Problem 4 Consider a process \(\left\\{X_{... [FREE SOLUTION]

Chapter 4: Problem 4

Consider a process $\left\\{X_{H}, n=0,1, \ldots\right\\}$ which takes on the values 0,1 , or 2 . Suppose $$ \begin{aligned} &P\left\\{X_{n+1}=j \mid X_{n}=i, X_{n-1}=i_{n-1}, \ldots, X_{0}=i_{0}\right\\} \\ &\quad=\left\\{\begin{array}{ll} P_{i j}^{\mathrm{I}}, & \text { when } n \text { is even } \\ P_{i j}^{\mathrm{II}}, & \text { when } n \text { is odd } \end{array}\right. \end{aligned} $$ where $\sum_{j=0}^{2} P_{i j}^{\mathrm{I}}=\sum_{j=0}^{2} P_{i j}^{\mathrm{II}}=1, i=0,1,2 .$ Is $\left\\{X_{n}, n \geqslant 0\right\\}$ a Markov chain? If not, then show how, by enlarging the state space, we may transform it into a Markov chain.

Short Answer

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The given process $\left\\{X_{n}, n \geqslant 0\right\\}$ is not a Markov chain because the probability of the next state depends on whether the current step is even or odd, in addition to the current state. To transform this process into a Markov chain, we can enlarge the state space to include information about whether the current step is even or odd. We define a new process $\left\\{Y_{n}, n \geqslant 0\right\\}$, where $Y_n = (X_n, k)$ with $k \in \{0, 1\}$ indicating if the current step is even ($k=0$) or odd ($k=1$). The enlarged process $\left\\{Y_{n}, n \geqslant 0\right\\}$ is a Markov chain, with transition probabilities: $$ P\left\\{ Y_{n+1} = (j, k') \mid Y_n = (i, k) \right\\} = \left\\{ \begin{array}{ll} P_{ij}^{\mathrm{I}}, & \text{ when } k = 0 \text{ and } k' = 1 \\ P_{ij}^{\mathrm{II}}, & \text{ when } k = 1 \text{ and } k' = 0 \\ 0, & \text{ otherwise } \end{array} \right. $$

Step by step solution

To check whether a process is a Markov Chain, we need to verify that it satisfies the Markov Property: $P\left\\{X_{n+1}=j \mid X_{n}=i, X_{n-1}=i_{n-1}, \ldots, X_{0}=i_{0}\right\\} = P\left\\{X_{n+1}=j \mid X_{n}=i\right\\}$. This means the probability of the next state should only depend on the current state and not on previous states. #Step 2: Check the Markov Property#

In this case, the probability of the next state depends on whether the current step ($n$) is even or odd: $$ P\left\\{X_{n+1}=j \mid X_{n}=i, X_{n-1}=i_{n-1}, \ldots, X_{0}=i_{0}\right\\} = \left\\{\begin{array}{ll} P_{i j}^{\mathrm{I}}, & \text { when } n \text { is even } \\\ P_{i j}^{\mathrm{II}}, & \text { when } n \text { is odd } \end{array}\right. $$ Since the probability of the next state depends on whether the current step is even or odd (in addition to the current state), the process does not satisfy the Markov Property. Hence, the process is not a Markov chain. #Step 3: Enlarge the State Space#

To transform this process into a Markov chain, we can enlarge the state space to include information about whether the current step is even or odd. We can define a new process $\left\\{Y_{n}, n \geqslant 0\right\\}$, where $Y_n = (X_n, k)$, with $k \in \{0, 1\}$, indicating whether the current step is even ($k=0$) or odd ($k=1$). Now, the state space of the new process is $\{(0, 0), (0, 1), (1, 0), (1, 1), (2, 0), (2, 1)\}$. We need to find the transition probabilities for this new enlarged process. #Step 4: Transition Probabilities#

For the enlarged process, the transition probabilities are: $$ P\left\\{ Y_{n+1} = (j, k') \mid Y_n = (i, k) \right\\} = \left\\{ \begin{array}{ll} P_{ij}^{\mathrm{I}}, & \text{ when } k = 0 \text{ and } k' = 1 \\ P_{ij}^{\mathrm{II}}, & \text{ when } k = 1 \text{ and } k' = 0 \\ 0, & \text{ otherwise } \end{array} \right. $$ Now, for the enlarged process, the probability of the next state $(j, k')$ only depends on the current state $(i, k)$ and is independent of previous states. Thus, the enlarged process $\left\\{Y_{n}, n \geqslant 0\right\\}$ is a Markov chain.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Markov Property

The Markov property is a defining characteristic of a particular sequence of random variables, known as a Markov chain. Understanding the Markov property is essential for studying systems that undergo transitions from one state to another. In simple terms, the Markov property states that the future state of a process depends only on the present state, not on how the process arrived at that state.

For example, consider a board game where your next move is determined by rolling a die. In this case, your future position on the board depends only on your current position -- the roll of the die -- and not on the path you took to get there. This is an example of the Markov property in action. In the context of the given exercise, it's important to note that the probabilities of moving to the next state should be independent of past states for the process to be a Markov chain.

Transition Probabilities

Transition probabilities are essential components of Markov chains. They represent the likelihood of transitioning from one state to another in a single step. These probabilities are usually presented in a matrix format known as a transition matrix, where each entry represents the probability of moving from the state corresponding to its row to the state corresponding to its column.

In our board game analogy, think of it as having a list that tells you the odds of landing on any given square after rolling the die from your current location. Typically, these probabilities are constant over time in a traditional Markov chain. However, the exercise describes a situation where the transition probabilities alternate based on whether the step number is even or odd, which initially violates the consistent transition probability requirement for Markov chains.

State Space Enlargement

State space enlargement is a method used to convert a non-Markovian process into a Markov chain. By expanding the state space, we include additional information that makes the future state dependent only on the present state. It's a way of encapsulating past information relevant to the future transition into the current state description.

In the exercise, this is achieved by defining a new, larger state space that not only considers the system's state but also whether we are at an even or odd step. By doing so, the enlarged state space effectively captures the alternating nature of the transition probabilities and restores the Markov property, since the state now incorporates all necessary information to determine future evolution without referring back to any additional historical data.

Probability Models

Probability models are mathematical descriptions of random phenomena. They give us a formal way to study uncertainty and make predictions about the behavior of complex systems. Markov chains themselves are probability models, specifically used to model the sequence of events where the outcome of a future event depends only on the current state and not on the sequence of preceding events.

In our exercise, we see a process that does not initially adhere to the Markov chain model due to its dependency on an external factor (whether the step is even or odd). However, by enlarging the state space to include this information within the states themselves, we effectively create a new probability model. This model now satisfies the Markov property and can be studied and analyzed as a Markov chain.

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Short Answer

Step by step solution

Key Concepts

Markov Property

Transition Probabilities

State Space Enlargement

Probability Models

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