Stochastic processes (with either discrete or continuous time dependence) on a discrete (finite or countably infinite) state space in which the distribution of the next state depends only on the current state. For Markov processes on continuous state spaces please use (markov-process) instead.

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165 views

Generating a stochastic matrix with a given second dominant eigenvalue

I need a procedure (iterative or otherwise) that, given a positive integer $N$ and a (possibly complex) number $\lambda$ such that $0 < \vert \lambda \vert < 1$, will be able to generate an $N ...
5
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0answers
83 views

Confusion in the proof of properties for $\psi$-irreducibility

Let $P$ be a stochastic kernel on a measurable space $(\mathsf X,\mathfrak B(\mathsf X))$. The kernel $P$ is called $\varphi$-irreducible if for a positive measure $\varphi$ and for all measurable ...
5
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238 views

Potential theory: discrete-time Markov processes

Recently I've found lecture notes on "Analysis on Graphs" where the potential theory methods were used to study discrete-time, time-reversible Markov chains (i.e. the state space is countable). ...
4
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49 views

Markov chains and natural filtration

I have the following problem Consider a homogeneous Markov chain $(X_n)$ with countable state set $E$. Suppose that $A$ is a proper subset of $E$ and consider the stopping times $\tau^0=0 $ and ...
4
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46 views

Standard deviation of a quantum walk?

The standard deviation of a classical random walk with $n$ steps is $\sqrt n$ - Standard deviation of a random walk. I have read in many places that the standard deviation of a quantum walk $n$ with a ...
4
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76 views

A Continuous-Time Markov Process Taking All Possible Values

Let $\mathbb{N}$ be the set of positive integers. For each $n \in \mathbb{N}$, let $X^{(n)}=\{ X^{(n)}(t): t \geq 0 \}$ be a Markov chain with state-space the two point set $\{0,1\}$ and $Q$-matrix ...
4
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133 views

Conditional probability and integrating out part of a random walk

Suppose that I have a random walk process defined by $\alpha_{t+1}$ ~ N$(\alpha_t, \omega^2)$. Given $\alpha_t$ and $\alpha_{t+2}$, I understand why the conditional formula for ...
4
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187 views

estimation of transition probabilities from aggregate data

Please, O mathematicians, help me understand the approach to the problem of estimating transition probabilities given only aggregate data in Kalbfleisch & Lawless' 1984 paper "Least-Squares ...
3
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0answers
37 views

Why a positive recurrent Markov chain implies positive limiting probability?

Let $X=\{X_0,X_1,\ldots\}$ be an irreducible, positive recurrent, and aperiodic Markov chain with the state space $S=\{0,1,2,\ldots\}$ then how do we show that the probability $$ ...
3
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32 views

A basic doubt on Markov chain/ergodicity

Consider a finite state (no. of state $N$) Markov chain $\{X_n\}$ (all the random variables are bounded) such that there is a state $i*$ such that $$ \sum_{i=1}^{N}p_{ii*}^{(n)} > 0$$ for all ...
3
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53 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 \begin{equation} dX_t = b(X_t)dt + \sigma(X_t)dW_t, ~~ X_0 = x_0, \end{equation} where ...
3
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91 views

Markov Chain Alternate Expectation

Consider a Markov chain defined by transition matrix $P$ such that for each transition from state $i\rightarrow j$ the probability is $p_{ij}$. Now say there is an associated value for each transition ...
3
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98 views

SLLN of Markov chains .

Let $X_1$, $X_2$,... be a finite state, irreducible and aperiodic Markov chain with initial state $X_0=i$. It is known that \begin{equation} \mathrm{P}\Big\{\lim_{n\rightarrow\infty} ...
3
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98 views

Prove the 2 definitions of the periodicity of Markov Chain are equivalent.

In many textbooks, there are basically 2 ways of defining the periodicity of Markov Chain. One is by partitioning the graph in to subgraph such that transition in one group of state leads to the other ...
3
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117 views

Monotonic convergence of powers of a stochastic matrix

Let $P$ be a stochastic matrix (nonnegative and each row summing to 1). Assuming that $P^n$ converges to $\textbf{1}\pi$ as $n \rightarrow \infty$, where $\pi$ is a row vector (stationary distribution ...
3
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96 views

maximum renewal rate of a Markov chain

Consider a Markov chain $(X_t)$ on a state-space with a countably generated $\sigma$-algebra and assume this Markov chain allows for small sets of order one. This means there exist sets $\mathfrak{S}$ ...
3
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133 views

Generalization of Dobrushin's Ergodic Decomposition for continuous Markov Chains

Let $T$ be the shift transformation. Let $P$ be invariant for $T$ and also define a discrete state space Markov Chain. Let $C_{1},\ldots,C_{n}$ be the connected components of the Markov Chain. It ...
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33 views

Name for maximum transition probability

Let $p(x,y)$ denote the transition probability of a markov chain. Similarly, let $p^n(x,y)$ be the n-step transition probability. My question is, is there a formal name for $S(x,y):=\sup_n p^n(x,y)$. ...
3
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62 views

When is this reversible diffusion on the integer lattice non-exploding?

