selftransitions

Self-transitions refer to transitions in a stochastic process where the next state is the same as the current state. In discrete-state Markov chains, the probability of a self-transition from state i to itself is denoted p_ii and appears on the diagonal of the transition matrix P, with P_ij = Pr(X_{t+1} = j | X_t = i).

Self-transitions determine how long the process tends to remain in a given state. The number of consecutive steps spent in i before moving to another state follows a geometric distribution with parameter 1 − p_ii, giving an expected dwell time of 1/(1 − p_ii). A higher p_ii implies greater persistence in that state and longer sequences of repeats; p_ii = 1 yields an absorbing state.

Estimating self-transition probabilities from data is standard in Markov modeling: p_ii is estimated as N_{ii} / N_i, where N_{ij} is the count of observed transitions from i to j and N_i = sum_j N_{ij}. In hidden Markov models, self-loops are common and influence the inferred state sequences and emissions. In continuous-time Markov chains, holding times rather than discrete self-transitions characterize persistence, with diagonal rates determining the likelihood of remaining in a state over a time interval.