All Questions
141 questions
2
votes
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Borel-Cantelli Lemma on MCs (absorbing states)
hi, I'm sorry if the question is silly, but I couldn't get my head around it for a while now.
In Markov Chains (MC) proving that a state is either recurrent or transient is through Borel-Cantelli ...
- 345
2
votes
2
answers
1k
views
Counterexample Markov process
Let $X$ be a homogeneous Markov process in a continuous time with value in the set $E$. Suppose that for some $T>0,x\in E, A\subset E$ we have
$$
P_x[X_t\in A] = 0
$$
for all $t\in [0,T]$ but
$$
...
- 1,655
2
votes
1
answer
59
views
The ranked mass process associated with a Lambda-coalescent
I am reading a paper by Pitman (1999), and I am confused by his Corollary 7. First some notation so that I can explain my confusion:
$\mathcal{P}_\infty$ is the space of partitions of $\mathbb{N}$, $\...
- 203
2
votes
1
answer
228
views
When is a stationary measure of a Markov chain "exponentially localized"?
Here exponentially localized can be thought in a non-rigorous manner as a measure that is mostly supported on a sparse number of nodes.
Some intuition can gained by thinking about a diffusion process, ...
- 2,281
2
votes
1
answer
201
views
Joint irreducibility and aperiodicity of two independent Markov chains
Let $(X_i)_i, (Y_i)_i$ be two independent Markov chains on Polish state spaces $\mathcal{X}, \mathcal{Y}$ and with kernels $P, Q$. Suppose that they are both $\psi$-irreducible and aperiodic and have ...
- 203
2
votes
2
answers
237
views
is this process a Markov one?
Here is the problem I can't solve.
Let $\xi_n$ $(n=1,2,3,\dots)$ be a sequence of i.i.d. random variables on $\mathbb{R}$ with density $p(x)>0$, let $\eta_n=\sum_{i=1}^{n}\xi_i^2$. Define $$\...
- 21
2
votes
1
answer
185
views
Entropy rate problem of ergodic Markov process with non-ergodic joint
I have a problem with the entropy rate when two ergodic Markov processes who are independent of each other having a non-ergodic joint. More specifically let us consider two finite-state Markov ...
- 333
2
votes
1
answer
187
views
Stationary distribution for a Markov Chain on an uncountable space
Suppose $X_n$ are i.i.d. random variables on $\mathbb{R}$ with compact support, and define the Markov chain $Y_n=X_n +\frac{1}{Y_{n-1}}$ on $\Omega=\mathbb{R}\cup \{\infty\}$. Does the chain $Y_n$ ...
- 339
2
votes
2
answers
184
views
Asymptotic Growth of Markov Chain
I asked the following question one week ago at math.stackexchange but didn't receive a response, so I want to give it here another try:
I'm interested in the following problem: We have got a time-...
- 111
2
votes
1
answer
412
views
Does random walk have more concentration surrounding the origin?
Consider a simple random walk $S_n$ on one dimension, starting at $0$. In this case, $S_n$ fluctuates between $-\infty$ and $\infty$, but intuition says that it might stay more often in an interval ...
- 502
2
votes
1
answer
665
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 ...
- 519
2
votes
1
answer
186
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 ...
- 2,133
2
votes
1
answer
101
views
Preservation of the Markov Property under Conditioning
Let $(X_t,Z_t)_t$ be an $\mathbb{R}^{n}\times \mathbb{R}^m$-valued time-homogeneous Markov process on a filtered probability space $(\Omega,\mathcal{F},(\mathcal{F}_t)_t,\mathbb{P})$ with transition ...
- 77
2
votes
1
answer
176
views
Monotonicity of Dirichlet form of Markov chain
Consider a continuous-time, irreducible Markov chain $X_t$ on a finite state space $E$. Assume the jump rates are $R(x,y)$ for $x,y\in E$, the generator is $L$, i.e for any function $f$ on E,
$$Lf(x)=\...
- 59
2
votes
1
answer
114
views
Strong Data Processing Inequality for capped channels
Let $X$ and $Y$ be two $\rho$ correlated Gaussian vectors, such that $X,Y\sim N(0,1)^n$ and $E[X_iY_i]=\rho$.
Let $M_X = f(X)$ and $M_Y = f(Y)$ be $k$-bit functions of $X$ and $Y$, that is $H(X)=H(Y)=...
2
votes
1
answer
168
views
Random Walk 2D with dependent weights [closed]
I have spent a lot of time trying to solve this problem but have had no luck so far! Any help would be highly appreciated!
Suppose I have a 3x3 grid as shown below.
(3,1) (3,2) (3,3)
(2,1) (2,2) (...
- 23
2
votes
1
answer
421
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 ...
