All Questions
141 questions
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Asymptotic mixing time and Euclidean probability distance for path graphs
We are given a simple path graph $P(V,E)$ with vertex set $V$ and edge set $E$, having $n=|V|$ nodes. Given an initial distribution $\mathbf{\mu}$ over $V$, let $d_t(\mathbf{\mu},\pi)$ be defined as $\...
- 1,677
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114
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An urn model with weighted objects and replacement
Consider the following game:
In an urn, there are $K$ balls, $x_0$ of them are blue and light (mass $m_0$), $x_1$ are blue and heavy ($m_1$), $x_2$ are red and light ($m_2$), the rest $x_3$ are red ...
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1
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72
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Bounding expectation of switching stochastic process
I am analyzing the behavior of an 1D stochastic dynamic system, where the state can vary randomly within a small magnitude. However, when the state deviates too much from zero, its expected magnitude ...
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1
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115
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Concatenation of Markov processes and independence
In chapter 14 of Sharpe's General Theory of Markov Processes the concatenation of Markov processes $X^1$ and $X^2$ is described. I've posed the relevant part at the bottom of this post.
It is rather ...
- 167
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110
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Birth and death process $M/M/\infty$
I was reading about continuous time Markov chains, when I met for the first time the theory of queue processes. In particular, I considered the following situation which I found on Wikipedia, called M/...
- 445
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181
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Random walk on 2d lattice with obstacles
Consider a random work on $L=\mathbb Z^2$ endowed with obstacles (i.e each cell $(x,y)$ of $L$ may contain a obstacle, i.e the random walk halts whenever it hits such a cell). Let $P(x,y) = 1$ if cell ...
- 6,853
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276
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Path dependent Markov property
Let's consider a function $\Psi\in \mathcal{C}_B(\mathcal{C}[t,T])$ continuous and bounded
\begin{align*}
\Psi \colon \mathcal{C}[t,T] \longrightarrow [0,+\infty)
\end{align*}
Then my question is:...
- 159
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61
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Convergence of empirical measure to Mc-Kean Vlasov equation for mean-field model with jumps
I am interested in the following mean-field model introduced in the reference below:
There are $N$ particles. At each instant of time, a particle's state is a particular value taken from the finite ...
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1
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1
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170
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Stationary distribution of Markov Chain with departure
I have a Markov Chain of $N$ states. Such states represent the energy levels in a molecule.
The states' connectivity is as follows:
States $j\in\{0,\ldots,N\}$ transition to $k\in\{\max(j-M,0),...,\...
- 151
1
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149
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Construction of Feller's pseudo-poisson process
Let
$(\Omega,\mathcal A,\operatorname P)$ be a probability space
$(E,\mathcal E)$ be a measurable space
$(Y_n)_{n\in\mathbb N_0}$ be a $(E,\mathcal E)$-valued time-homogeneous Markov chain on $(\...
- 167
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44
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Validating a probability density distribution forecast model for a Markov process
Let's say we have a Markov process $X_t$, and we come up with a forecast model that takes some information from outside world and says: "value $X_{t+1}$ has probability density distribution $P_t(x)$". ...
- 93
1
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101
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How to fit a stochastic matrix to given data.?
Given a data sequence of noisy observations of a 3-state Markov chain $X$ -- $y_1$,$y_2$,...$y_n$, with two transition matrices $A_1$ and $A_2$ corresponding to different regions (**) in the (unit) ...
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111
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Conditional probabilities in epidemic model
I was contemplating an epidemic model where infection and recovery rates are determined by links. Here node $i$ is infected first and recovers at a rate $\mu_i$. For all other nodes, the recovery is ...
- 519
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196
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The problem of the drunkard in a valley [closed]
We consider a Markov chain on a subset of positive integers S = {0, 1, 2, 3, .......N}, with transition probabilities defined as follows:
The chain jumps only one unit to the left or right.
p(i, j) =...
