# Tagged Questions

110 views

### A doubt on Balaji Meyn's ergodic theorem paper

I have a question regarding the classic paper by Balaji and Meyn: "Multiplicative ergodicity and Large Deviations for an Irreducible Markov Chain". Consider a recurrent aperiodic irreducible Markov ...
64 views

### 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 ...
38 views

### DTMC random walk model [closed]

For a discrete Markov chain random walk with p < 0.5 with state space S= {0,1,2..} What is the stationary distribution? I could use any help. Thank you
117 views

### Approximating Markov chains by Brownian motion

I would like a result along the following lines to be true, but haven't been able to locate it in the literature; pointers would be welcome. Let $X_t$ be a finite-state, irreducible, aperiodic Markov ...
94 views

### random walk with reflecting barriers [closed]

Consider a random walk on the line 1,...,d. You start at point 1. At each step you flip a coin: heads means go left, tails means go right. If you're at 1 and get a heads, just stay where you are (same ...
468 views

### How to explain “Feller process” to an undergraduate student?

I had to explain in informal terms what a Feller process was, to undergraduate students who understand Markov property, Poisson processes and such. It was easy to define Levy process as generalisation ...
31 views

### Markov decision processes: action set revealed at point of decision

I have a problem which looks like a finite horizon Markov decision process (MDP), except the action space at each time is revealed at the decision making point. There is no way to know before hand the ...
447 views

### References for a physicist migrating to stochastic processes

I've studied "Markov Chains" - Norris and "Measure, Integral and Probability" - Capinski, Kopp. Now, I'm looking for a couple of books (or other references) that help me bridging these two topics. ...
19 views

### Examples of POMDPs where the actions impact the transitions of the underlying markov Chain

I am not sure if the following is a legitimate question for this board. I am looking for examples of Partially observed Markov decision processes (preferably infinite horizon, Discrete time, Discrete ...
112 views

### Invariant measure of Euler-Maruyama Discretisation of an Ito diffusion

Let $(X_t)_{t \geq 0}$ be a diffusion process with dynamics governed by the stochastic differential equation $$dX_t = b(X_t)dt + \sigma(X_t)dW_t, ~~ X_0 = x_0,$$ where ...
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### Nonlinear Markov process

Consider the following nonlinear $\mathbb{R}$-valued stochastic recursive sequence: $X_{n+1} = F(X_n) + W_{n+1}, \quad (W_n)_{n\ge1} \stackrel{ \scriptsize \mathrm{i.i.d.} }{ \sim } \phi.$ How can ...
225 views

### 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 $Z$ ...
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### CLT for a Markov Renewal Process

Suppose $(X,T)=\{(X_n,T_n)\}_{n\geq0}$ is a Markov renewal process, where $X$ is a finite-state, discrete-time Markov chain with state space $\{1,2,...,R\}$. $T$ is the additive component, more ...
135 views

### Quasi-stationary distribution for a death process

In the paper, Survival in a quasi-death process by van Doorn and Pollett, the quasi-stationary distribution of a transient CTMC is discussed and QSD for a simple death process is derived. Consider a ...
136 views

### Transition probabilities in coupled Markov chains

I know that for a continuous-time Markov chain, the probability of transition from time $0$ to $t$ is given by $P(t)=e^{Q(t)t}$. I have a system of $N$ interdependent continuous-time Markov chains ...
47 views

### Relative vulnerabilities in SIS epidemic model

Consider the SIS model of epidemic spreading. There is a finite graph $G(V,E)$, link infection rates $\lambda_{ij}$ and node recovery rates $\mu_i$. There are a few initial nodes which are infected at ...
144 views

### Supremum in a Markov chain model

A Markov chain $X$ with finite state space $\{1,2,\cdots,N\}$ is defined on a probability space $(\Omega, P, \mathcal{F})$ equiped with filtration $\{\mathcal{F}_t\}$. And we assume that we can reach ...
335 views

### Does the variance of a continuous time, time homogeneous, Markov process starting from one point necessarily not decrease?

