Stochastic Processes · 免费预览
耦合与收敛定理
并行运行该链的两个独立副本，直到它们相遇，然后让它们一起移动，即可将其在时刻 n 的分布之间的距离限制为尚未相遇概率的两倍，这就是收敛定理的证明。
1. Two copies on one clock
Two copies on one clock
2. The meeting time is almost surely finite
The meeting time is almost surely finite
3. After the meeting the futures agree
After the meeting the futures agree
4. The theorem falls out
The theorem falls out
5. What a coupling is
In this proof, what exactly is the coupling that is constructed?
6. Why the copies must meet
Why is the coupling time T finite with probability one?
7. The key equality after meeting
Why do the events X_n equal y and Y_n equal y have the same probability once the copies have met by time n?
8. Reading the coupling inequality
What does the coupling inequality say about the distance between the two time-n distributions?
9. The role of the stationary copy
Why is Y_n guaranteed to have distribution pi at every time n?
10. Where aperiodicity enters
Where exactly does aperiodicity enter the coupling proof?
11. A bound from a slow meeting
If the coupling time T satisfies P of T greater than 20 equal to 0.03, use the coupling inequality to bound the absolute difference between p to the 20th of x, y and pi of y, and write the number as a decimal.
12. A tighter window
If P of T greater than 10 equals 0.05, bound the absolute difference between the distribution of X ten and pi of y at any state y, and write the number as a decimal.
13. The stationary target of the two-state chain
For the two-state chain with rows 0.5 and 0.5, then 0.3 and 0.7, the coupling theorem says p to the n of 1, 2 approaches the stationary weight of state 2. Compute that limit and write it as a fraction or decimal.