Consider the two-state Markov chain with the following...

Consider the two-state Markov chain with the following transition matrix.
$$
\mathbf{P}=\left[\begin{array}{cc}
p & 1-p \\
1-p & p
\end{array}\right]
$$
Find a simple expression for $P_{0,0^{\circ}}^{t}$

Key Concepts

Markov Chain

A Markov chain is a stochastic process that undergoes transitions from one state to another on a state space. It is characterized by the memoryless property, meaning that the probability of a future state depends solely on the current state and not on the sequence of events that preceded it.

Transition Matrix

The transition matrix is a mathematical representation that describes the probabilities of moving from each state to every other state in a Markov chain. Each entry in the matrix corresponds to the probability of transitioning from one state to the next in a single time step.

t-Step Transition Probabilities

The t-step transition probabilities describe the likelihood of moving from one state to another after t transitions. These probabilities can be obtained by raising the transition matrix to the power t, which accumulates the effect of the single-step transitions over multiple steps.

Eigenvalues and Diagonalization

Diagonalization is a matrix factorization method used to simplify the calculation of matrix powers. By expressing the transition matrix in terms of its eigenvalues and eigenvectors, the t-step transition probabilities can be derived in a compact form. In many symmetric or two-state Markov chains, one eigenvalue is 1 (representing the stationary aspect of the chain), while the other is related to the transition probability, allowing for a neat closed-form solution.

Transcript

00:01 Okay, we want to use the induction to prove the power of a matrix, of a special matrix.

00:10 Okay, first notice when n is equal to 1, they distribute because by expression, p of 1 will be equal to 1 half plus 2 over 2p minus 1, and 1 half minus 2 over 2p.

00:36 Here we have the same thing.

00:46 And just do some simple computation, we know this is equal to p.

00:52 Here we have 1 minus p, we have p here, and 1 minus p here.

00:59 So this is equal to p.

01:01 So when n is equal to 1, the result is trivial.

01:06 Now let's, we want to do the induction.

01:08 So we just suppose for any s which is greater or equal to 1 and less equal to n, our statement is true.

01:17 We want to consider s is equal to n plus 1.

01:23 Okay, by the definition, p to the power n plus 1 can be always written as p times p to the power n.

01:34 Okay, and by our assumption, when p is equal to n, then the inductive hypothesis applies.

01:43 I mean the statement is true, so it can be written as p is equal to, this is our p, and p to the power n, for p to the power n we can use our formula.

01:58 So it is equal to 1 over 2 plus, to make it easy, 1 plus 2p plus n, here we have 1 minus 2p minus n to the power n.

02:24 We have 1 plus 2p minus 1 to the power n over 2.

02:30 And we only need to do this multiplication between matrices.

02:37 Okay, let's do it explicitly.

02:40 So the first term will be equal to 2p plus p times 2p minus 1 to the power n plus minus, and for the second one, we have the similar expression.

03:39 Okay, here we have plus p minus p times 2.

04:05 This is the whole term.

04:10 The last term is equal to 2p minus 1 to the power n plus 2 over p...

Please give Ace some feedback