Let $U\in C^{\infty}(\mathbb R^n;\mathbb R)$ and consider a continuos time Markov chain on the scaled integer lattice $\delta\mathbb Z^n$ with jump rates given by $r_{\delta}(x,y) := \begin{cases} ...
3
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240 views

Probability question about change in vending machines; maybe markov chain?

Suppose there are vending machine that sells its goods for $3$. It's known that a third of the buyers use three coins of $1$, a third of the buyers use $2$ and $1$, and the last third use $5$. The ...
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0answers
13 views

Proving that an inductively defined function is a Markov chain

Let $X_0$ be a random variable with values in a countable set $I$. Let $Y_1,Y_2,\ldots$ be a sequence of independent random variables, uniformly distributed on $[0,1]$. Suppose we are given a function ...
2
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35 views

Geometric ergodicity and mixing - stationary case

I have this theorem: The Markov Chain {$X_n$} is stationary and geometrically ergodic if and only if {$X_n$} is stationary and absolutely regular with $\beta_n=O(\gamma^n)$ for some $\gamma\in ...
2
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0answers
44 views

Powers of (large) lower triangular matrix

Consider the following "game" of chance. Each time the player pushes a button he is awarded a random (finite, integer, non-negative) number of points. The probability of receiving any particular score ...
2
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0answers
54 views

Markov Chain with Normal Transition Matrix

Consider a (sub)-stochastic matrix $P$, and the associated Markov chain $X$ with \begin{align*} \mathbf P [X_n =y | X_0 = x] = P_{xy}^n. \end{align*} Suppose we have the condition $P^T P = P P^T$, ...
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14 views

$\psi$-irreducibility of m-skeletons.

In Proposition 5.4.5 of Meyn and Tweedie's Markov Chains and Stochastic Stability, it is said that if a chain $\Phi$ is $\psi$-irreducible and aperiodic, then every $m$-skeleton of it is also ...
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32 views

markov spectral radius independent of states?

Let $\Pi$ be a stochastic matrix of an irreducible markov chain. We define the spectral radius of $\Pi$ as: $\rho(\Pi) := \limsup_{n \to \infty} \left( \pi^{(n)}_{(a,b)} \right)^{\frac{1}{n}}$ Why ...
2
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71 views

simple proof of the $L^2$ weak law for discrete-time ergodic Markov processes

Let $\{X_t\}_{t\in\mathbb{Z}}$ be a stationary and ergodic stochastic process with finite second moment. Von Neuman's ergodic theorem implies that the time average $(1/N)\sum_{j=0}^{N-1} X_j$ ...
2
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0answers
61 views

Intuition behind criterion for an irreducible Markov chain to be transient

I have been looking over my notes for Markov chains, and I have come across the following: Theorem: An irreducible Markov chain is transient iff for some state $i$ there exists a nonzero vector $y$ ...
2
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0answers
271 views

Random walk, Cat and mouse

Here is the problem. In graph G, on different vertices there is cat and mouse. Cat and mouse do independent random walk, but time is synchronous, in one unit of time both cat and mouse do one step. ...
2
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0answers
97 views

Markov chain from Poisson

Let $K_t$ be a Poisson process with rate $1$ and $X_n=K_n-n$ $, \ \ \ n\in \mathbb{N}$ am asked to determine whether it is null or positive recurrent, we already know it is recurrent. I ...
2
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0answers
163 views

Boundedness of expected reward Markov chain (may be related to discret $M/M/\infty$ queue)

[EDIT]: I read a bit on $M/M/\infty$ queue and it may not be the right comparison and my notation may be confusing (I'm in discrete time and $\lambda,\mu$ look likes rates when they are probability). ...
2
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0answers
26 views

Ruin time with a maximum purse size

Imagine I have a gambler's ruin scenario where I start with $m$ dollars and I cannot have more than $N$ dollars. For each of however many rounds, I flip a coin, and with probability $p$ I win a ...
2
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0answers
26 views

Broken disk head

Broken disk head: We would like to read 1 byte = sequence of 8 bits from a disk, starting from bit 0. Our disk head reads 1 bit at a time. Disk head can only move forward, but after reaching bit 7 ...
2
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0answers
72 views

Is the following interpretation for the stationary distribution of a Markov process correct?

