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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The expectation of total number of different states in N time points

[Conditions] (1) An object has K possible states. (2) This object can have only one state at a single time point. (3) The probability of each state at any single time point is 1/K, and each time ...
3
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1answer
199 views

A basic question on irreducible periodic markov chain

For an irreducible periodic (period $2$) Markov Chain I know that both of the following two quantities are same and equal to $\pi(i)$: $$ \lim_{n\to \infty} \frac{1}{2}(p_n(j,i) + p_{n+1}(j,i))$$ $$ ...
3
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1answer
397 views

Proof about Steady-State distribution of a Markov chain

Consider $A$ as a matrix, that when normalized represents an finite-state time-homogeneous Markov chain $M$ with entries $0\leq p_{i,j}\leq 1$, where each row sums up to $1$. We can also assume that ...
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1answer
149 views

what's the generalized approach to this infinite state markov chain problem

Say, a bag has 10 balls, in which 9 are red, 1 is black. Each red ball is worth 1 point, each black is worth 4 points. I have 8 picks from the bag to start with (the bag refills itself after each ...
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1answer
383 views

Countable state Markov chain: detailed balance consequences

Let $S$ be a countable set and $\pi$ a probability distribution on $S$. A discrete-time Markov chain $(X_n)$ with state space $S$ is said to be in detailed balance with respect to $\pi$ (or simply in ...
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1answer
785 views

Probability distribution of markov chain

I have a Markov chain with state space $E = \{1,2,3,4,5\}$ and transition matrix below: $$ \begin{bmatrix} 1/2 & 0 & 1/2 & 0 & 0 \\ 1/3 & 2/3 & 0 ...
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1answer
1k views

coin flips and markov chain

Consider the case of an infinite (or finite $n$) string of coin tosses, and let $q$ and $1-q$ be the probabilities that the coin comes up tails and heads, respectively. (For simplicity, we can take ...
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1answer
91 views

What happens to a regular Markov matrix that has more than one steady state/stationary distribution?

It is known that for a regular Markov matrix $M,$ $M^{n}$ has the steady-state vector as all of its columns as $n \to \infty.$ I learned this in class, but what if there is more than one steady-state ...
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1answer
235 views

Identifying states in Markov chains

I just started learning about Markov processes and got the following homework question. Classify all the states as recurrent or transient for the Markov chain below $$\begin{matrix} ...
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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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69 views

2D random walk variation

If a point on a 2D lattice is allowed to take a random walk by taking a unit step either up, down, left or right, there is probability $1$ of reaching any point (including the starting point) as the ...
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47 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 ...
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78 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 ...
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95 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} ...
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1answer
74 views

expected hitting time with two absorbing states

Consider a Markov chain in a finite space and with two absorbing states, each of which is accessible from the other, transient states. Is the expected number of transitions to reach any single ...
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94 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 ...
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111 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 ...
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95 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 $\mathfrak{S}$ ...
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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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0answers
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 $S(x,y):=\sup_n p^n(x,y)$. ...
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60 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} ...
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3answers
189 views

Markov chain stationary probability simulation

Having a defined markov chain with a known transition matrix, rather than to calculate the steady state probabilities, I would like to simulate and estimate them. Firstly, from my understanding there ...
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238 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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2answers
62 views

Transition Matrix eigenvalues constraints

I have a Transition Matrix, i.e. a matrix whose items are bounded between 0 and 1 and either rows or columns sum to one. I would like to know if it is possible that in any such matrices the ...
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2answers
311 views

The second largest eigenvalue for Perron-Frobenius matrix

The Perron-Frobenius theorem is about the largest eigenvalue and eigenvector of a non-negative (irreducible) matrix. My question: Is there any estimation of the difference between the first and ...
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1answer
247 views

If $P$ is a regular transition probability matrix then $P^{n^2}$ has no zero element

A transition probability matrix $P\in M_{n\times n}$ is regular if for some $k$ the matrix $P^k$ has all of its elements strictly positive. I read that this can be tested by using the following ...
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1answer
2k views

What does the steady state represent to a Markov Chain?

I'm a little confused as to the interpretation of the steady state in the context of a Markov chain. I know Markov chains are memoryless, in that each state only depends on its immediate predecessor, ...
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1answer
3k views

How can I compare two matrices?

I have a matrice A. It is model probability matrice for some process (Markov chain). Then, I have estimated matrice B. I have to somehow compare these two matrices to tell whether process that gave ...
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2answers
122 views

Using Markov - Chain to find average and probability

Suppose a computer generate a random vector of n positions where each position appears on of the numbers from 1 to n. The generation is performed uniformly on the $n!$ possibilities. In the problem we ...
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2answers
150 views

Professor has 4 umbrellas, Markov chain and Probability

OK this problem is making me tear my hair out. I need someone to walk me through this in baby-steps method like 1 + 1 = 2. I am trying to figure out what I don't understand. I know this is going to be ...
2
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2answers
104 views

Independence of the first passage time of a Markov chain being less than or equal to $n$ and $X_n$

I am reading my lecture notes on Markov chains, and in the proof of one proposition the following statement is made: "For $n = 1,2, \dots$ the event $\{n \leq T_k\}$ depends only on $X_0, \dots, ...
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2answers
51 views

Is there a proof that the observations of a hidden Markov chain is not itself a Markov chain?

