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Part BCSIR NET June 2025the-state-space-is-the-2025-residues-not-the-2024-step-values

The state space is the 2025 residues not the 2024 step values

Let Z1,Z2,…Z_{1}, Z_{2}, \dots be a sequence of independent and identically distributed random variables having discrete uniform distribution over {1, 2, …, 2024}. Let Yn=∑iY_{n} = \sum_{i}₌1nZi,n≥2_{1}^{n} Z_{i}, n \ge 2. Further, let XnX_{n} be the remainder when YnY_{n} is divided by 2025. Then, which of the following statements is true?

  1. A.limn→∞P(Xn=0)=1/2026lim_{n}\to\infty P(X_{n} = 0) = 1/2026
  2. B.limn→∞P(Xn=0)=1/2025lim_{n}\to\infty P(X_{n} = 0) = 1/2025✓ Correct
  3. C.limn→∞P(Xn=0)=1/2024lim_{n}\to\infty P(X_{n} = 0) = 1/2024
  4. D.limn→∞P(Xn=0)=1/2023lim_{n}\to\infty P(X_{n} = 0) = 1/2023

You have the answer. Trap Analysis is why the other three were written.

Not a worked solution repeated four times — the specific reasoning error each wrong option was built to reward, and why this one tests standard counterexample.

See pricing

50 are analysed free — try those first.

The trap it tests

Standard counterexample

There is a canonical object that settles this. Recognising it is the whole question.

Drill statements like this

Related counterexample: An irreducible chain with a stationary distribution converges to it

More on this topic

The chapter behind this: Markov chains: classification and stationary behaviour — free to read

From Limit Theorems and Markov Chains › Markov chains: classification of states, stationary distributions

Last revised . Found a mistake? Tell us — corrections are the fastest thing we act on.

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