markov-process's questions - Chinese 1answer

1.246 markov-process questions.

I have a problem setup that reduces to finding the stationary state of a discrete time continuous state stationary Markov process. Is that state unique, as it is in the discrete state case? ...

I want to understand exactly what my title asks "Why is the Stochastic Process for the short rate in the HJM model of interest rates non-Markovian?" That process is the following: $r(t)=F(0,t)+\int^{t}...

Im getting a bit confused with the terminology of continous time markov processes. Now let $\{X_t | t \geq 0 \}$ be a stochastic Process with state space $\mathbb{R}$, and $(F_t)$ be its ...

I just read a paper by the authors Kohlberg and Neyman stating that "A single-person stochastic game is known as a Markov Decision Process (MDP)." Does anyone know if the following extension to $n$ ...

I have just started reading about the Ergodicity of Markov processes, and I'm particularly interested in showing that time averages of certain diffusions converge as time grows, and in characterizing ...

When we have a countably infinite state space what can we use to define the dynamics of our process, and how can we use this to then give a definition of a process in an equilibrium distribution /and ...

Let $(X_t)_{t \geq 0}$ be a time-homogeneous Markov process on a Polish state space $E$ and for a state $x$ denote by $P_x$ and $E_x$ the probability measure resp. expectation for $x$ as initial state....

What's the difference between discounted cost , total expected cost and average expect cost MDP? Are they just MDP problems with different objective function? When the discounted factor equals 1, then ...

Can someone please provide a reference to a detailed proof of the CLT for MC? I have a paper MC's for MCMC'ists but I have a problem understanding a crucial part. Namely, the transition from line 2 to ...

I've a problem that I almost finished but I can't complete. This is the problem: "Consider the continous version of the Ehrenfest model: a particle moves on a grid consisting of points $x=ma$, where m ...

I know that a reversible distribution is a stationary distribution, but is there a case where we have a stationary distribution but it's not reversible distribution? In other words, every stationary ...

I have some problems in solving this problem: "Two independent walkers move on a discrete ring with N state (N odd) and periodic boundary conditions, with transition probability: $$P_{n\rightarrow n\...

The following question is based on the slides 3-5 in http://www.maths.lancs.ac.uk/~belton/www/notes/23iv14.pdf. The material is consistent with the book "Levy Processes and Stochastic Calculus", ...

I came across this question while studying for a probability midterm, and I can't figure out whether the statement is true or not. The main point of confusion I am having is if a stationary ...

Suppose that the weather tomorrow depends only on the weather today and that there are $3$ states for the weather: Sun, Cloudy, Rainy. Suppose today is day $1$ and is Sun, what's the probability of ...

I am wondering if somebody can tell me anything about the practical differences between using Markov Decision Processes and and Bayesian Networks in reasoning about probabilistic processes?

Right so I found these questions at the end of our chapter on discrete markov chains. The question asks whether they are Markov chains. I can see some being continuous markov chains and wonder if the ...

$\newcommand{\E} {\mathbb E}$I'm working on the following queueing problem. Group of customers arrive according to Poisson Process with rate $3$ per hour. With probability $2/3$ only one customer ...

I'm learning about continuous-time Markov chains and am trying to hone my intuition. Let $T_1, T_2, \ldots$ be i.i.d random variables. Let $S$ be some state space, we can take it to be finite. Let $...

I'm a little bit confused about the terms of arrival and service processes in queuing systems. I know about Kendall's notation but it is often explained slightly different in literature. As far as I ...

I've been given the following definition: For a THMC with one step transition matrix $\mathbf{P}$, the row vector $\mathbf{\pi}$ with elements $(\pi_{i})_{i \in S}$ (where $S$ is the state space) ...

After answering [this question] (Expectation of a stopping time uniquely determined by a function) I was looking for the literature on the mean hitting/exit time for a discrete-time Markov process. ...

Suppose I keep rolling a die and I stop once I got two consecutive 6. What is the expected number of time to roll the die under the condition that all the rolls are even number? So, the sample space ...

Let a Markov Chain be given with the following transition probabilities; $$p_{0,0} = \frac{1}{2}, p_{0,1} = \frac{1}{2}, P_{i, i+1} = \frac{i + 1}{i+2} (i \geq 1), p_{i,0} = \frac{1}{i+2} (i \geq 1)$$ ...

Suppose $X_n$ is an ergodic Markov chain (i.e.,there exists an invariant probability measure $\pi$ s.t. $||P^n(x,\cdot)-\pi(\cdot)||_{TV}$) on a noncountable compact space. Define $\phi:\mathbb{R}\...

