All Questions
Tagged with pr.probability markov-chains
117 questions with no upvoted or accepted answers
9
votes
0
answers
239
views
Is the P.M.F. of the first return time of a random walk monotone?
Suppose $X_1,X_2,\ldots$ are i.i.d. $\mathbb Z$-valued random variables such that the random walk
$$S_n=\sum_{i=1}^nX_i$$
is recurrent with some period $k\geq1$ (i.e., $\Pr[S_n=0]>0$ if and only if ...
- 891
8
votes
0
answers
130
views
Functions between Markov chains that preserve local harmonicity
Given two Markov chains with respective state-spaces $S$ and $T$, say that a function $\phi$ from $S$ to $T$ is holomorphic iff for all states $t \in T$, every real-valued function $f$ on $T$ that is ...
- 19.7k
7
votes
1
answer
391
views
Idempotent splitting for Markov kernels
Let $X$ be a standard Borel space and $e : X \to X$ a Markov kernel. Suppose that $e$ is idempotent, that is $e \circ e = e$, or written out using the Chapman-Kolmogorov equation,
$$e(A|x) = \int_X e(...
- 6,406
6
votes
0
answers
608
views
'Permutation Coupling' for Markov Chains
Suppose I have a Markov chain (discrete time, finite state space) on $[N] = \{1, 2, \cdots, N\}$, with Markov kernel given by a doubly stochastic matrix $P$. The double-stochasticity guarantees that ...
- 801
6
votes
0
answers
337
views
Maximal inequalities for square of partial sums
Let $S_n = \sum_{i \leq n} X_i$ be the partial sums of a nice sequence of random variables $X_i$. In my application, $X_i$ is a functional of a finite-state, irreducible, aperiodic Markov chain, so ...
- 1,069
6
votes
0
answers
1k
views
Coin Toss Probabilities like Penney's Game
Generate a binary number, using coin toss. Until you receive a predefined terminating sequence. What is the probability that the number is a multiple of some $k$.
For example, the terminating ...
- 91
6
votes
1
answer
218
views
Finding cohesive (low exit probability) sets in a Markov process
The following is a fact about Markov chains that came up in a game theory paper. The purpose of this question is to ask if related notions or similar results are found elsewhere in probability, or are ...
- 1,068
5
votes
0
answers
272
views
How to play golf in one dimension?
One-dimensional golf is a function $g$ on $\mathbb R$ such that
$g(x)= 1+\min_\mu E[g(x+N(\mu,c\mu^2))]$ if $|x|>1$ and 0 if $|x|\le 1.$
Here $N$ is the normal distribution, whose mean $\mu$ you ...
- 18.9k
5
votes
0
answers
485
views
Hierarchical Random Walk (also known as Hierarchical Hidden Markov Model)
Let us consider the following hierarchical (recursive) random walk model, which is also known as the hierarchical hidden Markov model in computer science (https://en.wikipedia.org/wiki/...
- 1,127
5
votes
0
answers
95
views
Most visited vertex in a random walk with place dependent drift
Consider the following Markov chain on $\mathbb{Z}$:
$$
P(x,x+1)=1-P(x,x-1)=\frac{1}{2}+e^{-|x|}\cdot \mathbf{1}_{\{x\neq 0\}}
$$
Do there exist constants $c,C>0$ such that
$$
c\cdot P^t(z,z) \...
- 527
4
votes
0
answers
61
views
Mixing times for the exclusion process with rejection
Consider the following Markov chain on $k$-subsets of $\{1,\ldots, L\}$, equivalently, sequences $x\in \{0,1\}^L$ with $k$ 1's.
Let $p_1,\ldots, p_L\in (0,1)$ and $q_i=1-p_i$.
At each step, choose an ...
- 371
4
votes
0
answers
589
views
Optimal transport between two distributions in a Markov chain
In a previous question, given an ergodic Markov chain, I'm interesting in sampling as short a path as possible with prescribed distributions for its endpoints. In a comment, I propose that the ...
- 6,853
4
votes
0
answers
153
views
Mixing time for dimers on the square-octagon graph
Consider the "fortress graph" of order $n$ (see Figure 9 of http://faculty.uml.edu/jpropp/tiling/www/mdblum/arctic.html). It's been known empirically for twenty years that if one turns the set of ...