- 340
2
votes
1
answer
447
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 $\...
2
votes
2
answers
861
views
Spectral gap of a product of Markov processes
For $m \in [N] \equiv \{1,\dots, N\}$, let $Q^{(m)}$ be the generator of a (well-behaved) continuous-time Markov process on a finite state space $[n_m]$. Write $J \equiv (j_1,\dots,j_N) \in \prod_m [...
- 15.4k
2
votes
0
answers
54
views
Including fixed-time transitions into a continuous time Markov chain system
I have system which is mostly described by a CTMC (Continuous-time Markov chain) with a single absorbing state and a large but tractable and sparse transition matrix. However, at a fixed set of "...
- 21
2
votes
0
answers
112
views
Embedding a Markov chain in a Markov process
Let $X_{t\ge 0}$ be a Markov process with values in a metric space $(\mathcal{X},d)$ defined on a probabiltiy space $(\Omega,\mathcal{F},\mathbb{P})$ and let $(\tau_n)_{n=1}^{\infty}$ be a sequence of ...
- 21
2
votes
0
answers
155
views
Can a diffusion process admit an invariant measure with a non-differentiable density?
The precise domain of the generator $A$ of an Itō diffusion on a Hilbert space $H$ (assume $H=\mathbb R^d$, if that's easier for you to work with) can usually not be determined explicitly$^1$. Usually,...
- 167
2
votes
0
answers
175
views
Representing a continuous time-inhomogeneous Markov chain by a stochastic integral
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 ...
- 31
2
votes
0
answers
37
views
Exclusion processes from point of view of a tagged particle
I'm interested in the simple exclusion processes on $Z^d$ and the ergodic theorems that can be proved from the point of view of the particle. Ellen Saada proved the following in 1987 (Annals of Prob): ...
- 941
2
votes
0
answers
416
views
How can we treat the generator of a discrete-time Markov chain as the generator of a Markov-jump process?
In the popular paper Weak Convergence and Optimal Scaling of Random Walk Metropolis Algorithms by Roberts, Gelman and Gilks, the authors state (see below) that "in the Skorokhod topology, it does not ...
- 167
2
votes
0
answers
37
views
Reference request: semimarkov processes
What are some good modern introductions to the theory of semimarkov processes? To be clear, by a semimarkov processes, I mean a Markov chain, together with "waiting times" between transitions, the ...
- 505
2
votes
0
answers
440
views
Hitting time of a specific Markov chain using martingale approach (or otherwise)
Let $0 < c < 1$. Consider the Markov chain $(X_i)$ on $\{0, 1, \dots, n\}$, with transition probabilities
$$ P(k,k+1) = \left(1 - \tfrac {k}{n} \right)(1-c), \quad k = 0, \dots, n-1, $$
$$ P(k,...
- 985
2
votes
0
answers
32
views
$\mbox{Var}(\sum \delta\{X_n > i_n\} )$ i.f.o. correlation of $(X_n)_n$
Question
Suppose we have an ergodic positive stochastic process $(X_n)_{n \in \mathbb{N}}$ (in particular I'm mainly interested in the case where $(X_n)_n$ is an aperiodic, irreducible, positive ...
- 277
2
votes
0
answers
74
views
Literature/Book on counting processes
I seek literature that makes a rigorous treatment of counting processes. In particular im interested in a precise treatment of the conditional intensity $\lambda_t$ which is often informally defined ...
- 315
2
votes
0
answers
207
views
markov processes and ergodic theory
For an ergodic Markov Chain
$$
\frac{1}{N}\sum_{i=1}^n f(X_i) \rightarrow E_\pi[f]
$$
where $\pi$ is the invariant distribution. I am also dealing with a Markovian process (a state space model to ...
- 169
2
votes
0
answers
166
views
Must rows of a transition matrix be distinct?
Is it true that for all continuous time Markov processes on a countable state space $S$, we have
all rows of the transition matrix $\mathbf{P}_t$ are distinct for all time $t\in[0,\infty)$ ?
This ...
- 165
2
votes
0
answers
199
views
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 ...
- 41
1
vote
1
answer
306
views
English translation of a Russian paper by Gordin and Lifšic
Unfortunately I can't read Russian, I was wondering if there is an English translation of this paper
“The central limit theorem for stationary Markov processes”, Dokl. Akad. Nauk SSSR, 239:4 (1978), ...
- 757
1
vote
1
answer
4k
views
First passage time of a 1D simple random walk in a discrete time infinite markov chain [closed]
If we consider a simple Random Walk on the positive integers (discrete Markov chain), with symmetric transition probabilities. We start at time $0$ at the integer $i_0 = m$ and at each time step $P(...
