# Tagged Questions

Stochastic processes (with either discrete or continuous time dependence) on a discrete (finite or countably infinite) state space in which the distribution of the next state depends only on the current state. For Markov processes on continuous state spaces please use (markov-process) instead.

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### spectral gap of the graph / Markov chain

Let $G=(V,E)$ be a graph. The gradient of a function $f:V\longrightarrow R$ is defined on the edges of the graph, given by the discrete derivative $$\nabla f(e)=f(y)-f(x), \quad e=(x,y) \in E$$ ...
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### Gradient of a function on the vertices of a graph

Let $G=(V,E)$ be a graph. The gradient of a function $f:V\longrightarrow R$ is defined on the edges of the graph, given by the discrete derivative $$\nabla f(e)=f(y)-f(x), \quad e=(x,y) \in E$$ ...
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### Markov chain supplementary litterature

I'm studying markov chains through Durrett and I'm finding it quite hard to read. Does anyone have a good idea to supplementary book, preferably one on his level of generality and which studies some ...
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### How to sample the walk which visits each vertex of a graph specific number of times?

Is there any MCMC mathod that allow me to uniformly sample from all feasible walks where the following restrictions apply: ...
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### How to apply the strong Markov property in this case?

I'm trying to understand the following proof: Theorem: Let $(X_n)$ be an irreducible $(\alpha, \mathbf p)$-Markov chain with a finite state space $S$. Then $(X_n)$ is positive recurrent. ...
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### A discrete time Markov chain with such a transient state that $\mathbb P(T_i<\infty \ | \ X_0=i) \neq 0$

All examples of discrete time Markov chains my text provides are where $S$ is finite, and as far as I can tell, it makes all transient states have $$\mathbb P(T_i<\infty \ | \ X_0=i) = 0.$$ Are ...
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Here's my question: Take three numbers $x_1$, $x_2$, and $x_3$, and form the successive running averages $$x_n = \frac{x_{n-3} + x_{n-2} + x_{n-1}}{3}$$ starting with $x_4$. Prove that $$\lim_{n \to \... 2answers 174 views ### Calculate average time to empty the router Consider a buffer, in which every second the number of packets increases by 1 with probability .4 and decreases by 1 with probability .6. Currently there are n packets in the router. ... 1answer 509 views ### Markov chains - transition matrix - probabilty formula and application help? Consider the 3\times3 transition matrix$$ \begin{array}{c|ccc} &A &B &C \\ \hline A &0.2 & 0.3 &0.5\\ B &0.3 & 0.5 &0.2\\ C &0.3 & 0.3 & 0.4 \end{...
I'm doing a branching process problem and am not sure I did it correctly. Suppose $X_0 = 1$ and $p_0 = .5, p_1 = .1,$ and $p_3 = .4$ represent the probabilities that zero, one, and three individuals ...