- 19.7k
4
votes
0
answers
264
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 ...
- 18.8k
4
votes
0
answers
282
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 ...
- 141
4
votes
0
answers
1k
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 ...
- 509
3
votes
0
answers
58
views
Infinitesimal generators of random evolutions
Consider two state spaces $X$ and $Y$ and infinitesimal generators of Markov processes $(A_y)_{y\in Y}$ and $B$, on $X$ and $Y$ respectively. We assume that $A_y$ share the same domain $D(A)$, and ...
- 31
3
votes
0
answers
241
views
A few questions on Feller processes
Update. Most of my questions have been answered in the comments. I am adding these answers to the post.
There are at least three definitions of Feller semigroup and the corresponding processes: $C_0 \...
- 620
3
votes
0
answers
54
views
Multi-type Galton-Watson-like process where only majority-type is allowed to reproduce
Are you aware of any research papers that have explored a multi-type Galton-Watson process in which only particles of the majority type are permitted to reproduce in each generation?
I've been unable ...
- 56
3
votes
1
answer
373
views
Concentration of very dependent Markov chains
Consider the following simple Markov chain $ X_1\to X_2\to\cdots\to X_n $ where each $X_i$ is $\{-1,1\}$-valued and $X_1\sim\mathrm{Unif}(\{-1,1\})$ (such that the chain is stationary).
The flip ...
- 161
3
votes
0
answers
90
views
How does one define the gradient of a Markov semigroup?
In the context of functional inequalities for Markov semigroups $(\mathcal P_t)_{t\ge0}$, what is one denoting by $\nabla\mathcal P_tf$? For example, I've found the following assumption in this paper:
...
- 167
3
votes
0
answers
88
views
Joint drunkard walks
The drunkard walk is a game where two players have $a$ and $b$ dollars, respectively, and they play a series of fair games (both risking one dollar in each game) until one of them goes broke.
My ...
- 131
3
votes
0
answers
106
views
Find the generator of a markov process with constant decay and exponential jumps
Suppose we have a continuous time Markov process $(X_t)_{t\in [0,\infty)}$. This Markov process represents the queue length in amount of work left, therefore its state space is given as $S = [0,\infty)...
- 277
3
votes
0
answers
115
views
Approximating the *conditional* probability of 1D discrete random walk not having revisited the origin given last position
I'm looking for a good closed form approximation to the following conditional probability, with provable approximation guarantees.
Consider a 1D random walk on the integers, starting at the origin, ...
- 101
3
votes
0
answers
182
views
Spectral radius of infinite substochastic upper triangular matrix
Let $M$ be a Markov chain on $\{0, 1, 2, \dots\} \cup \{\delta\}$, where $\Pr(i \to j) > 0$ for $i, j \in \mathbb{N}$ only if $j > i$, and $\Pr(\delta \to \delta) = 1$. This represents a birth-...
- 131
3
votes
0
answers
112
views
Conditional expectation with respect to paths of a Markov jump process
I'm having some trouble detangeling how the conditional expectation in equation (2.13) in the article https://arxiv.org/abs/cond-mat/9811220 (Lebowitz, Spohn) is defined.
The context is as follows: ...
- 53
3
votes
0
answers
151
views
Sequential generation of any random graph
The high-level question is: can we generate any random graph with size $d$ using a Markov chain?
For example, let $X^{(0)} = (1,0,\ldots,0) \in R^d$ be the initial state, and $X^{(t+1)} = f^{(t)}(X^{...
- 1,127
3
votes
0
answers
144
views
The spring Markov chain on $\mathbb{N}$
I'm trying to understand and learn more about "almost surely bounded" Markov chains on countable state spaces. I'm looking for references where I can learn how to work with more complicated examples ...
- 1,961
3
votes
0
answers
158
views
Worst-Case Solution to (Stochastic) Matrix Inequality
EDIT: Some specific conjectures added.
This problem comes with an associated stochastic process, but I phrase everything as linear algebra in case somebody from a non-probability community has seen ...
- 31
3
votes
0
answers
305
views
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 ...
