The Markov process is one of the part of stochastic process in which it is necessary for us to satisfy a Markovian equation:

P {X_{t + 1} = j | X_{0} = k_{0}, X_{1} = k_{1}… , X_{t-1} = k_{ }_{t-1} ,X_{ }_{t}= i}

= P {X_{ }_{t+1} = j | X_{t} = i} for t = 0, 1, …

This equation shows that the environment that is going to occur in future depends directly and only on the environment of the next day.

**There are two of transition probability**:

- First one is known as
**one step transition probability**which is given by the equationP { X_{t+1}= j | X_{t}= i} which shows the process between time period t and t+1.

Further a one-step conditional probability is divided into **stationary** and **non-stationary** probabilities. The stationary one-step conditional probability is shown by the equation:

P {X_{ }_{t+1} = j | X_{t} = i} = P {X1 =j | X_{0} = i} for all t = 0, 1, …

Where transition remain fixed with time. This equation is denoted by P_{ij. }

- Another type is known as
**n-step transition probability**which is shown by the equation:

P {X_{t+n}=j | X, = i} = P {Xn = j | X_{0}= i} for all t = 0, 1, ..,

And is denoted by p_{ij}^{(n)} for all i and j.

Both the equation will further satisfy two important properties:

- p
_{ij}^{(n )}≥ 0 for all i and j and n = 0,**1,**2, … - ∑ p
_{ij}^{(n )}=1 for all i andn = 0, 1, 2, … , and M =No. of states.

**Matrix form of transition probabilities**:

Transition of conditional probabilities can also be represented in matrix form.

**N-step transition probabilities**:

**One-step transition probabilities**:

Thus, we see from the above matrix forms that P denotes one-step transition probabilities and P^{(n) } is called n-step transition probabilities.

Finite state Markov chain:

A countable number of forms, the Markovian property, stationary transition probabilities and the set of probabilities P {X_{0} = i} together satisfy the finite state of Markov theorem of a stochastic process {X1} (t=0,1,2,…).