Stochastic Processes · 免费预览
无限状态空间
当状态数无限时，遍历性不再能保证平稳分布的存在；反射随机游走分化为正递推、零递推和暂留三种情形，而分支过程则通过生成函数的最小不动点呈现同样的三分法。
1. A walk that can drift forever
A walk that can drift forever
2. The borderline and beyond
The borderline and beyond
3. Extinction as a fixed point
Extinction as a fixed point
4. Three regimes, one theorem
Three regimes, one theorem
5. A geometric equilibrium
For the reflecting random walk with p below one half, what does detailed balance give for the stationary distribution?
6. The meaning of null recurrence
The reflecting walk at p equal one half returns to zero with probability one, yet is called null recurrent. What does null recurrence mean?
7. Certainty of return in the fair case
How does the fair ruin formula P of x of V N less than V 0 equal x over N imply that the fair reflecting walk returns to zero with probability one?
8. Drift means escape
For the reflecting walk with p above one half, what is the probability of ever returning to zero from a starting height x?
9. Which fixed point to pick
The extinction probability theta satisfies phi of theta equal to theta, but the equation can have several roots. Why is theta the smallest solution in the interval from zero to one?
10. The critical tangency
For a branching process with mean offspring exactly one, p one below one, what happens?
11. A patient return to zero
For the reflecting random walk with p equal to 0.4, compute the mean return time to zero, E of zero T zero, and write the number.
12. A family line on the brink
In binary branching each individual has two children with probability 0.6 and none with probability 0.4. Compute the extinction probability and write it as a fraction or decimal.
13. Three children, fair coins
Each individual has a number of children with the binomial distribution Binomial of 3 and one half, so phi of x is the quantity one plus x cubed over eight. The extinction probability is the smallest fixed point, equal to the square root of five minus two. Compute it and write the answer as a decimal.