- 61
3
votes
0
answers
494
views
Maximization of a total variation distance subject to another total variation distance in Markov chain
Suppose two dependent random variables $X$ and $V$ from finite alphabets $\mathcal{V}$ and $\mathcal{X}$ with known joint and marginal distributions are given. Let $P_{XV}$ and $P_X$ and $P_V$ are the ...
- 1,109
3
votes
0
answers
95
views
Best convergence rate for convolutions on $\mathbb{Z}_p$
Suppose, that we have sequence of i.i.d variables $X_1,\ldots,X_n$ taking values in $\mathbb{Z}_p$ such that $d_{TV}(X_1,U) < \delta$.
How fast, in terms of $\delta$ and $n$ does the sum $X_1+\...
- 696
3
votes
0
answers
770
views
Kullback-Leibler Divergence of Stationary Distributions of Markov chains
Consider two finite Markov chains on the same state space, both assumed to be irreducible, with transition matrices $P$ and $Q$ and associated stationary distributions $\pi$ and $\tilde \pi$. Is it ...
- 418
3
votes
0
answers
341
views
maximum variance unfolding
Consider positive weights $\pi_1, \ldots, \pi_n$ (one can suppose that they add up to $1$) and $n-1$ lengths $d_1, \ldots, d_{n-1}$.
Is there an analytical solution to the following problem:
find the ...
- 2,133
2
votes
0
answers
124
views
dimensionality reduction of Markov chains
Suppose that $M$ is a time-homogeneous (and, for simplicity, stationary) Markov chain on $d$ states, which induces the probability measure $P$ on paths of length $n$. I seek a Markov chain $M'$ on $d'&...
- 6,541
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
237
views
Ball games: How to allocate $N$ balls into $M$ boxes so as to maximize the expected number of taken balls
Consider the following ball games, which looks like very intuitive and simple but I have tried for a long time.
Assuming we have $M$ identical boxes and $N$ identical balls, we distribute these $N$ ...
- 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
57
views
Right spectral gap of vector of two independent Markov chains
Let $(X_i)$ be a stationary Markov chain on $S$ (a potentially uncountable space with a Borel sigma algebra) with stationary distribution $\pi$ and transition kernel $P$. Let $(Y_i)$ be a stationary ...
- 203
2
votes
0
answers
123
views
Probability of a finite cylinder set in a free group
Let $\mathbb{F}_n$ be the free group (each elemen is in its reduced form) generated by the set $\Sigma_n = \{a_1, a_2, \cdots, a_n, a_1^{-1}, a_2^{-1}, \cdots, a_n^{-1}\}$ and let $e$ denote the ...
- 307
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
321
views
Why do we assume that $\mathcal{A}$ is an algebra in this 2003 paper of Bobkov and Tetali?
In Bobkov and Tetali - Modified Log-Sobolev Inequalities, Mixing and Hypercontractivity (extended version Modified Logarithmic Sobolev Inequalities in Discrete Settings), at the beginning of section 3,...
- 105
2
votes
0
answers
113
views
Characterizing the relationship between element-wise Markov transitions and the full-conditionals of the stationary distribution
Consider a $p$ dimensional random variable with a discrete support. Consider a Markov transition kernel on the state space that is defined in terms of element-wise transition distributions.
One can ...
- 2,791
2
votes
0
answers
70
views
If $X^n$ is a sequence of càdlàg processes whose FDDs converge to a continous process $X$, does $X^n$ converge to $X$ in the Skorohod topology?
Let $(\Omega,\mathcal A,\operatorname P)$ be a probability space, $E$ be a complete locally compact separable metric space, $(X^n_t)_{t\ge0}$ be an $E$-valued càdlàg process on $(\Omega,\mathcal A,\...
- 167
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
74
views
Random contractions and contractions on the space of measures
Let $(S,d)$ be some separable and complete metric space, and let $\mathbb{F}$ be some collection of functions from $S$ to $S$. Endow $\mathbb{F}$ with a suitable sigma algebra such that everything I ...
2
votes
0
answers
102
views
Verifying if Markov process's dependency on time is not random
I am simulating two objects on a grid, and checking how long they can run until the two objects meet. The two objects move randomly, and choose any block (up, down, left, right) randomly. If the side ...
- 21