# Tagged Questions

A stochastic process satisfying the Markov property: the distribution of the future states given the value of the current state does not depend on the past states. Use this tag for general state space processes (both discrete and continuous times); use (markov-chains) for countable state space ...

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### Poisson: Conditional Probability on Pizza order

I am not sure about my answer. In particular, part b of the following question. Pizza orders arrive according to a Poisson process of rate 20 per hour. Orders are independently for a vegetarian ...
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### estimation of transition probabilities from aggregate data

Please, O mathematicians, help me understand the approach to the problem of estimating transition probabilities given only aggregate data in Kalbfleisch & Lawless' 1984 paper "Least-Squares ...
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### Correct steps in rewriting expectation to a probability

My knowledge of measure theory and probability spaces is limited, so please keep it relatively simple. Let $\{X(t), ~ t \ge 0\}$ be a Markov process on the countable state space $\mathbb{N}_0$ with ...
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### Why are inter arrival times in the continuous version of discrete-time Markov chains always exponentially distributed?

I am curious whether there exist continuous time Markov processes for which the times between jumping times (which I call inter arrival times) are not exponentially distributed, but have some other ...
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### Fractional powers of Markov generators

Let $H$ be the generator of a symmetric Markov semigroup on $L^2(\mathbb{R}^n).$ Why the fractional power $H^\alpha$ (defined on a proper domain) with $0 < \alpha < 1$ turn out to be the ...
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### Is the infinitesimal generator for Lie groups the same as the infinitesimal generator of a Markov semigroup?

Is the infinitesimal generator for Lie groups related to the infinitesimal generator of a Markov semigroup? Or are they totally different concepts? https://en.wikipedia.org/wiki/Lie_group#...
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### Uniform convergence of the action of a Feller semigroup in one variable.

Assume we have two subsets of the some euclidean spaces $X\subset \mathbb{R}^m$ and $Y\subset\mathbb{R}^n$ and a a Feller semigroup $(Q_t)_{t\geq 0}$ on $Y$. Suppose also that we have a continuous ...
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### Is this transformation of a Markov process again Markovian?

Let $(X_t)_{t\in\mathbb{N}_0}$ be a stationary Markov process valued in $\mathbb{R}$ and $c\in\mathbb{R}$. Is the process $(Y_t)_{t\in\mathbb{N}_0}$ defined by $$Y_t={\bf 1}{(X_t<c)}$$ again a ...
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### Strategy for selling/buying a stock by average reward value iteration

At beginning of any day $t$, I may own $0$ or $1$ share. The price of the share follows the Markov chain in the table below. At the beginning of a day where I own a share, I may either sell at today’s ...
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### Strong markov property vector valued process from independent strong markov components

I want to prove that if $X$ and $Y$ are (continuous time) independent strong markov $\mathbb{R}$-valued processes w.r.t. their natural filtrations $\mathcal{F}^X_t$ and $\mathcal{F}^Y_t$, that the ...
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### Moving Average of an Ergodic Markov processes.

Let $\{X(t); t\geq 0\}$ be an ergodic Markov process, and let G be a positive integrable function with $\int_0^\infty G(x)dx=1$. Does $$F(t)\doteq G*X(t) = \int_0^t G(t-s)X(s)ds$$ converge in ...
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### Good introductory book coupling methods

I am very interested in coupling methods, can you recommend me a good introductory books on this subject? Thanks
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### Gambler's ruin problem - expected time

I have troubles seeing the following. Consider the classical gambler's ruin problem, betting 1 at each time $t\in \mathbb{N}$, and losing or winning -1 respectively +1 at each time till the fortune of ...
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### Electrostatic capacity of two spheres with changing radii

Although I have read a lot of questions and answers here, this is my first time actually posting. Feel free to suggest needed edits. My question is the following (in a simplified setting). All this ...
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### Discrete-Time Stochastic Calculus and Stopping Times: Resources

In my measure-theoretic probability course we covered what the professor called "discrete-time stochastic calculus". Essentially, it was a three part method for computing certain quantities such as ...
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### Limit of decreasing sequences of markov time (stopping time) is markov time?

Let $(\Omega, \mathcal{F}, \{\mathcal{F}_t\}_{t \geqslant 0}, \mathbb{P})$ be a filtered probability space and let $\tau_n \geqslant \tau_{n+1}$ be a markov time (stopping time) with respect to ...
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### Define Markov chain and rewrite to recursively solve

Customers arrive at a server with rate $\lambda$ and are served at rate $\mu$. The server breaks down with rate $\gamma$, which causes all customers to leave. New customers can only arrive once the ...
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### Model as a continuous time Markov Chain

A system consists of two machines, of which one works and the other is standby. Only the working machine can break down (with rate $\lambda$). If it breaks down the other machine takes over (if it not ...
### Probability of going from a set $S$ to its complement on a Markov chain
I need to show that if $\pi$ is the stationary distribution of a Markov chain $M$, then for every set of vertices $S$, the probability to choose a random node in $S$ according to $\pi$ and then going ...