of moving to a different state, using the strong law of large numbers. The strong law of large numbers states that as the number of trials in a probability experiment increases, the difference between the theoretical probability of an event and the relative frequency probability approaches zero. This implies that for a random walk Markov chain, if the probability of moving in one direction is greater than 0 and less than 1, the process will never settle into a long-term equilibrium, indicating that the chain is transient. In other words, the probability of returning to the original or a previous state does not converge to 1 if p satisfies conditions a (p > 0), b (p = 1), c (p = 0.5), or d (p is less than 0) since the process has a drift or no tendency to return.