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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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
39 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
126 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
196 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 ...
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2answers
74 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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4k views

Expected number of steps/probability in a Markov Chain?

Can anyone give an example of a Markov Chain and how to calculate the expected number of steps to reach a particular state? Or the probability of reaching a particular state after T transitions? I ...
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179 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 ...
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2answers
308 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 ...
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2answers
298 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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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
708 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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20 views

Markov Chain: Moving on a circle

A particle moves on 12 points situated on a circle. At each step it is equally likely to move one step in the clockwise or in the counterclockwise direction. Find the mean number of steps for ...
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2answers
46 views

Verifying the Markov property

We throw a dice infinitely often. Define $U_n$ to be the maximal number shown up to time $n$. How can I verify that $$ ...
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2answers
27 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 ...
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28 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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127 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 ...
2
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1answer
82 views

time-homogeneous continuous time Markov chain

I have a question about the continuous time Markov chain. In the Poisson process we have independent and stationary increments. Do we have this in a continuous time Markov chain that is ...
2
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1answer
43 views

Is the following Markov Chain a martingale?

Say I have a finite, ergodic Markov chain with states ${0,1,2,3}$ and with the following transition matrix: $$\begin{bmatrix} \frac{7}{10} & \frac{3}{10} & 0 &0\\ \frac{1}{10} & ...
2
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1answer
61 views

Probability distribution of Poisson process

Let $X_t$ and $Y_t$ be two independent Poisson process with rate parameter $\lambda_1$ and $\lambda_2$, respectively, measuring the number of customers arriving in stores $1$ and $2$, respectively. ...
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2answers
49 views

Generate random sample with three-state Markov chain

I have a Markov chain with the transition matrix $$\pmatrix{0 & 0.7 & 0.3 \\ 0.8 & 0 & 0.2 \\ 0.6 & 0.4 & 0}$$ and I would like to generate a random sequence between the three ...
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239 views

Markov chains: is “aperiodic + irreducible” equivalent to “regular”?

I have two books on stochastic processes. In one book, it says that the limiting matrix is possible to find if the matrix is regular, that is, if for some $n$ $P^n$ has only positive values. The ...
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1answer
87 views

Probability problem with markov property

Problem: In a test paper, the questions are arranged so that 3/4's of the time a True is followed by a True and 2/3's of the time a False is followed by a False. You are confronted with a 100 ...
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1answer
166 views

Proof that Markov Chains converges to the stationary distribution

Let $P$ is a transition matrix of a Markov Chain, which is irreducible, aperiodic and lets assume $\pi$ is its stationary distribution: $\pi = \pi P$. Does anyone knows the proof for the following ...
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425 views

Expected time for winning in biased Gambler's Ruin

Consider the random walk $X_0, X_1, X_2, \ldots$ on state space $S=\{0,1,\ldots,n\}$ with absorbing states $A=\{0,n\}$, and with $P(i,i+1)=p$ and $P(i,i-1)=q$ for all $i \in S \setminus A$, where ...
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2answers
103 views

Continuous-time finite-state Markov chain as a subordinated Brownian motion

I think I read somewhere that every semimartingale is representable as a time changed Brownian motion (sorry, I don't have a reference). This suggests that in particular a continuous-time finite-state ...
2
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3answers
112 views

why is this Markov Chain aperiodic

I have this Matrix: $$P=\begin{pmatrix} 0 & 1 \\ 0.3 & 0.7 \end{pmatrix}$$ this markov chain is said to be aperiodic, I dont understand how it comes to it. Period $\delta$ is the gcd of ...
2
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1answer
182 views

Construction of positive recurrent Markov chain

Let $\{X_i\}_{i\geq 1}$ be i.i.d. with values in $\mathbb N_0$. Define a Markov chain via the following transition matrix: $$p(0,n) = \mathbb P(X_1 = n-1) \qquad p(m,n) = \mathbb P\left(\sum_{k=1}^m ...
2
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2answers
346 views

Finding the exact stationary distribution for a biased random walk on a bounded interval

Imagine we have a biased random walk on an interval $[0, L]$, where the probability of taking a $+1$ step is $p$ and the probability of taking a $-1$ step is $(1-p)$. At the reflecting boundary $0$, ...
2
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1answer
196 views

Random walk with 3 possible steps

I have i.i.d. random variables with following distribution: $$ P(\xi_i =1) = p_1, \ P(\xi_i = 0) = p_0, \ P(\xi_i = -1) = p_{-1}; \quad S_n = \sum^n_{i=1}\xi_i.$$ I am interested in probability of ...
2
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1answer
136 views

Markov Property

this relates to an unanswered question I posted a few days ago: Let $\{ X_t : t = 1, 2, 3 \dots \}$ follow a 2-state Markov chain with transition matrix P. Does the Markov property mean I can break ...
2
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1answer
84 views

How does this Markov process involving balls and bins behave?