Suppose we have a Ehrenfest Markov chain with $d + 1$ states $0, 1, \dots, d$. With transition probabilities $p(i, j) =1 - (i/d)$ if $j = i + 1$ and $p(i, j) = i/d$ when $j = i - 1$ (otherwise $p(i, j ...

The following is from my class lecture notes (handwritten) on Markov chain. I am having trouble understanding some parts of it. Trying to summarize what is written: The $m$-step transition function: $...

Given a probability $p$ and a row $i$ of a stochastic matrix $M$, what is the maximum number of columns $j$ such that $M_{ij} \gt p$? ...

I was learning about Markov chains from my lecturer's handwritten notes, but I got stuck at "transition functions". It will be a quite a while until I get to ask the lecturer about what he meant. So ...

Let $X(n)_{n \geq 0}$ be a homogeneous Markov chain with state space $S$. Define $$p_{ij}(n):= \mathbb{P}( X(n+m)=j|X(m)=i),$$ where $i, j \in S$, $n \geq 1$, and $m \geq 0$. Let $d(i)$ be the period ...

The task Let $\xi_i, i \in \mathbb{Z}_+$ be i.i.d. random variables on $\mathbb{R}$ with a probability density function $\rho(x) > 0$. Denote by $\eta_n$ the minimum of r.v.s $\xi_i$ for $i \leq n$...

There are $C$ i.i.d processes running in parallel. Each process can be in any one of the $k$ states and is a birth-death process (with only one step transitions allowed). That is, for any process, the ...

Let $X_1(t)$ and $X_2(t)$ be two Continuous Markov Process with the same initial boundary conditions. Let their density functions for a given $t$ be $P_{1,2}(x,t\,|\,x_0,0)$ and their infinitesimal ...

Suppose $(X_n)_{n\in \mathbb N_0}$ is a discret-time Markov process on state space $(0,1]$ with transition kernel $\kappa(x,\cdot)$ possessing an absolute continuous (Lebesgue-)density for all $x\in (...

I've read the answer of George Lowther on this question and I don't understand his proof. I know just a few things on Markov Process. If I define $\left(X_n\right)_{n \in \mathbb{N}}$ with $X_0=1$ and ...

Let $\xi_t$, $t \in \mathbb{R}$ be a stochastic process, and $\mathcal{F}_{=t}$, $\mathcal{F}_{\geq t}$, $\ldots$ are $\sigma$-algebras induced by it. I want to find out whether the two following ...

Theorem: i is recurrent then $\mu_{ii}=\sum_{n=1}^\infty nf_{ii}^n = \lim_{s\rightarrow 1^{-}} \frac 1 {(1-s)p_{ii}(s)}$ Proof: Let $i$ be recurrent. Then, note: \begin{align} & \lim_{s\...

Claim:$i\leftrightarrow j$ implies $d\left ( i \right )=d\left ( j \right )$ Suppose $i\leftrightarrow j$: This implies that j is accessible from i and i is accessible from j. Then, $\exists n_{1}$ ...

Let a Markov Chain be irreducible and aperiodic. j is null-recurrent IFF $\sum_{n=1}^{\infty}p_{jj}^{n}=\infty$ and $p_{jj}^{n}\rightarrow 0$ as $n\rightarrow \infty$. Suppose $\sum_{n=1}^{\infty}p_{...

Let P be a transition matrix of a Markov Chain. Let $p_{ij}^{n}=P\left [ X_{n}=j | x_{0}=i\right ]$ be the transition probability of a Markov Chain from an initial state i to a final state j in n-...

I'm dealing with Pardos Proof on page 13, which shows the construction of a selfsimilar process $X$ with independent increments, while there is only $\nu_{X(1)}$ given. I asked myself, how he does ...

In the proof of the FCLT Theorem 7.4.1 of Ethier and Kurtz (1986) "Markov Processes", the last four lines on p.355 write ($\tau_n^r = \inf\{t: |X_n(t)|\geq r\}$): "Consequently, if $X$ is a solution ...

Let the initial state of a MC, $X_{0}=i$, be fixed. Definition: $T=\left\{n \in \mathbb{Z^{+}}:X_{n}=i \right\}$ is the time of the first visit from i to i. Definition: $\pi\left(i\right)$ is ...

Claim: Let P be an irreducible transition matrix and let P has period $d>1$. Then, the state space S splits into d sets $A_{1}, \cdot \cdot \cdot A_{d}$ I am unclear why this is true. Since P ...

What is a simple definition of Stationary Increments? I saw that: $X_t - X_s = X_{t-s}$ How does this relate to the property of being a stationary process? Or even a weakly stationary process, ...

The proof in the book goes something like this: for all times $s_1 < s_2 < s_3 < ...< s < t$ and all states $x_1, x_2,....,x_n$ and $x$ in $S$ the State Space and all subsets of $A$ of ...

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