Stochastic Processes · 免费预览
极限行为
具有平稳分布的不可约非周期链，其矩阵幂逐行收敛到平稳向量；即使周期性阻碍了收敛，平均矩阵幂和停留时间占比仍会稳定于该向量。
1. What can block convergence
What can block convergence
2. The convergence theorem
The convergence theorem
3. Fractions of time and average cost
Fractions of time and average cost
4. A machine's repair bill
A machine's repair bill
5. The period of a state
How is the period of a state x defined, and what makes a state aperiodic?
6. Ehrenfest never converges
Why do the matrix powers of the three-ball Ehrenfest chain fail to converge?
7. One self loop is enough
An irreducible chain has p(x, x) greater than 0 at one single state x. What follows?
8. Reading the convergence theorem
For an irreducible aperiodic chain with stationary distribution pi, what does the convergence theorem say about p to the n of x, y?
9. Long-run fraction of time
For an irreducible recurrent chain with stationary distribution pi, the long-run fraction of time spent in state y converges to which quantity?
10. Cost accounting with the stationary vector
A chain has stationary distribution pi and a cost function f. Which expression gives the long-run average cost per step?
11. The limit of the rainy column
For the weather chain with stationary distribution one third rain and two thirds sun, compute the limit of p to the n of sun, rain as n grows and write the answer as a fraction or decimal.
12. Ehrenfest's return rhythm
Compute the period of each state of the three-ball Ehrenfest chain and write the number.
13. Type 1 parts over five years
In the repair chain, the stationary weights of states 12 and 13 are 22 over 8910 and 60 over 8910, and one type 1 part is used on each visit to either state. Compute the long-run average number of type 1 parts used per day and write the answer as a fraction.