Sage-Tips

Is a Markov chain a random walk?

A random walk on a graph is a very special case of a Markov chain. Unlike a general Markov chain, random walk on a graph enjoys a property called time symmetry or reversibility.

What defines a Markov chain?

Definition of Markov chain : a usually discrete stochastic process (such as a random walk) in which the probabilities of occurrence of various future states depend only on the present state of the system or on the immediately preceding state and not on the path by which the present state was achieved.

Are random walks ergodic?

Theorem 1 A random walk on a graph G is ergodic if and only if G is connected and not bipartite.

Is random walk mean stationary?

In fact, all random walk processes are non-stationary. Note that not all non-stationary time series are random walks. Additionally, a non-stationary time series does not have a consistent mean and/or variance over time.

Are Markov chains useful?

Introduction. Markov Chains are exceptionally useful in order to model a discrete-time, discrete space Stochastic Process of various domains like Finance (stock price movement), NLP Algorithms (Finite State Transducers, Hidden Markov Model for POS Tagging), or even in Engineering Physics (Brownian motion).

Are random walks independent?

The definition of a random walk uses the concept of independent random variables whose technical aspects are reviewed in Chapter 1.

What is lazy random walk?

We will often consider lazy random walks, which are the variant of random walks that stay put. with probability 1/2 at each time step, and walk to a random neighbor the other half of the time.

What is the meaning of ergodic?

Definition of ergodic 1 : of or relating to a process in which every sequence or sizable sample is equally representative of the whole (as in regard to a statistical parameter) 2 : involving or relating to the probability that any state will recur especially : having zero probability that any state will never recur.

Can you predict random walk?

A random walk is unpredictable; it cannot reasonably be predicted.

What is a Markov chain?

A Markov chain is a specific kind of random process such that: – the state of the [math](n+1)^{th}[/math] random variable depends only on the state of the [math]n^{th}[/math] random variable.

How do you find the transition matrix of a Markov chain?

very important special case is the Markov chain that corresponds to a random walk onan undirected, unweighted graph. Here, the random walk picks each step a neighbor chosenuniformly at random and moves to that neighbor. Hence, the transition matrix isP=D1,whereDis the diagonal matrix with1Di;i=deg(i).

Does Z N = S n + Sn + 1 define a Markov chain?

Let S n = ∑ k = 1 n X k. Now we define Z n = S n + S n + 1 and ask if this defines a markov chain. My efforts so far tell me that this is true, but other sources suggest that the Z n do not define a markov chain.

Is a random walk on a graph reversible?

Random walks on graphs and random walks on edge-weighted graphs are always reversible. (A simple example for a non-reversible Markov chain is a Markov chain for which there are two states with P x;y>0 but P x;y= 0.)