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0answers
14 views

Unique Stationary Distribution of A Markov Chain

I have a Markov Chain like $Y_i=\sum_n\pi_{n,i}(Y)Y_n$, i=1,2,3...N. So the Markov chain has N states and the transition matrix depends on the vector $\textbf{Y}$. I am wondering what conditions $\pi$ ...
2
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2answers
91 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-...
7
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0answers
188 views
+50

Hierarchical (Recursive) Random Walk (also known as Hierarchical Hidden Markov Model)

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/...
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1answer
34 views

Convergence of an inhomogeneous markov chain

A markov chain is defined as $X_t=F(X_{t-1})X_{t-1}$, where $X_t$ and $X_{t-1}$ are both vector. So the transition matrix depends on the current states. I want to show that for any given initial ...
10
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3answers
292 views

How to sample a uniform random polyomino?

A polyomino is formed by joining finitely many unit squares edge to edge. It may be regarded as a finite subset of the regular square tiling with a connected interior. In particular, for us, ...
2
votes
1answer
260 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 ...
0
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0answers
22 views

showing that a matrix has repetitive values?

Here my primary aim is to calculate the stationary distribution of a DTMC using left-eigen values i.e, $ \pi = \pi*P$. But for some matrices, I observe that some states a same stationary probability. ...
5
votes
2answers
171 views

Frequency of visiting states in Markov chains

Given a finite, ergodic Markov $\{X_i\}$, and two natural numbers $a>b$. Let $$p=P\left[\forall n, \sum_{k=n}^{n+a-1} \mathbf{1}_m(X_k)\leq b\right]$$ where $\mathbf{1}_m(X_k) =1$ if $X_k=m$ and 0 ...
5
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2answers
1k views

Random walk to stay in an interval forever

Consider a random walk on the real time, starting from $0$. But this time assume that we can decide, for each step $i$, a step size $t_i>0$ to the left or the right with equal probabilities. To ...
6
votes
1answer
110 views

Basic Definition and Notations in RWRE

From the definition of Zeitouni's lecture notes on RWRE: $(V, E)$ is a special graph, and $N_v:= \{k \in V: (v,k) \in E\}$ is the neighborhood of $v \in V$. $\Omega = \prod_{v \in V} M_1(N_v)$ ...
0
votes
1answer
101 views

Exact formula for computing n-step transition probability of random walks with self-transitions

Consider a semi-infinite random walks $X_n$, $n=0,1,2,\ldots$, whose state space is a set of consecutive integers and whose one-step transition probabilities are $P_{ij}=\mathrm{Pr}\{X_{n+1}=j|X_n=i\}$...
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0answers
76 views

Looking for an exposition of a certain theorem of Talagrand

The following is a theorem by Talagrand (as stated here, http://arxiv.org/pdf/1511.08609v1.pdf), Let $(X, \mu)$ be a probability space. Let $F : X \rightarrow \{0,1\}$ be a family of functions ...
1
vote
1answer
63 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, \...
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0answers
26 views

Dependency of the error term on the states, in the definition of the transition rates of a continuous time Markov chain

I think this is certainly not a research or graduate level question. But I didn't get any answer from math.stackexchange.com. I'm studying G.F.Lawler's stochastic process book. There he defines the ...
1
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2answers
69 views

How can I efficiently approximate the stationary distribution of an infinite CTMC with a sparse rate matrix?

I am looking for methods to approximate the stationary distribution of an infinite CTMC with a sparse rate matrix. Each row and column of the rate matrix has a finite number of non-zero elements. ...
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3answers
374 views

Why does the overhand shuffle converge to the uniform distribution on $S_n$?