Let $x_t$ be a zero mean, time homogeneous Markovian process (chiefly look at the case where the value is in $1$ dimension) over time $t$ starting from $x_0=0$. Is it necessary that, in continuous ...
132 views

### 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) ...
179 views

### Generalized Markov Processes on CW complexes of dimension > 1

Markov processes have a large variety of applications to physics and chemistry (as well as many other fields). Such processes are formulated on graphs, i.e., CW complexes of dimension one. It is ...
242 views

### 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 ...
142 views

### Markov operators and existence of ergodic measures

My question refers to the yesterday's question (see here) of John Learner and goes as follows: Can we deduce the existence of an ergodic measure if we know that an invariant measure exists, but the ...
104 views

### 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 ...
87 views

### Empirical distribution of a collection of iid Markov chains

Suppose we have $N$ independent 2-point Markov chains each having a rate matrix $Q = [-1,1;1,-1]$ and stationary distribution $\pi = [0.5,0.5]$. At time $t=0$, we initiate the chains so that the ...
497 views

### Hitting time probability in a Random Walk with possibility to die.

A Random Walker can move of one unit to the right with probability $p$, to the left with probability $q$ and it can jump again to the starting point with probability $r$ and die. Naturally $p+q+r=1$. ...
207 views

### Extending Wald's equation to two classes of i.d. random variables?

I try to adopt Wald's equation to a slightly more complex problem. In fact, after a full day, I found some solution now, but it has a confusing argument in the middle. Perhaps somebody can help me at ...
575 views

### Markov chains: invariant measures and explosion

The following seems like such an elementary question, but I didn't get anywhere with it. Suppose you are considering a Markov chain in continuous time which is transient and has an invariant measure ...
356 views

### MCMC with progressive demollification of delta distributions

Edit: I simplified the example to a canonical case for clarity. Given an integral $\int_{\Omega}{g(\mathbf{x})}$ with a well-posed integrand $g(\mathbf{x})$ defined on some multidimensional space ...
369 views

### The spectrum of a Markov Operator and Invariant Measures

Suppose I have a discrete-time Markov Chain (in an infinite dimensional state space $\Omega$) with Markov operator $P$, a linear operator on the space of bounded measurable functions on $\Omega$. (Or ...
191 views

### Stochastic processes having Markov kernels

Let $(\Omega_1, \mathcal{F}_1, P_1)$ and $(\Omega_2, \mathcal{F}_2, P_2)$ be probability spaces and suppose $(X_t)$ and $(Y_t)$ are real-valued stochastic processes defined on the respective spaces. ...
581 views

### Difference in probability distributions from two different kernels

Let $(E,\mathscr E)$ be a measurable space and $P,\tilde P$ be two stochastic kernels on that space. I wonder how the induced measures $\mathsf P_x$ and $\tilde{\mathsf P}_x$ differ on the space of ...
183 views

### One point on $\phi$-irreducibility

Let $P(x,A)$ be a stochastic kernel on a measurable space $(E,\mathcal E)$ and $G = \sum\limits_0^\infty P^n$ be its potential kernel. A $\sigma$-finite measure $\phi$ is called the irreducibility ...
372 views

### stochastic processes conditional on other stochastic processes

Problem: I'm working in reliability field and have seen papers written on the topic like process of failures when systems are functioning under unobservable (or observable) Markov-like environment, ...
511 views

### Ergodicity of a Markov chain

Hi, I'd appreciate some help on a Markov chain result I'm trying to show. I believe the following is sufficient for a continuous time Markov chain $(X_t)$ with a countable state space to be ergodic: ...
316 views

### Probability-one event for Markov chain

Let $X$ be a Markov chain, with countable state space $I$ and transition probability matrix $P$. $X$ is irreducible, but need not be recurrent. Let $S$ be a fixed subset of $I$. Define a subset $K$ ...
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### Reachability for Markov process

Let $X$ be a Markov process (in continuous or discrete time) and define an event $$R(T,A) = (\exists t\leq T: X_t \in A).$$ I have seen in one paper that  \Pr[R(\infty,A)] = \sup\limits_{\tau} ...
156 views

### scalar diffusions are reversible

It is well known that under mild assumptions a scalar diffusion $dX_t = a(X_t) dt + \sigma(X_t) dW_t$ with invariant probability distribution $\pi$ is reversible. This is indeed not true for ...
I'm trying to understand a 1954 paper of Kubo intitled "Note on the stochastic theory of resonance absorption". The specific problem can be stated mathematically as follows: let $X(t)$ be a random ...