Suppose $\{X_n\}$ is the hidden Markov chain, and $\{Y_n\}$ is the series of observations, where $\mathbb{P}\{Y_n = j| X_n = i\}$ is the same for all $n$ (please correct me if I have not stated the ...
2
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1answer
541 views

A problem on Expected value using the survival function

Let $X$ be a random variable denoting the number of times needed to roll ( including the last roll) a fair six-sided die until we obtain 4 consecutive six's. I would like help in computing ...
2
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1answer
68 views

Random walk on connected graph: show $E_vT_w \ne E_wT_v$

Let $G$ be a connected graph on at least 3 vertices in which the vertex $v$ has only one neighbor, namely $w$. Let $(X_t)_{t \ge 0}$ be a simple random walk on $G$, where $X_t$ is the current vertex ...
2
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1answer
35 views

Finding the probability of ever visiting a transient state for a zero-seeking device for a Markov Chain?

A zero-seeking device operates as follows: if it is in state $j$ at time $n$, then at time $n+1$, its position is $0$ with probability $\frac{1}{j}$ or $k$ with probability $\frac{2k}{j^2}$, where $k$ ...
2
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1answer
29 views

Proving that a HMC state is recurrent or transient?

Looking at the HMC $$\begin{bmatrix} 1-\alpha & \alpha \\ 0 & 1 \end{bmatrix} $$ How do I prove that the state 2 is recurrent and that state 1 is transient? What does it actually mean by ...
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2answers
119 views

How do you prove that the second largest number $Y_n$ shown among the first $n$ rolls is not a Markov Chain?

How do you prove that the second largest number $Y_n$ shown among the first $n$ rolls is not a Markov Chain? My attempt: Consider the case, $P(Y_{n+1}=3|Y_n=1)=\frac{1}{6}$ if the current ...
2
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2answers
135 views

Wald's equation example controversy

I'm trying to get a grip of Wald's equation, applying it to the following example. Suppose, we have a simple sequence of fair coin flips, where heads wins us a dollar, while tails means loss of a ...
2
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2answers
73 views

Random walk on one-dimensional lattice - understanding the expression $pe^{i\theta} + qe^{-i\theta}$

I've started reading the book - First Steps in Random Walks and in the very first example in Chapter 1 they talk about a random walk on a one-dimensional lattice. If we consider a particle starting ...
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2answers
1k views

Markov chain with infinitely many states

I understand that a Markov chain involves a system which can be in one of a finite number of discrete states, with a probability of going from each state to another, and for emitting a signal. ...
2
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1answer
174 views

Some basic questions on Markov chains (Durrett)

If you have a state space $S$, usually I think of a Markov chain $X_n$ on it as $X_n$ takes values in $S$ and satisfies the obvious Markov property and so on. In Durrett's book, he says one should ...
2
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2answers
284 views

What does it mean for MCMC to converge?

I know that a Markov Chain is a discrete random process where the current state decides the next and in a random walk, the probability that we move from node u to v is 1/N(u). An MCMC sample will ...
2
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2answers
281 views

Calculating probabilities (Markov Chain)

Let $\mathcal{X}=(X_n:n\in\mathbb{N}_0)$ denote a Markov chain with state space $E=\{1,\dots,5\}$ and transition matrix ...
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1answer
81 views

Irreducible MCs

Why is it that theorems for (discrete) Markov chains always require that the MC concerned is irreducible? Can problems with reducible MCs can be simplified to considering the irreducible components? ...
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3answers
631 views

How to show Martingale property for sum of $S_k-E(S_k)$-summands where $S_k$ is a function of two RV's

EDIT: new formulation of the question (old version below). In a paper I found the statement that a certain sum $M_n =Y_1+\dotsb Y_n$ is a martingale, $Y_i=f (X_k, Z_k) - E ( f(X_k, Z_k) | X_k)$. (The ...
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1answer
74 views

Probability of a substring occurring in a string

Consider a random string of length $n<\infty$ where each digit can be between 0-9 with equal probability and a substring of length $k<n$ consisting of only zeros. What is the probability of ...
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2answers
23 views

Probability of returning to a given state after n transitions-Markov chains

Let us denote $f_j^{(n)}$ denote the probability of the first return to state $j $after n transitions. Let $p_{jj}^{(n)}$ be the probability of returning to the state $j$ after $n$ transitions when ...
2
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1answer
23 views

Applying transition matrix to a probability vector seems to ruin its normalization

I had a little bit about stochastic processes during my "Statistical Physics" course and on my exam I got a problem with a Markov chain. My solution seems to be without computational mistakes (checked ...
2
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1answer
30 views

Markov chains by hand

If I have a starting point: $A_T=[0,1]$ at $T=1$ and a one step transition matrix of: $B=\left[ \begin{align} &\frac34 & \frac14& \\& \frac1{20}& \frac {19}{20} &\end{align} ...
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1answer
77 views

Inverse of a regular stochastic matrix

Is it true that the inverse of a regular stochastic matrix is also regular? Are there any other interesting features that the inverse may have of a regular stochastic matrix? Hope someone could answer ...