Imagine I have some Markov process with stationary distribution $\pi$ and a mixing time of $\tau$ after which $|Prob[x=s_i] - \pi(s_i)| \leq \epsilon$. Can I assume the following: A state $(x=s_i)$ ...
2
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0answers
19 views

Problem with the uniform transience

Let $X$ be a Borel space and let us consider a Markov Chain $(\Phi_n)_{n\geq 0}$ on this space given by the stochastic kernel $$ P(x,\mathrm dy) = p(x,y)\mu(\mathrm dy) $$ where the density $p$ is ...
2
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0answers
209 views

In finite-state Markov chain state $i$ is transient

Can you help me please with proof of this question: Prove, that in finite-state Markov chain state $i$ is transient if and only if is exist state $k$ such that $i\rightarrow k$ but k $\nrightarrow ...
2
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0answers
191 views

Computing the stationary distribution of a markov chain

I have a markov chain with transition matrix below, $$\begin{bmatrix} 1-q & q & & & \\ 1-q & 0 & q & & \\ & 1-q & ...
2
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0answers
174 views

Ergodicity and mixing

From MathOverflow, R W said: Unfortunately, the way the term "ergodic" is used in the theory of (finite) Markov chains is completely misleading from the point of view of general ergodic ...
2
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0answers
586 views

Classifying a Stochastic Process as transient, null-recurrent or positive-recurrent

For a discrete time Markov chain with state-space the non-negative integers, for $j>0$, $$ p_{j,k} = \begin{cases} p/j & \text{for } k = j+1 \\ 1 - 1/j & \text{for }k=j \\ (1-p)/j & ...
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0answers
19 views

Is there a single matrix norm such that for all stochastic $P$, $\| P \| = 1$?

By a stochastic matrix I mean a square real non-negative matrix with rows summing to one. Denote the set of all such matrices $\mathcal{S}$. By matrix norm I mean a norm in the vector space of ...
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0answers
17 views

Problem involving periodic Markov Chains — probability of being in a given state at time $n$

I'm working on the following problem: I believe that the simplest possible irreducible periodic Markov Chain would be one with two states and no self-loops? Does this seem correct? However, I'm ...
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0answers
17 views

Random Walk and strong law

I want to prove that a Random Walk in 1 dimension is transient when $p\neq\frac{1}{2}$ but i want to prove it by the strong law of large numbers, so i have this: Define a random variable $$X_i = ...
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0answers
48 views

Can we find a correlation between states of a Markov chain?

I have a fair bit of knowledge on Markov chains but I recently wondered if there is a way to find out a correlation between the states of a finite Markov chain. I could not find any material on this. ...
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43 views

Linear Filtering Problem (Keynman Fac/Particle Model)

$lienar Filtering Problem $$X_n^1 = X_{n-1}^1 + \epsilon_n *W_n $$ $$X_n^2 = (1-\alpha* \delta) X_{n-1}^2 + \beta*\delta X_n^1 $$ $$X_n^3 = X_{n-1}^3 + \delta*X_n^2$$ above is $$\approx$$ $$dX_n^1 ...
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0answers
27 views

Stochastic process using Markov chain (thief on the run!!)

I'm given an exercise where we are to simulate a thief escaping from an officer. The thief (let's call him/her T for simplicity) and an officer (O) have four cities to be in. Let's call the cities A, ...
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0answers
55 views

Decomposing a stochastic matrix into a product of stochastic matrices.

It is well-known that any square real matrix of small rank $k$ can be decomposed into a product of a skinny matrix with $k$ columns and a fat matrix with $k$ rows by means of an SVD. This question is ...
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0answers
16 views

Calculate ultimate survival when more than 1 survival curve is needed to determine outcome

I would like to know if it is possible to combine multiple survival curves via an equation (e.g., via matrix multiplication or whatever) rather than stepping through multiple equations. E.g., assume ...
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0answers
22 views

Cereal boxes - Mean time spent in transient states

Problem: A cereal company gives 2 images in each cereal box it has. There are a total of 5 images. Once a buyer have 5 images she wins a prize. No box contains 2 images that are the same. What is the ...
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0answers
56 views

Follow-up on solution to Markov process equation

I asked a question here about solving a system related to an absorbing Markov chain. I now have a variation where there are $m$ types (of student, job seeker, etc) each of which applies to ...
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0answers
30 views

Conditional return time of simple random walk

Consider a simple random walk on $\mathbb{Z}$, $(S_t)_{t \geq 0}$, with $S_0 = 0$. The probability to jump to the right neighbour is $p \geq \frac{1}{2}$. Call $\tau_k = \min\{t \in \mathbb{N}\, : \, ...