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Transition Probability

The probability that a Markov chain moves from one specific state to another, over one step or over $n$ steps.

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In Plain English

A transition probability is the single number answering the question, if I'm in state $i$ right now, what's the chance I land in state $j$ next? Collect all of these numbers for every pair of states and you get the transition matrix; look several steps ahead instead of just one, and you get the $n$ -step transition probability, the chance of landing in $j$ after $n$ jumps starting from $i$ .

Definition

The one-step transition probability $p_{ij} = P (X_{n + 1} = j ∣ X_{n} = i)$ is the probability that a Markov chain currently in state $i$ moves to state $j$ at the next time step; for a time-homogeneous chain this does not depend on $n$ . The $m$ -step transition probability $p_{ij}^{(m)} = P (X_{n + m} = j ∣ X_{n} = i)$ generalizes this to $m$ steps ahead, and is computed via the Chapman-Kolmogorov equation (equivalently, the $(i, j)$ entry of $P^{m}$ ). Time-homogeneity, that $p_{ij}$ does not change with $n$ , is the standard default assumption; a chain whose transition probabilities do vary over time is called time-inhomogeneous.

Formula

p_{ij}^{(m)} = P (X_{n + m} = j ∣ X_{n} = i)

Notation

p_{ij}

—One-step transition probability from state $i$ to state $j$

p_{ij}^{(m)}

—$m$-step transition probability, from $i$ to $j$ in exactly $m$ steps

Properties

Individual transition probabilities are the atoms of Markov chain theory, but nearly every practical question, stationary distributions, hitting probabilities, expected return times, is answered by combining them via matrix algebra rather than reasoning about single transitions in isolation.
For a chain to be time-homogeneous the transition probabilities must be constant over time; many real applications (a queue whose arrival rate changes by hour) are only approximately homogeneous, and violating this assumption invalidates simple closed-form results.
In continuous-state settings the transition probability $p_{ij}$ generalizes to a transition density or transition kernel $p (x, d y)$ , describing the probability of the next state landing in an infinitesimal neighborhood of $y$ given the current state $x$ .

At a Glance

One-step case

p_{ij} = P (X_{n + 1} = j ∣ X_{n} = i)

n-step case

p_{ij}^{(n)}

, computed via

P^{n}

or the Chapman-Kolmogorov equation

Time-homogeneous chain

p_{ij}

is the same at every time step

n

Collected intothe transition matrix (finite state space) or transition kernel (general state space)

Last updated September 8, 2026← Back to the Dictionary