Pemantle 1989 proves, among other things, that the Markov chain on $S_n$ induced by repeatedly and independently performing an overhand shuffle on a deck of $n$ cards is ergodic and has limiting ...
2
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0answers
132 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 ...
3
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2answers
168 views

The necessary sufficient condition for recurrence of a Markovian random walk

Suppose $\sigma_{1},\sigma_{2},...$are i.i.d random variables.$S_{0}=0$. Define $S_{n}=S_{0}+\sum_{i=1}^{n}\sigma_{i}$, then ${S_{n}}$ is a Markovian random walk. I want to figure out the necessary ...
-1
votes
1answer
39 views

Discrete time hidden markov process [closed]

I am dealing with a hidden Markov model for variable $X_{t+1}$ where $X_{t+1}$ = $\alpha_{t}$$X_{t}$ + $(1-\alpha_{t})$$Z_{t}$ $X_{t}$ is an indicator variable indicating wether an individual is ...
1
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1answer
130 views

Neat definition of Harris Ergodicity

I can't find any reference where the definition of Harris Ergodicity for Continuous time Markov processes is defined. a) What would be exactly the definition? b) What reference could be helpful? ...
6
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1answer
325 views

Does every (generalized?) Markov chain admit transition probabilities?

To pose the question let us start by recalling the following notions: Transition Probabilities. A transition probability matrix between two measurable spaces $(S,\mathcal{S})$ and $(V,\mathcal{V})$...
1
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1answer
70 views

Is there an easy way to convert a non-deterministic optimal policy to a deterministic optimal policy for a given MDP?

For a MDP (Markov Decision Process) is there an easy way to convert a non-deterministic optimal policy into a deterministic optimal policy? The trivial way will take $O(|\mathcal{A}|^{|\mathcal{S}|}$...
2
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0answers
153 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 ...
5
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0answers
69 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) \...
1
vote
1answer
60 views

Choose uniformly from fixed-length paths in $[0,n]\cap\mathbb{Z}$ with fixed start and end

Let $X_k$ be a symmetric (discrete time) random walk on $\mathbb{Z}$ and let $m,n\in\mathbb{N}$. I want to chose uniformly from the paths of $X_k$, which start at $0$ stay in $[0,n]\cap\mathbb{Z}$ ...
0
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0answers
30 views

Sufficient moment conditions to make $E[\sup_n |X_n|]< \infty$ for Markov process $X_n$

Is there any Markov process $X_n$ for which we can impose sufficient moment condition which will imply $E[\sup_n |X_n|]< \infty$
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0answers
34 views

Strong Markov vector-valued process from component strong Markov process and independence

I want to prove that if $X$ and $Y$ are (continuous time) independent strong markov $\mathbb{R}$-valued processes w.r.t. their natural filtrations $\mathcal{F}^X_t$ and $\mathcal{F}^Y_t$, that the ...
1
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0answers
46 views

Expected number of forward jumps to reach a given quantile of a rv [closed]

I'm a noob in randomized algorithm and ran into a problem(definitely not home work. I'm doing a self study out of my interest with help of my friends. I'm pursuing research career in a machine ...
0
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1answer
30 views

Does an infinite chain of a.s. eventual transitions between states necessarily implies a.s transitions along the whole chain?

Given a Markov process among a (possibly infinite) set of states $S$, with possibly infinite depth (that is, the transition probabilities from $s_i \to s_j$ at time $t$ are permitted to depend not ...
2
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1answer
111 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) (...
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0answers
57 views

Maximal inequality for Markov process

For a Markov process $\{X_n\}$ is there any inequality available for $$ E[\sup_{0 \leq n \leq k} X_{n}]$$ in terms of moments of $E[X_n], 0 \leq n \leq k$
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0answers
19 views

Can MDPs over functions be solved?

I understand that dynamic programs are difficult to be solved in general. However I have an MDP, for which intuitively I have a solution, I am curious to know if there is a formal approach to get a ...
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votes
1answer
34 views

how to resolve the infinite nesting of interactive POMDP

I am reading papers about I-POMDP. I cant understand the finitely nested I-POMDPs given in these papers. The belief update of the algorithm has a problem that agents' belief updates mutually depend ...
1
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0answers
101 views

Markov Chains and Simple Machine Learning [closed]

Suppose I have a large training set consisting of many strings of symbols. $TS = \{Str_0, Str_1, ..., Str_n\}$ $Str_i = \{Sym_0 ... Sym_{len}\}$ These strings of symbols are each generated by the ...
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0answers
23 views

Properties of a map regarding the space of invariant probability measures for controlled Markov process

