# Birth process

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{{Short description|Type of continuous process in probability theory}}
{{for|the biological process|birth}}
thumb|alt=birth process|A birth process with birth rates <math>\lambda_0, \lambda_1, \lambda_2, ...</math>.
In [probability theory](/source/probability_theory), a '''birth process''' or a '''pure birth process'''{{sfnp|Upton|Cook|2014|loc=birth-and-death process}} is a special case of a [continuous-time Markov process](/source/continuous-time_Markov_process) and a generalisation of a [Poisson process](/source/Poisson_process). It defines a continuous process which takes values in the [natural numbers](/source/natural_numbers) and can only increase by one (a "birth") or remain unchanged. This is a type of [birth–death process](/source/birth%E2%80%93death_process) with no deaths. The rate at which births occur is given by an [exponential random variable](/source/exponential_random_variable) whose parameter depends only on the current value of the process

==Definition==
===Birth rates definition===
A birth process with birth rates <math>(\lambda_n, n\in \mathbb{N})</math> and initial value <math>k\in \mathbb{N}</math> is a minimal right-continuous process <math>(X_t, t\ge 0)</math> such that <math>X_0=k</math> and the interarrival times <math>T_i = \inf\{t\ge 0: X_t=i+1\} - \inf\{t\ge 0: X_t=i\}</math> are independent [exponential random variable](/source/exponential_random_variable)s with parameter <math>\lambda_i</math>.{{sfnp|Norris|1997|p=81}}

===Infinitesimal definition===
A birth process with rates <math>(\lambda_n, n\in \mathbb{N})</math> and initial value <math>k\in \mathbb{N}</math> is a process <math>(X_t, t\ge 0)</math> such that:
* <math>X_0=k</math>
* <math>\forall s,t\ge 0: s<t\implies X_s \le X_t</math>
* <math>\mathbb{P}(X_{t+h}=X_t+1)=\lambda_{X_t}h+o(h)</math>
* <math>\mathbb{P}(X_{t+h}=X_t)=o(h)</math>
* <math>\forall s,t\ge 0: s<t\implies X_t-X_s</math> is independent of <math>(X_u, u < s)</math>

(The third and fourth conditions use [little o](/source/little_o) notation.)

These conditions ensure that the process starts at <math>i</math>, is non-decreasing and has independent single births continuously at rate <math>\lambda_n</math>, when the process has value <math>n</math>.{{sfnp|Grimmett|Stirzaker|1992|p=232}}

===Continuous-time Markov chain definition===
A birth process can be defined as a [continuous-time Markov process](/source/continuous-time_Markov_process) (CTMC) <math>(X_t, t\ge 0)</math> with the non-zero Q-matrix entries <math>q_{n,n+1}=\lambda_n=-q_{n,n}</math> and initial distribution <math>i</math> (the random variable which takes value <math>i</math> with probability 1).{{sfnp|Norris|1997|p=81–82}}

<math>Q=\begin{pmatrix}
-\lambda_0 & \lambda_0 & 0 & 0 & \cdots \\
0 & -\lambda_1 & \lambda_1 & 0 & \cdots \\
0 & 0 & -\lambda_2 & \lambda_2 & \cdots\\
\vdots & \vdots & \vdots & & \vdots \ddots
\end{pmatrix}</math>

===Variations===
Some authors require that a birth process start from 0 i.e. that <math>X_0=0</math>,{{sfnp|Grimmett|Stirzaker|1992|p=232}} while others allow the initial value to be given by a [probability distribution](/source/probability_distribution) on the natural numbers.{{sfnp|Norris|1997|p=81}} The [state space](/source/state_space) can include infinity, in the case of an explosive birth process.{{sfnp|Norris|1997|p=81}} The birth rates are also called intensities.{{sfnp|Grimmett|Stirzaker|1992|p=232}}

==Properties==
As for CTMCs, a birth process has the [Markov property](/source/Markov_property). The CTMC definitions for communicating classes, irreducibility and so on apply to birth processes. By the conditions for recurrence and transience of a [birth–death process](/source/birth%E2%80%93death_process),{{sfnp|Karlin|McGregor|1957}} any birth process is transient. The transition matrices <math>((p_{i,j}(t))_{i,j\in\mathbb{N}}), t\ge 0)</math> of a birth process satisfy the [Kolmogorov forward and backward equations](/source/Kolmogorov_equations_(Markov_jump_process)).

