What Are Transition Matrices, Irreducibility, Periodicity, Stationary Distribution and Convergence ?

Learn Stochastic Processes- Simplified , Conceptual And Exam Focused Way- 2 Hours Live Power Session- CS2 Actauarial Science

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0:00 – CS2 April–May 2026 focus. Why exam questions build intuition
1:00 – Valid vs invalid transition matrices. Row sums, square matrices, non-negativity
4:30 – Drawing transition graphs from matrices
5:30 – Irreducibility and periodicity introduced
7:00 – Path-based definition of irreducibility
9:20 – Sales staff Markov chain exam problem
11:00 – Discrete time interpretation. Weekly observation
12:30 – Multi-step transitions using matrix powers
14:40 – Long-term proportions and equilibrium idea
16:20 – Key theorem. Finite state space guarantees stationarity
18:45 – Irreducibility implies uniqueness
20:15 – Reducible chains and trapping states
23:30 – Population flow intuition before equations
27:45 – Solving stationary equations
30:00 – Why infinite stationary solutions occur
33:00 – Reducible vs irreducible contrast
37:30 – Long-run behaviour without convergence
42:45 – Meaning of “at least one” stationary distribution
45:00 – Final irreducibility check logic
47:30 – Irreducible two-state example
50:50 – Periodic chain example introduced
52:10 – Stationary distribution exists but oscillation continues

53:10 – Most confusing CS2 concept explained
55:30 – Stationarity vs convergence distinction
57:00 – Hospital example with initial distribution
58:45 – Aperiodic chain reaches stationarity
1:00:40 – Periodic chain never reaches stationarity
1:03:15 – Stationary does not mean reachable
1:05:00 – Convergence formally defined
1:07:30 – House to airport analogy
1:12:30 – Final theorem. Irreducible + aperiodic ⇒ convergence
1:14:30 – Period calculation using return times
1:17:00 – Periodic vs aperiodic comparison
1:20:00 – Why one chain settles and the other oscillates

1:22:00 – No Claim Discount system introduction
1:24:15 – States as discount levels
1:25:00 – Promotions and demotions intuition
1:27:30 – Modified NCD rules
1:29:30 – Why original model is not Markov
1:32:30 – State expansion idea
1:34:15 – Creating plus/minus states
1:36:00 – Converting to valid Markov chain
1:38:50 – High-value exam insight
1:44:30 – Preview of Poisson process and random walk

You are preparing for the IFoA CS2 April–May 2026 exam.
This session builds core Markov chain intuition from exam-style questions.

You start with validity checks for transition matrices.
You learn how to draw transition graphs step by step.
You develop irreducibility and periodicity using logic, not memorisation.
You connect short-term transitions with long-term behaviour.
You understand when stationary distributions exist, when they are unique, and when there are infinitely many.
You see why reducible chains break uniqueness.
You see why irreducible chains guarantee uniqueness.
You see why periodic chains still have stationary distributions but behave differently over time.

This class is designed to help you score marks, avoid common traps, and think like an examiner.

TOPICS COVERED
• Valid vs invalid transition matrices
• Square matrix condition
• Non-negativity of probabilities
• Transition graphs from matrices
• Irreducibility definition with paths
• Reducible chains and trapping states
• Exam-style sales staff Markov chain problem
• Multi-step transitions using matrix powers
• Long-term proportions and equilibrium
• Stationary distribution equations
• Finite state space theorem
• Unique vs multiple stationary distributions
• Periodic vs aperiodic chains
• Intuition before algebra
This session covers the most misunderstood CS2 topic.
Over 90 percent of students struggle here.

You learn the difference between having a stationary distribution and converging to it.
You see why some chains never converge even though a stationary distribution exists.
You understand periodicity using intuition, numbers, and exam logic.
You learn why irreducible is not enough.
You see why aperiodicity is required for convergence.
You convert a real insurance No Claim Discount system into a valid Markov chain by state expansion.
This part is high-yield for exams and industry intuition.