Let us consider a controlled Markov process with the transition kernel $p(dy|x,\theta)$ ($\theta$ being the control parameter. Now, consider the map $\theta \to I(\theta)$ where $I(\theta)$ is the ...
0
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0answers
30 views

Markov Modulated Markov Chain

Consider a discrete time Markov chain $X_t$ on some finite state space $\mathcal{S}$ with transition matrix $P$. Now consider a process $Y_t$ also on $\mathcal{S}$, which conditioned on $X_{t}=s$ ...
2
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1answer
80 views

Eigenvectors of a perturbed reducible stochastic matrix

Let $Q$ be a $n\times n$ reducible stochastic matrix. Let $J$ be such that $[J]_{ij}={1 \over n}$. Now for a small positive constant $\alpha\in [0,1]$, consider the matrix $$\tilde{Q}\,=\,(1-\alpha)Q+\...
0
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0answers
56 views

Circular process ergodic?

Let us define a continuous-time Markov process on a circle consisting of $m-$ equally spaced points, i.e. every point has two neighbours. Now, we define a space of functions $S:= \{-1,1\}^{\{1,...,m\}...
0
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1answer
176 views

Mixing time of lazy random walk on the directed cycle $C_n$

Briefly: A hint (if this is easy), reference or derivation would be of great help. The question Let $C_n$ be the directed cycle with loops in each of its $n$ vertices, and consider the random walk ...
1
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0answers
17 views

Bounding Hidden Markov model Bayesian filter error with inexact models

In context of a hidden Markov model, I am interested in bounding the error of a Bayesian filter when using inexact state transition and observation models. Consider a hidden Markov model (HMM) with ...
0
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0answers
48 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\...
0
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0answers
68 views

Order statistic of Markov chain sample path and related probabilities

Consider a 1D sample path, denoted as $\{X(1), ..., X(t), ..., X(n)\}$, generated from a discrete time finite state (time homogeneous) Markov chain over states $\{1,...,m\}$, with transition ...
1
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0answers
37 views

steady state of a continuous-time birth-death process

we consider a continuous-time birth-death process $\{X(t),t\geq 0\}$ with discrete state space taking non0negative integer values $\{0,1,2,3,...\}$. The transition rates of the process $\{X(t)\}$ are ...
3
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1answer
101 views

Similarity transformation of transition matrix of reversible Markov chain (reference request)

If $P$ is the transition matrix of a reversible Markov chain, and $\pi$ is its stationary distribution, and let $R$ be defined by: $$R_{ij} = \sqrt{\frac{\pi_i}{\pi_j}}P_{ij}~.$$ By reversibility, ...
2
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0answers
46 views

Spectral gap of two step Markov chain

Suppose $X_1,X_2,\ldots$ is a reversible Markov chain with state space of size $k$ and absolute spectral gap $\gamma_*.$ What is the spectral gap of the (non-reversible) Markov chain $Y_1,Y_2,\ldots,$ ...
0
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0answers
32 views

steady state distribution for a jump Markov chain

Consider a queueing process with the following transition matrix: $\mathbf{P}=\left( \begin{smallmatrix} 1-\lambda & \lambda & & & & & & &\\ \mu & 1-\...
0
votes
1answer
88 views

Finite hitting time implies hits at any finite time?

I was wondering about the following problem: Assume we have a state space $S:=\mathbb{Z}$ and a Markov chain, such that we can go from any state $x$ to some state $y$ with positive probabilities, i.e....
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0answers
27 views

steady state distribution of the following infinite-state Markov chain

Given the following state transition equation: $P_0(n+1)=P_0(n)(1-\lambda \Delta t)+ P_1(n)\mu \Delta t$ $P_j(n+1)=P_{j}(n)(1-\lambda \Delta t-\mu \Delta t)+\lambda \Delta t P_{j-1}(n)+ \mu \Delta t(...
0
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1answer
111 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 $...
0
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0answers
223 views

Calculate the KL divergence between two transition matrices

I want to calculate how different two markov transition matrices are. For example: $\begin{pmatrix} .2 & .8 \\ .1 & .9 \end{pmatrix}$ and $\begin{pmatrix} .3 & .7 \\ .1 & .9 \...