Stochastic Processes · 免费预览
返回固定状态
访问某个固定状态之间的时间独立同分布，因此由强大数定律，它们的均值就是长期访问率，即平均返回时间的倒数；该值同时也是平稳权重。
1. Cycles cut the path into pieces
Cycles cut the path into pieces
2. Counting visits with a sandwich
Counting visits with a sandwich
3. Stationary weights are reciprocal return times
Stationary weights are reciprocal return times
4. The cycle trick and rewards
The cycle trick and rewards
5. Why return gaps are i.i.d.
Why are the times between successive returns to a fixed state y independent and identically distributed?
6. Reading the return-time law
If R of k is the time of the k-th return to y, what does the strong law of large numbers conclude about R of k over k?
7. A quick return for the weather
The weather chain has stationary distribution one third rain and two thirds sun. What is the mean return time to rain, E of rain, T of rain?
8. Averages beat oscillation
The Ehrenfest chain with three balls has period 2, so its matrix powers do not converge. What happens to the fraction of time spent in state 1?
9. Uniqueness from the bridge identity
Why does the identity pi of y equals one over the mean return time imply that an irreducible chain has at most one stationary distribution?
10. The sandwich argument
In the proof of the frequency theorem, what role do the inequalities R of N_n(y) at most n less than R of N_n(y) plus 1 play?
11. Days between rainy returns
For the weather chain with stationary weights one third for rain and two thirds for sun, compute the mean return time to rain and write the number.
12. Days between sunny returns
For the same weather chain, compute the mean return time to the sunny state and write the answer as a decimal or a fraction.
13. The urn's slowest state
For the Ehrenfest chain with three balls, whose stationary weights are one eighth, three eighths, three eighths and one eighth, compute the mean return time to state 0, the empty left urn, and write the number.