I have some set $S_1,\ldots,S_k$ ($k \geq 3$) of bins, each initially with $N_0(S_i)$ balls ($N_t(S_i)$ denotes the number of balls in $S_i$ at time $t$). A bin can contain a negative number of balls. ...
2
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1answer
102 views

Expected number of jumps in regular jump HMC

Consider a homogeneous Markov Chain $X$ on a countable state space, ie a jump process. It is said to be regular (does not explode) if there are only a finite number of jumps in every finite interval. ...
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1answer
494 views

Understanding a Markov Chain

I am using a Markov Chain to get the 10 best search results from the union of 3 different search engines. The top 10 results are taken from each engine to form a set of 30 results. The chain starts ...
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2answers
190 views

Markov chain basic positive recurrency question

If a discrete markov chain is stationary (as far as I know: doesn't modify itself with time), irreducible (doesn't have transient states) and aperiodic (no periodic states), is it positive recurrent? ...
2
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1answer
454 views

Expected time of mouse's survival (stochastic matrix)

In the following wikipedia page explaining stochastic matrices, there is an example with 5 boxes and a cat and a mouse where they jump to a left or right box at every turn and it explains how to ...
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2answers
638 views

Proof of Markov property for Ehrenfest urn

[the question got downvoted on MO with the recommendation to ask here] In many books Ehrenfest Urn is used as an example of a homogeneous Markov chain, where entries in transition probabilities ...
2
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2answers
422 views

Markov Chain: Pensioner Problem

A pensioner receives 2000 dollars at the beginning of each month. The amount of money he needs to spend during a month is independent of the amount he has and is equal to i (i.e. i thousand dollars) ...
2
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1answer
78 views

Invariance on the product space

Let us consider two spaces $\mathbb{X},\mathbb{Y}$. For simplicity we put $\mathbb{X} = \mathbb{Y} = \mathbb{R}$. On the product space $\mathbb{S} = \mathbb{X}\times \mathbb{Y}$ we consider a Markov ...
2
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1answer
62 views

A Markov Chain Flea Problem

A Flea moves around the vertices of a triangle in the following manner: Whenever it is at vertex i it moves to its clockwise neighbor vertex with probability $p_i$ and to the counterclockwise neighbor ...
2
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1answer
75 views

Counterexample for $\| (\Phi^\top \Xi \Phi)^\dagger \Phi^\top \Xi P \Phi \|_2 \leq 1$

All matrices are real. Let $P$ be any stochastic matrix (i.e. square, non-negative, rows summing to one). Let $\Xi$ be a diagonal matrix containing on its diagonal any left eigenvector of $P$ ...
2
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1answer
46 views

Find conditions on the distribution on $X$, but what is meant by $X$?

Let $X, X_{n,k}$ for $k,n\in\mathbb{N}$ denote independent random variables with values in $\mathbb{N}_0$. Define $N_0:=1$ and for $n\in\mathbb{N}$ set $$ N_n:=\begin{cases}0, & \text{ ...
2
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1answer
41 views

Layman perspective of mean time spent in transient state of a Markov chain.

Let $X=\{X_n\}$ be a finite state Markov Chain with the state space $\{0,1,2,\ldots,N\}$ such that $0$ is the single absorbing state and all the rest states are transient. The following is the ...
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1answer
51 views

Why left multiplication when it comes to Markov chains?

When working with Markov chains and transition matrices $P$ we multiply from the left, meaning that for example $\mu^{(n)} = \mu^{(0)}P^n$ or that the stationary distribution satisfies $\pi = \pi P$. ...
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1answer
43 views

probability that a game finishes at $n$th step

A coin is flipped sequentially. The game finishes when the sequence TTH is formed(player X wins) or the sequence HTT is formed(player Y wins). I can find the expected time until absorption by X or Y ...
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1answer
71 views

Safe small wins vs. risky large wins at roulette

Short statement of problem : Two players play roulette at a casino. They both start with the same initial amount. Each player always plays his favorite bet each time, and stops playing as soon as he ...
2
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1answer
84 views

What does a customer see when it begins to be served in $M/M/1$ queue?

In queueing theory, the PASTA (Poisson Arrivals See Time Averages) principle [wiki] justifies $a_n = P_n$ where $$a_n = \text{proportion of customers that find } n \text{ customers in the system when ...
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1answer
32 views

How to get transition matrix of markov process?

I am monitoring a Markov process with ~21 states. I know all the states, initial state and what states transitions can/cannot be, so that zero elements of transition matrix are known. I know the ...
2
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1answer
67 views

continuous time Markov chain, something the book does not explain

I have a problem with something in my book, under the chapter of continuous time Markov chains. I will post a link to what the book does. They do something which they seem to take for granted, but I ...
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2answers
353 views

Mean recurrence time and stationary distribution of a Markov chain?

In a Markov chain is there a theorem relating the existence of the stationary distribution and the mean recurrence time? E.g. impossible for stationary distribution to exist therefore mean recurrence ...