Question

Consider a Markov chain on \( \mathbb{Z} \) with transition probability \( p_{i, i+1}=1 \), for all \( i \in \mathbb{N} \). Show that the invariant measure is of the form \( \mu(i)=c \) for all \( i \

Answer 1

The invariant measure for the Markov chain on
`\( \mathbb{Z} \)` with transition probability
`\( p_(i, i+1)=1 \)`for all
`\( i \in \mathbb{N} \)` is
`\( \mu(i)=c \)`for all i , where c is a constant.

Step-by-step explanation:

In a Markov chain, an invariant measure is a probability measure that remains unchanged by the transition probabilities. For the given Markov chain on
`\( \mathbb{Z} \)`, with the transition probability
`\( p_(i, i+1)=1 \)`, the chain only moves to the right, indicating that the probability mass is shifted to the right at each step. This implies that the probability distribution does not depend on the specific position i leading to a constant invariant measure.

Mathematically, for an invariant measure
`\( \mu(i) \)`, we require
`\( \mu(i+1) = \mu(i) \)` for all i . Since
`\( p_(i, i+1) = 1 \),` the transition probability from i to i+1 is certain. Therefore,
`\( \mu(i+1) = \mu(i) \),` ensuring that the probability mass remains constant as the chain progresses.
`\( \mu(i+1) = \mu(i) \)`

The constant
`\( c \) in \( \mu(i) = c \)` represents the invariant measure, and its value can be determined by normalization, ensuring that the probabilities sum up to 1.