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1
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173
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Spectral gap of a Markov chain on the nonnegative integers
Let $\lambda_k,\mu_k\in\mathbb R_{\ge0}$ $(k\ge1)$ be nonnegative real numbers such that $\sum_{k=1}^\infty k\lambda_k<\infty,$ let $S=\mathbb Z_{\ge0}$ be the nonnegative integers, let $T=\mathbb ...
- 276
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2
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804
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Convergence of stationary distributions of a sequence of Markov Chains
I fairly new in the field of Stochastic Processes and Markov Chains so excuse my ignorance.
My question is: If we have a sequence of Markov chains such that each one has a stationary distribution $\pi^...
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2
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306
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Lower bounds on discrete time finite Markov chains hitting probabilities
I am interested in some general theorems related to lower bounds on discrete time finite Markov chains hitting probabilities (preferably ergodic chains , but not necessarily ), with references . ...
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1
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203
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Law of large numbers for Harris recurrent Markov chains
I'm trying to familiarize myself with the details of the proof that the Markov chains produce by the Metropolis-Hastings algorithm have a law of large numbers. I've found a half dozen or more ...
- 2,791
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2
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128
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Markov with epsilon memory and Quantitative Strong Markov property
We have a process $\{X_{t}\}_{t\geq 0}$ ,with fixed parameter $\epsilon>0$, starting from zero that satisfies
The process is strictly monotone $X_{t+r}-X_{t}>0$ with moments existing $p\in(-\...
- 5,474
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204
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How is the Cauchy-Schwarz inequality used in the proof of Lyapunov's criterion in the book "Analysis and Geometry of Markov Diffusion Operators"
Let $(E,\mu,\Gamma)$ be a full Markov triple (see definition below), $J\in\mathcal A$ with $J\ge1$ and $g\in\mathcal A_0$. In the proof of Theorem 4.6.2 of the book "Analysis and Geometry of Markov ...
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1
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96
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What is the significance of Blumenthal and Getoor's result on the boundedness of paths of a standard Markov process?
In the book Markov processes and Potential Theory of Blumenthal and Getoor we can find the following result:
I don't understand the significance of this result. If I don't misinterpret the assertion, ...
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262
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Construction of a Markov process with prescribed local behavior and state-dependent jump distribution
Let
$(E,\mathcal E)$ be a measurable space
$\mathcal E_b:=\left\{f:E\to\mathbb R\mid f\text{ is bounded and }\mathcal E\text{-measurable}\right\}$
$(\kappa_t)_{t\ge0}$ be a Markov semigroup on $(E,\...
- 167
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1
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2k
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Markov Chain: state reduction
Hi I am trying to understand a proof in a paper (written by Isaac Sonin), I don't know if anyone could give me a clarification on the following:
Firstly we have a Markov chain $\{Y_k\}$ with finite ...
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85
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Does a 2d random walk hit 0 for increasing distances AND time spans?
Question: For a simple symmetric random walk $(Z_t)_{t\geq 0}$ in $\mathbb{Z}^2$, does
$$\lim_{\beta\rightarrow 0}\mathbb{P}^{x_\beta}(Z_t=0\text{ for some }t\leq h(\beta)T)=0\quad (2.8)$$
where $|x_\...
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101
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Simulation of Markov processes with exponential timestepping
Let $(Y_t)_{t\ge0}$ be a time-homogeneous Markov process with transition semigroup $(\kappa_t)_{t\ge0}$. Numerical simulation of $(Y_t)_{t\ge0}$ can be done in the following way:
Choose an initial ...
- 167
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34
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Does the definition of mixing time work for general non-Markovian processes?
A definition of the mixing time for Markov chains is given by
\begin{equation}
\tau_{\text{mix}}\equiv\inf{\{t>0: \sup_i\left\vert \frac{\boldsymbol{p}(t|p_j(0)=\delta_{ij})}{\boldsymbol{\pi}}-\...
