4
votes
How long will it take for a coin to repeat a certain behavior?
I think you did this analysis correctly, some thoughts:
The study of mixing times is very interesting :) if you want are looking for a rigorous source, have a look at this book: Markov Chains and ...
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2
votes
How long will it take for a coin to repeat a certain behavior?
This is not exactly in line with your approach (Markov chains and mixing times), but it might be useful to gain some intuition.
First, notice that your initial question ("How long will it take ...
- 64.4k
2
votes
For any row stochastic matrix M, is $P_n=\frac{1}{n}\sum_{i=1}^n M^i$ always converging?
The matrix $P_n$ you've defined is called the Cesáro sum, and should rather be defined as $P_n = \frac{1}{n}\sum_{i=0}^{n-1} M^i$.
Let $\lvert\lvert{\cdot}||$ be the operator norm on matrices induced ...
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1
vote
What is the mean of the stochastic differential equation $dX=K dt + \sigma X dW$ and how to find it?
I will answer your comment before turning to your main question:
For your first result: $X(t)=X(0)e^{-\frac{\sigma^2}{2} t} e^{\sigma W_t}$. The average of $e^{\sigma W_t}$ is $e^{\sigma^2 t/2}$. This ...
- 243
1
vote
Optimal permutation of transition probabilities in random walk to minimize expected stopping time
I have a way of reformulating the problem that may be helpful:
Let $t_r$ be the expected stopping time starting from position $r$. Then we have, for $2 \le r \le n$:
$$t_r=p_r t_{r+1}+(1-p_r)t_{r-1}+1$...
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