# Tagged Questions

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### Can the transition probabilities of an inhomogeneous Markov chain be written as an exponential?

If $Z_t$ is a homogeneous continuous-time Markov chain with finite state space $E=\{1,\ldots,p\}$, transition matrices $(P(t))$ and intensity matrix $Q$, it holds that $$P(t) = \exp(tQ),$$ see for ...
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### Always null recurrence at the boundary between positive recurrence and transience?

I have the following theorem: Let $\rho$ be the traffic intensity. a) If $\rho<1$, then $X$ is positive recurrent. b) If $\rho>1$, then $X$ is transient. c) If $\rho=1$, ...
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### Discrete and Continuous Time Markov Properties

$\newcommand{\indep}{\perp\!\!\!\perp}$ In discrete time, the Markov property is $$P[X_{n+1}\in A\mid X_n=s_n,X_{n-1}=s_{n-1}\dots ]=P[X_{n+1}\in A\mid X_n=s_n]$$ On the other hand, the "general ...
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### Q-matrix vs. P-matrix description of a Markov chain

Consider a continuous time Markov chain $(X_t)_{t \geq 0}$ on some state space $S$ with transition matrix (P-matrix) $p_t(x,y)$, the probability density of jumping from $x$ to $y$ in time $t$. The ...
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### Exercise on Markov chain

Prove, or give an explicit counterexample to refute, the following assertion: if $\{X_n\}$ is a Markov chain, then $\{X_n^2\}$ is also a Markov chain. It's easy to show that ...
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We know that if $\{X_n\}$ is a Markov chain, then $X_{n+1}$ is independent with the past states $X_0,\ldots,X_{n-1}$ given current state $X_n$, that is ...
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### Markov chain problem with finite states

Consider any Markov chain on a state space with exactly $r$ states. We want to find the largest $N>0$ such that there exists states $i,j$ for every $n < N$ we have $p_{ij}^{(N)} > 0$ and ...
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### Markov Chain depicting unruly customer behavior

A store has 2 bins of balls. 1 bin is red, and contains 3 red balls. The other bin is gray and contains 2 gray balls. Every minute, on the minute, exactly one customer comes by the bins, picks up ...
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### Library chain stationary distribution

This is an exercise 1.47 from Richard Durrett's Essentials of Stochastic Processes p.85 (doi: 10.1007/978-1-4614-3615-7_1 or Google Books). On each request the ith of the $n$ possible books is the ...
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### random walk in a certain environment

Consider the following random walk in one dimension, starting from $r(0)=0$. $$r(i+1) = r(i) + \xi,$$ where $\xi(i, r(i))$ is an increment with distribution $P(\xi=1) = \frac{c^{r(i)}}{i-r(i)+1}$ ...
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### A theorem in the paper “Noncommuting Random Products” by Furstenberg

I have a question concerning the proof of theorem 2.5 at page 395 of the paper Noncommuting Random Products, by H. Furstenberg, Trans. Amer. Math. Soc., 1963. The statement is as follows: Let $\mu$ ...
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### Log-likelihood function

I'm not sure if this could be asked here, or in math overflow... In the following paper Cho, Jin Seo, and Halbert White. "Testing for regime switching." Econometrica 75.6 (2007): 1671-1720. doi: ...
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### Markov Chains Proof using Statistics

Source: This came from "Introduction to probability" by Charles Miller Grinstead, and James Aurie Snell. It was located on page 407 and is Theorem 11.1 in the section 11.1 Introduction. Theorem: The ...
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### Transition function is a Markov semigroup?

How does the transition function in a Markov process become a Markov semigroup in time homogeneous Markov processes? Thanks a lot.
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### Question on the proof of the simple Markov property of a Brownian motion

Today we proofed the (simple) Markov property for the Brownian motion. But I really don't get a crucial step in the proof. The theorem states in particular that for $s\geq0$ fixed, the process ...
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### How is a simple birth process is time-homogeneous?

Why is it that a simple birth process is time-homogeneous? The incidence of a birth in a small time interval depends on the population size at the time of start of the interval. Doesn't this ...
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### Interpretation of potential kernels for Markov processes

One can associate a strongly continuous contraction semi group (SCCSG) to a Markov process with state space $S$ through its transition function, say $P_t$. Now one can interpret $P_t$ as a linear ...
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### Why do these expressions tend to zero?

In a chapter on Markov chains, it is claimed that $\binom {2n}{n} p^n(1-p)^n$ (where $p$ is a probability) tends to $0$ as $n$ tends to $\infty$. But why is this so? It is also claimed that for an ...
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### What is the Expectations of all 3 ants meeting at same point?

Say we have 3 ants in three corner's of triangle. What is the expectations that all 3 ants meeting together given that the ant moves in any direction. So by just seeing it I figured out that in 2 ...
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### Is it possible to reverse probabilistic automaton?

Is it possible to reverse probabilistic automaton (PA), i.e. calculate the probability of previous state given current state? Will reversed automaton be a PA (Markov?), i.e. will next probability ...
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### Markov property of a random process (a solution of piece-wise deterministic equations)

Consider a piece-wise deterministic (Markov!) process
I saw this on Liggett's Book (P.95). Let $S=% %TCIMACRO{\U{2115} }% %BeginExpansion \mathbb{N} %EndExpansion ,$ and suppose $\left( X_{t}\right) _{t\geq 0}$ is a continuous-time Markov process with ...
Let $m$ be a probability measure on $Y \subseteq \mathbb{R}^p$, so that $m(Y)=1$. Consider a function $f: \mathbb{R}^n \times Y \rightarrow \mathbb{R}^n$, continuous on the first arguments, ...