3 votes
Accepted

Why does the Wiener process use $\sqrt{dt}$ instead of $dt$? Why does simulation of random walk in continuous-time not occur as expected?

Essentially because the variance scales quadratically, e.g. $\operatorname{Var}(tX) = t^2\operatorname{Var}(X)$. I'm only going to talk about discrete time processes, rather than something like ...
daisies's user avatar
  • 1,548
2 votes
Accepted

In a discrete-time Markov chain, is the probability of not being able to get from a state $u$ to $v$ a rational number?

If the transition probabilities are all rational, and the Markov chain is finite, then essentially all probabilities describing the Markov chain's behavior will also be rational, and that extends to ...
Misha Lavrov's user avatar
1 vote

Requirements for a Markov chain to converge to its stationary distribution.

For an example, see my comment on the discrete time Markov chain (DTMC) with $S=\{1,2\}$ and $P_{12}=P_{21}=1$, which is irreducible, recurrent, and periodic and has a solution $\pi = (1/2,1/2)$ to $\...
Michael's user avatar
  • 24k
1 vote

Properties of a transient state in a Markov Chain

As you have pointed out, $\sum\limits_{n=1}^{\infty}P_{jj}^{n}<\infty$, and that $\mathbb{P}_{ij}(s)=\mathbb{F}_{ij}(s)\mathbb{P}_{jj}(s)$, what you need follows directly from Abel's theorem, since ...
Sparkle-Lin's user avatar

Only top scored, non community-wiki answers of a minimum length are eligible