The backwards equations are:{{sfnp|Ross|2010|p=386}}
:<math>p'_{i,j}(t)=\lambda_i (p_{i+1,j}(t)-p_{i,j}(t))</math> (for <math>i,j\in\mathbb{N}</math>)

The forward equations are:{{sfnp|Ross|2010|p=389}}
:<math>p'_{i,i}(t)=-\lambda_i p_{i,i}(t)</math> (for <math>i\in\mathbb{N}</math>)
:<math>p'_{i,j}(t)=\lambda_{j-1}p_{i,j-1}(t)-\lambda_j p_{i,j}(t)</math> (for <math>j\ge i+1</math>)

From the forward equations it follows that:{{sfnp|Ross|2010|p=389}}
:<math>p_{i,i}(t)=e^{-\lambda_i t}</math> (for <math>i\in\mathbb{N}</math>)
:<math>p_{i,j}(t)=\lambda_{j-1}e^{-\lambda_j t}\int_0^t e^{\lambda_j s}p_{i,j-1}(s)\, \text{d} s</math> (for <math>j\ge i+1</math>)

Unlike a Poisson process, a birth process may have infinitely many births in a finite amount of time. We define <math>T_\infty=\sup \{T_n:n\in\mathbb{N}\}</math> and say that a birth process explodes if <math>T_\infty</math> is finite. If <math>\sum_{n=0}^\infty \frac{1}{\lambda_n}<\infty</math> then the process is explosive with probability 1; otherwise, it is non-explosive with probability 1 ("honest").{{sfnp|Norris|1997|p=83}}{{sfnp|Grimmett|Stirzaker|1992|p=234}}

==Examples==
[[File:Poisson process.svg|thumb|alt=Poisson process|A [Poisson process](/source/Poisson_process) is a special case of a birth process.]]
A [Poisson process](/source/Poisson_process) is a birth process where the birth rates are constant i.e. <math>\lambda_n=\lambda</math> for some <math>\lambda>0</math>.{{sfnp|Grimmett|Stirzaker|1992|p=232}}

===Simple birth process===
thumb|alt=Simple birth process|A simple birth process, where birth rates are equal to the size of the current population.
A '''simple birth process''' is a birth process with rates <math>\lambda_n=n\lambda</math>.{{sfnp|Norris|1997|p=82}} It models a population in which each individual gives birth repeatedly and independently at rate <math>\lambda</math>. [Udny Yule](/source/Udny_Yule) studied the processes, so they may be known as '''Yule processes'''.{{sfnp|Ross|2010|p=375}}

The number of births in time <math>t</math> from a simple birth process of population <math>n</math> is given by:{{sfnp|Grimmett|Stirzaker|1992|p=232}}
:<math>p_{n,n+m}(t)=\binom{n}{m}(\lambda t)^m(1-\lambda t)^{n-m}+o(h)</math>

In exact form, the number of births is the [negative binomial distribution](/source/negative_binomial_distribution) with parameters <math>n</math> and <math>e^{-\lambda t}</math>. For the special case <math>n=1</math>, this is the [geometric distribution](/source/geometric_distribution) with success rate <math>e^{-\lambda t}</math>.{{sfnp|Ross|2010|p=383}}

The [expectation](/source/Expected_value) of the process grows exponentially; specifically, if <math>X_0=1</math> then <math>\mathbb{E}(X_t)=e^{\lambda t}</math>.{{sfnp|Norris|1997|p=82}}

A simple birth process with immigration is a modification of this process with rates <math>\lambda_n=n\lambda+\nu</math>. This models a population with births by each population member in addition to a constant rate of immigration into the system.{{sfnp|Grimmett|Stirzaker|1992|p=232}}

==Notes==
{{reflist}}

==References==
* {{cite book | title=Probability and Random Processes | first2=D. R. | last2=Stirzaker | first1=G. R. | last1=Grimmett | author-link=Geoffrey Grimmett | year=1992 |edition=second | publisher=Oxford University Press|ISBN=0198572220 }}
* {{cite journal| url=https://www.ams.org/journals/tran/1957-086-02/S0002-9947-1957-0094854-8/S0002-9947-1957-0094854-8.pdf
 | title=The classification of birth and death processes | last1=Karlin |first1= Samuel | last2= McGregor |first2= James | year=1957 | journal=Transactions of the American Mathematical Society | volume=86 | issue=2 | pages=366-400 | author1-link= Samuel Karlin}}
* {{cite book |last=Norris |first=J.R. |title=Markov Chains |year=1997 |publisher=Cambridge University Press |isbn=9780511810633}}
* {{cite book |last=Ross |first=Sheldon M. |title=Introduction to Probability Models |edition=tenth |year=2010 |publisher=Academic Press |isbn=9780123756862}}
* {{cite book |last=Upton |first=G. |last2=Cook |first2=I. |year=2014 |title=A Dictionary of Statistics |edition=third |isbn=9780191758317}}

{{Stochastic processes}}

Category:Markov processes
Category:Poisson point processes

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Adapted from the Wikipedia article [Birth process](https://en.wikipedia.org/wiki/Birth_process) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Birth_process?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.