Stochastic Processes and Time Series
GATE ST tests Markov chains, Poisson processes, and stationary time-series models. You should move between transition probabilities, stationary distributions, autocovariances, and model identification.
Core rules
- A Markov chain satisfies the Markov property. The n-step transition matrix is Pⁿ. A stationary distribution satisfies π=πP with probabilities summing to 1.
- A Poisson process has independent stationary increments; the count in an interval of length t is Poisson(λt), and interarrival times are exponential(λ).
- Weak stationarity requires constant mean and autocovariance depending only on lag. Strict stationarity concerns the full joint distribution.
- For ARMA models, check stationarity and invertibility through characteristic roots. AR(1) Xt=φXt−1+εt is stationary when |φ|<1, with ACF φ^k.
- ACF and PACF patterns help identify candidate models, but estimation and diagnostics are still required.
Worked example
A two-state chain has P=[[0.9,0.1],[0.2,0.8]]. Let π=(π1,π2). Stationarity gives π1=0.9π1+0.2π2 and π1+π2=1. From the first equation, 0.1π1=0.2π2, so π1=2π2. Hence π2=1/3 and π1=2/3. The long-run fraction of time in state 1 is 2/3, provided the chain is irreducible and aperiodic.
Traps
Do not mix row-stochastic and column-stochastic conventions. A stationary distribution is not automatically the limiting distribution unless the chain is irreducible and aperiodic. For AR models, stationarity depends on roots lying outside the unit circle, not simply on the sign of a coefficient. Differencing can induce non-invertibility, so check the transformed series.