Suppose I repeatedly roll a standard six-sided die and...

Suppose I repeatedly roll a standard six-sided die and obtain a sequencé of independent random variables $X_{1}, X_{2} \ldots .$ where $X_{i}$ is the outcome of the $i$ th roll. Let $$ Y_{j}=\sum_{i=1}^{j} X_{t} \bmod 10 $$ be the sum of the first $j$ rolls considered modulo 10 . The sequence $Y_{j}$ forms a Markow chain. Determine its stationary distribution, and determine a bound on $\tau(\varepsilon)$ for this chain. (Hint: One approach is to use the method of Exercise 11.4.)

Suppose I repeatedly roll a standard six-sided die and obtain a sequencé of independent random variables $X_{1}, X_{2} \ldots .$ where $X_{i}$ is the outcome of the $i$ th roll. Let
$$
Y_{j}=\sum_{i=1}^{j} X_{t} \bmod 10
$$
be the sum of the first $j$ rolls considered modulo 10 . The sequence $Y_{j}$ forms a Markow chain. Determine its stationary distribution, and determine a bound on $\tau(\varepsilon)$ for this chain. (Hint: One approach is to use the method of Exercise 11.4.)

Key Concepts

Markov Chain

A Markov chain is a stochastic process that undergoes transitions from one state to another in a state space, with the key property that the probability of moving to the next state depends only on the current state and not on the sequence of states that preceded it. This memoryless property makes Markov chains particularly tractable and is fundamental to modeling processes where past history is summarized entirely by the present state.

Stationary Distribution

The stationary distribution of a Markov chain is a probability distribution that remains unchanged as the chain evolves over time. When the chain starts in the stationary distribution, it will have the same distribution at every subsequent step. For irreducible and aperiodic finite-state Markov chains, the stationary distribution is unique and often can be computed by balancing the flow between states or by exploiting symmetry, as is the case with symmetric additive processes like a modular arithmetic random walk.

Modulo Arithmetic Random Walk

A modulo arithmetic random walk is a process where the state is updated by adding a random increment and then reducing the result modulo some fixed number. This setup creates a periodic state space (a finite cyclic group), in which properties such as irreducibility and periodicity can be analyzed using group theoretic and probabilistic techniques. The uniformity of the stationary distribution often follows from the symmetry inherent in the addition operation modulo the chosen modulus.

Mixing Time

Mixing time is a measure of how fast a Markov chain converges to its stationary distribution. It quantifies the number of steps needed so that the distribution of the state of the chain is close to stationary within a specified tolerance, typically measured in terms of total variation distance. A shorter mixing time implies that the chain ’forgets’ its initial condition quickly.

Coupling Method

The coupling method is a probabilistic technique used to bound the mixing time of a Markov chain. It involves constructing two copies of the Markov chain on the same probability space, started from different initial states, and then designing a joint process (the coupling) such that the two copies eventually come together (or couple) and evolve identically thereafter. Analyzing the coupling time provides direct bounds on how quickly the chain converges to the stationary distribution.

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