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92
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MDP Average Reward independent of Initial State
Consider a Markov Decision Process where the state space $S$ and the action space $A$ are continuous and compact.
In state $s$, if action $a$ is chosen and the next state becomes $s'$, the ...
- 115
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142
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Calculating the expected hitting time of a specific birth and death chain
I am working with a specific birth and death chain, defined as follows.
Consider a set of states $X = \{0,1,2,...,n\}$, where $x^* \in (0,n)$ is a recurrent state. Transition probabilities are defined ...
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161
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Markov process with time varying transition kernels
I cross post this question from StackExchange as it may be more appropriate.
I am interested in studying the evolution of a variable $\alpha_t\in [0,1]$ governed by the following stochastic dynamical ...
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72
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If $\kappa$ is a Markov kernel with density $p$, does it generally hold $p(x,z)=\int p(x,y)p(y,z)\:{\rm d}y$?
Let $(E,\mathcal E)$ be a measurable space and $\kappa$ be a Markov kernel on $(E,\mathcal E)$. Assume that $$\kappa(x,B)=\int_Bp(x,y)\:\lambda({\rm d}y)\;\;\;\text{for all }(x,B)\in E\times\mathcal E$...
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83
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Constrained MDP
I have a question that is an extension of this one.
My question is: Can we say that for every policy, there exists a deterministic policy in case of a finite-state, finite-action infinite-horizon ...
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85
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If $W$ is a Markov chain and $N$ is a Poisson process, then $\left(W_{N_t}\right)_{t\ge0}$ is Markov
Let $(\Omega,\mathcal A,\operatorname P)$ be a probability space, $(E,\mathcal E)$ be a measurable space, $(W_n)_{n\in\mathbb N_0}$ be a time-homogeneosu Markov chain on $(\Omega,\mathcal A,\...
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1
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80
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A question about positive operator pregenerator [closed]
Thank you for reading.
My question was raised up when I tried to prove an example in the book of Liggett(1985), which is in P13 Example 2.3(a).
Here is a link of the page:
https://books.google.com/...
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72
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Invariant measures for a renewal process driven by Interarrival times bounded away from zero
Good morning, I apologize in advance if my question sounds too basic but after some research I was unable to come up with satisfactory answers to my doubts.
I am currently studying a model which ...
- 277
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169
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Behaviour of a Markov Chain, given a Lyapunov condition
I'm reading this notes from Martin Hairer about convergence of Markov Processes (on a discrete state space $S$ and in continuous time). On page 12, before presenting the so-called "Harris Theorem", ...
- 203
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78
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Core of direct product of Markov processes
Let $X$ and $Y$ be two diffusion processes. Suppose they have generators $G_X$ and $G_Y$ with domains $D(G_X)$ and $D(G_Y)$ and cores $C(G_X)$ and $C(G_Y)$. Let $Z$ be the product diffusion with ...
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355
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Summing up costs over a Markov chain
I apologize in advance if this question is too simplistic to be appropriate for MathOverflow. I have inquired in multiple places but have found little to indicate that this is a previously studied ...
- 101
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117
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Ergodicity property for continuous-time Harris positive Markov process
I have posted this question on there, but got no answer.
The following theorem is Theorem 13.3.3 of Meyn and Tweedie's Markov Chains and Stochastic Stability on page 328:
Theorem 13.3.3. If $\Phi$ ...
- 109
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408
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Generating independent random variable from two correlated random variables
Suppose two random variables $X$ and $V$ are given. I am wondering what kind of condition we need to impose on joint distribution of $V$ and $X$ to make sure that there exists a random variable $Z$ ...
- 1,109
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151
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Inequality relating stationary probabilities and transition probabilities
Let $P$ be the transition probability matrix of a aperiodic irreducible DTMC and let $\pi$ be its stationary distribution. I would like to know if there is any literature on types of Markov chains ...
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370
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What's the probability of two independent events in time domain?
Suppose there are two independent events A and B. The probability that A or ...
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