- 53
1
vote
2
answers
228
views
Is a linear combination of Markov generator a Markov generator?
Let $X$ be a compact metric set and $\mathcal{C}(X)$ be the set on continuous real functions over $X$ endowed with the supremum norm $\|\cdot\|_{\infty}$. Let $\Omega_1$ and $\Omega_2$ be two Markov ...
- 449
1
vote
1
answer
96
views
Asymptotic behavior of a Markov process on the set of $\{0,1\}$-polynomials
This question is cross-posted from https://math.stackexchange.com/questions/4711799/asymptotic-behavior-of-a-markov-process-on-the-set-of-0-1-polynomials
I am trying to study the asymptotic behavior ...
1
vote
1
answer
410
views
Occupation times for two-state Markov processes
Consider a two-state Markov process in continuous time, with states labelled $A$ and $B$. The transition rates for going from state $A$ to $B$, and state $B$ to $A$ are $\alpha$ and $\beta$ ...
1
vote
1
answer
305
views
Existence of a Lyapunov function for a log-concave measure
Let $d\in\mathbb N$, $f:\mathbb R^d\to\mathbb R$ be convex with $$\int e^{-f(x)}\:{\rm d}x<\infty\tag1$$ and $\mu$ denote the measure with density $e^{-f}$ with respect to the Lebesgue measure on $\...
- 167
1
vote
2
answers
302
views
how to derive stationary distribution of maximal entropy random walk
I was reading the paper 0810.4113v2, burda, which analyzed the stationary distribution maximal entropy random walk on the irregular lattice. I am confused on some of the steps.
Description:
The ...
- 211
1
vote
1
answer
404
views
Does Irreducibility holds for the Ergodic non-stationary Markov chain?
In the stationary case, I know that if the chain is irreducible and aperiodic, it is Ergodic. But in the non-stationary case, i can not comprehend the content deeply. I want to know if Irreducibility ...
- 121
1
vote
1
answer
222
views
Uniqueness of invariant measure for equivalent transition probabilities
Suppose $P(x,dy)$ and $Q(x,dy)$ are two Markov transition kernels on a topological space $E$ equipped with Borel $\sigma$-algebra $\mathcal B(E)$. Suppose for every $x \in E$, $P(x,\cdot)$ and $Q(x, \...
- 985
1
vote
1
answer
337
views
How can we determine the generator of this Markov process (at least formally)?
Let
$(\Omega,\mathcal A)$ be a measurable space;
$(E,\mathcal E)$ be a measurable space with $\{x\}\in\mathcal E$;
$(Y_t)_{t\ge0}$ be an $(E,\mathcal E)$-valued time-homogeneous Markov process on $(\...
- 167
1
vote
1
answer
233
views
Random walks on Galton–Watson trees
I am working on a paper of Elie Aidekon : ‘Speed of the biased random walk on a Galton–Watson tree’ and have a question about one transformation in a proof:
\begin{align}
& 1+\frac{1}{1-\lambda}+\...
- 39
1
vote
1
answer
175
views
Existence of Markov chain on nonnegative integers with specified rates
Let $\lambda_k,\mu_k\in\mathbb R_{\ge0}$ $(k\ge1)$ be nonnegative real numbers, let $S=\mathbb Z_{\ge0}$ be the nonnegative integers, let $T=\mathbb R_{\ge0}$ be the nonnegative real numbers and ...
- 276
1
vote
1
answer
143
views
Comparison of hitting probability of two Markov chains both with only one absorbing state version 2 under stronger condition
Let $N_n:=\{1,2,\cdots,n\}$. Given two finite states Markov chains $\big(X^{(j)}_i\in N_n\}\big)_{i=0}^\infty$ for $j\in\{1,2\}$, both of which have two absorbing states at $1$ and $n$.
$\text{Pr}\...
- 2,239
1
vote
1
answer
276
views
Number of deaths in birth-death process conditioned on start and end points
Say I have a simple linear continuous time birth-death process with state space the non-negative integers, where there are parameters $b$ and $d$, with the rate (as you'd see in a $Q$ matrix) of going ...
- 11
1
vote
1
answer
108
views
Regularity of the entrance measure of SRW
Let $S(n)$ be the discrete sphere of radius $n$ (i.e., the internal boundary of the Euclidean discrete ball $B(n)$) centered in the origin, and consider a simple random walk starting at some $x\in\...
- 1,897
1
vote
1
answer
370
views
Markov chain with Feller property
Does anybody know whether there is an analysis of when the monotone decreasing chain has the Feller property?
The monotone decreasing is defined as a chain on $\mathbb{N}$ and the rate of going down $...
- 13
1
vote
1
answer
687
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 ...
- 21
1
vote
1
answer
293
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 ...
- 1,034