What does it mean for a Markov chain to be time homogeneous?

What does it mean for a Markov chain to be time homogeneous?

The Markov chain X(t) is time-homogeneous if P(Xn+1 = j|Xn = i) = P(X1 = j|X0 = i), i.e. the transition probabilities do not depend on time n. This says that if it is sunny today, then the chance it will be sunny tomorrow is 0.8, whereas if it is rainy today, then the chance it will be sunny tomorrow is 0.4.

Is it homogeneous or homogeneous?

Is it homogeneous or homogenous? Homogenous is an older scientific term that describes similar tissues or organs. It has been replaced by homologous. Homogeneous is an adjective that describes similar or uniform characteristics.

What does homogeneous appearance mean?

adj. 1 composed of similar or identical parts or elements. 2 of uniform nature. 3 similar in kind or nature. 4 having a constant property, such as density, throughout.

Is homogeneously a word?

ho·mo·ge·ne·ous adj. 1. Consisting of parts that are the same; uniform in structure or composition: “a tight-knit, homogeneous society” (James Fallows).

What is positive recurrent?

A recurrent state j is called positive recurrent if the expected amount of time to return to state j given that the chain started in state j has finite first moment: E(τjj) < ∞. A recurrent state j for which E(τjj) = ∞ is called null recurrent.

What is the period of a Markov chain?

Pr(Xn′=i|X0=i)>0. Otherwise (k > 1), the state is said to be periodic with period k. A Markov chain is aperiodic if every state is aperiodic. My Explanation. The term periodicity describes whether something (an event, or here: the visit of a particular state) is happening at a regular time interval.

What is homogeneous example?

A homogeneous mixture appears uniform, regardless of where you sample it. Examples of homogeneous mixtures include air, saline solution, most alloys, and bitumen. Examples of heterogeneous mixtures include sand, oil and water, and chicken noodle soup.

What is absorbing state?

An absorbing state is a state that, once entered, cannot be left. Like general Markov chains, there can be continuous-time absorbing Markov chains with an infinite state space.

What is null recurrent?

A state that is recurrent but not positive recurrent is called null recurrent. A positive recurrent, aperiodic state is called ergodic. An example of a null recurrent state appears in the simple random walk.

Is absorbing state recurrent?

You are correct: an absorbing state must be recurrent. To be precise with definitions: given a state space X and a Markov chain with transition matrix P defined on X. A state x∈X is absorbing if Pxx=1; neccessarily this implies that Pxy=0,y≠x.

What is a recurrent state?

In general, a state is said to be recurrent if, any time that we leave that state, we will return to that state in the future with probability one. On the other hand, if the probability of returning is less than one, the state is called transient.

What is the definition of homogeneous?

Definition of homogeneous. 2 : of uniform structure or composition throughout a culturally homogeneous neighborhood 3 : having the property that if each variable is replaced by a constant times that variable the constant can be factored out : having each term…

What is a time homogeneous Markov process?

Klenke defines ( Definition 17.3, p. 346) a time-homogeneous Markov process independently, rather than as a special case of a stochastic process that possesses the Markov property ( Definition 17.1, p. 345) and in addition satisfies certain further constraints.

Is homogeneous culture the same as native culture?

Homogeneous culture, however, is of course not the same thing as native culture. Homogeneous: Structure uniform throughout all parts of the colony. And next, that the parts of such a body must be Homogeneous, or of the same kind.

What does it mean to homogenize milk?

To homogenize milk is to blend it in a way that results in the milk fat being evenly distributed throughout it. The state of being homogeneous is homogeneity. The general sense of homogeneous can be used interchangeably with the word homogenous (which is spelled without a second e and is pronounced differently).