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When studying Markov chains, the concept of a stationary distribution is fundamental. It's a probability distribution that remains unchanged as the system evolves over time. In simpler terms, once a Markov chain reaches its stationary distribution, the probabilities of being in each state no longer change with further steps.

In the case of the exercise, we define a stationary distribution for the sum of independent rolls of a fair dice as it relates to a multiple of 13. The chain comprises 13 states, representing the possible remainders when the sum is divided by 13. After a large number of rolls, the distribution of probabilities over these states stabilizes, and this is what we call our stationary distribution.For our fair die scenario, where the sum of rolls is divided by 13, the stationary distribution informs us about the long-term behavior of our probabilistic system. To find this stationary distribution, one typically solves the equation \[vP = v\]where 'v' is our stationary distribution and 'P' is the transition matrix. In practice, we look for a set of values for 'v' that doesn't change when it undergoes the transformation described by 'P' and whose elements sum up to 1. This represents the probabilities that the sum of dice rolls yields a remainder of 0,1,2,..., or 12 upon division by 13 after many trials.

In Markov chains, transition probabilities are all about chances: specifically, the chance of moving from one state to another in a single step. It encapsulates the essence of Markovian systems - their memorylessness. The future state depends only on the current state, not on the sequence of events that preceded it.In our dice rolling example, the transition probabilities are the chances that the remainder when dividing the sum of dice rolls by 13 changes from one value to another with an additional roll. Mathematically, this is represented as \[P_{ij} = P(X_{n+1} = j\,|\,X_n = i)\]where 'i' is the current state and 'j' is the next state. With a fair die, every face has an equal chance of appearing, affecting the transitions between states in a predictable manner. The transition matrix 'P' that we create holds all these probabilities and is the cornerstone for calculating the stationary distribution as well as other long-term behaviours of the Markov chain.

Understanding limiting probabilities bridges the gap between short-term randomness and long-term predictability in Markov chains. As the name suggests, limiting probabilities are what the transition probabilities approach as the number of steps goes to infinity. It's about the end game, the final set of probabilities when the process has stabilized and time is no longer a factor.Leveraging this concept, for n independent rolls of a die, we're looking to discover \[\lim_{{n \rightarrow \infty}} P(Y_n\, \text{{is a multiple of}}\, 13)\]This is essentially the probability that after many, many rolls, the sum of those rolls will be a multiple of 13. We arrive at this through the stationary distribution which, for large 'n', the system has virtually settled into. If you were to continue rolling the die indefinitely, the limiting probability provides the assurance of what to expect in the long run. In the exercise, by examining the stationary distribution found, we determine that the limiting probability of the sum being a multiple of 13 converges to approximately 0.02381.