**Definition**

Suppose that the complete state-space of a Markov chain is divided into disjoint subsets of states, where these subsets are denoted by *t _{i}*. This forms a partition of the states. Both the state-space and the collection of subsets may be either finite or countably infinite. A continuous-time Markov chain is

**lumpable**with respect to the partition

*T*if and only if, for any subsets

*t*and

_{i}*t*in the partition, and for any states

_{j}*n,n’*in subset

*t*,

_{i}where *q*(*i,j*) is the transition rate from state *i* to state *j*.

Similarly, for a stochastic matrix *P*, *P* is a **lumpable matrix** on a partition *T* if and only if and only if, for any subsets *t _{i}* and

*t*in the partition, and for any states

_{j}*n,n’*in subset

*t*,

_{i}where *p*(*i,j*) is the probability of moving from state *i* to state *j*.

Read more about this topic: Lumpability

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### Famous quotes containing the word definition:

“The physicians say, they are not materialists; but they are:MSpirit is matter reduced to an extreme thinness: O so thin!—But the *definition* of spiritual should be, that which is its own evidence. What notions do they attach to love! what to religion! One would not willingly pronounce these words in their hearing, and give them the occasion to profane them.”

—Ralph Waldo Emerson (1803–1882)

“Was man made stupid to see his own stupidity?

Is God by *definition* indifferent, beyond us all?

Is the eternal truth man’s fighting soul

Wherein the Beast ravens in its own avidity?”

—Richard Eberhart (b. 1904)

“The very *definition* of the real becomes: that of which it is possible to give an equivalent reproduction.... The real is not only what can be reproduced, but that which is always already reproduced. The hyperreal.”

—Jean Baudrillard (b. 1929)