Birth process

<h2 id="definition">Definition</h2>
<h3>Birth rates definition</h3>
A birth process with birth rates 
 
 
 
 (
 
 λ
 
 n
 
 
 ,
 n
 ∈
 
 N
 
 )
 
 
 {\displaystyle (\lambda _{n},n\in \mathbb {N} )}
 
 and initial value 
 
 
 
 k
 ∈
 
 N
 
 
 
 {\displaystyle k\in \mathbb {N} }
 
 is a minimal right-continuous process 
 
 
 
 (
 
 X
 
 t
 
 
 ,
 t
 ≥
 0
 )
 
 
 {\displaystyle (X_{t},t\geq 0)}
 
 such that 
 
 
 
 
 X
 
 0
 
 
 =
 k
 
 
 {\displaystyle X_{0}=k}
 
 and the interarrival times 
 
 
 
 
 T
 
 i
 
 
 =
 inf
 {
 t
 ≥
 0
 :
 
 X
 
 t
 
 
 =
 i
 +
 1
 }
 −
 inf
 {
 t
 ≥
 0
 :
 
 X
 
 t
 
 
 =
 i
 }
 
 
 {\displaystyle T_{i}=\inf\{t\geq 0:X_{t}=i+1\}-\inf\{t\geq 0:X_{t}=i\}}
 
 are independent <a href="/facts/Exponential_random_variable/yY3EdV0u">exponential random variables</a> with parameter 
 
 
 
 
 λ
 
 i
 
 
 
 
 {\displaystyle \lambda _{i}}
 
.<a class="footnote-ref" id="fnref:2" href="#fn:2">2</a>

<h3>Infinitesimal definition</h3>
A birth process with rates 
 
 
 
 (
 
 λ
 
 n
 
 
 ,
 n
 ∈
 
 N
 
 )
 
 
 {\displaystyle (\lambda _{n},n\in \mathbb {N} )}
 
 and initial value 
 
 
 
 k
 ∈
 
 N
 
 
 
 {\displaystyle k\in \mathbb {N} }
 
 is a process 
 
 
 
 (
 
 X
 
 t
 
 
 ,
 t
 ≥
 0
 )
 
 
 {\displaystyle (X_{t},t\geq 0)}
 
 such that:

<ul><li>
 
 
 
 
 X
 
 0
 
 
 =
 k
 
 
 {\displaystyle X_{0}=k}
 
</li>
<li>
 
 
 
 ∀
 s
 ,
 t
 ≥
 0
 :
 s
 <
 t
 
 ⟹
 
 
 X
 
 s
 
 
 ≤
 
 X
 
 t
 
 
 
 
 {\displaystyle \forall s,t\geq 0:s<t\implies X_{s}\leq X_{t}}
 
</li>
<li>
 
 
 
 
 P
 
 (
 
 X
 
 t
 +
 h
 
 
 =
 
 X
 
 t
 
 
 +
 1
 )
 =
 
 λ
 
 
 X
 
 t
 
 
 
 
 h
 +
 o
 (
 h
 )
 
 
 {\displaystyle \mathbb {P} (X_{t+h}=X_{t}+1)=\lambda _{X_{t}}h+o(h)}
 
</li>
<li>
 
 
 
 
 P
 
 (
 
 X
 
 t
 +
 h
 
 
 =
 
 X
 
 t
 
 
 )
 =
 o
 (
 h
 )
 
 
 {\displaystyle \mathbb {P} (X_{t+h}=X_{t})=o(h)}
 
</li>
<li>
 
 
 
 ∀
 s
 ,
 t
 ≥
 0
 :
 s
 <
 t
 
 ⟹
 
 
 X
 
 t
 
 
 −
 
 X
 
 s
 
 
 
 
 {\displaystyle \forall s,t\geq 0:s<t\implies X_{t}-X_{s}}
 
 is independent of 
 
 
 
 (
 
 X
 
 u
 
 
 ,
 u
 <
 s
 )
 
 
 {\displaystyle (X_{u},u<s)}
 
</li></ul>
(The third and fourth conditions use <a href="/facts/Little_o/weFFjSWg">little o</a> notation.)
These conditions ensure that the process starts at 
 
 
 
 i
 
 
 {\displaystyle i}
 
, is non-decreasing and has independent single births continuously at rate 
 
 
 
 
 λ
 
 n
 
 
 
 
 {\displaystyle \lambda _{n}}
 
, when the process has value 
 
 
 
 n
 
 
 {\displaystyle n}
 
.<a class="footnote-ref" id="fnref:3" href="#fn:3">3</a>

<h3>Continuous-time Markov chain definition</h3>
A birth process can be defined as a <a href="/facts/Continuous-time_Markov_process/h7ytJ6p3">continuous-time Markov process</a> (CTMC) 
 
 
 
 (
 
 X
 
 t
 
 
 ,
 t
 ≥
 0
 )
 
 
 {\displaystyle (X_{t},t\geq 0)}
 
 with the non-zero Q-matrix entries 
 
 
 
 
 q
 
 n
 ,
 n
 +
 1
 
 
 =
 
 λ
 
 n
 
 
 =
 −
 
 q
 
 n
 ,
 n
 
 
 
 
 {\displaystyle q_{n,n+1}=\lambda _{n}=-q_{n,n}}
 
 and initial distribution 
 
 
 
 i
 
 
 {\displaystyle i}
 
 (the random variable which takes value 
 
 
 
 i
 
 
 {\displaystyle i}
 
 with probability 1).<a class="footnote-ref" id="fnref:4" href="#fn:4">4</a>

 
 
 
 Q
 =
 
 
 (
 
 
 
 −
 
 λ
 
 0
 
 
 
 
 
 λ
 
 0
 
 
 
 
 0
 
 
 0
 
 
 ⋯
 
 
 
 
 0
 
 
 −
 
 λ
 
 1
 
 
 
 
 
 λ
 
 1
 
 
 
 
 0
 
 
 ⋯
 
 
 
 
 0
 
 
 0
 
 
 −
 
 λ
 
 2
 
 
 
 
 
 λ
 
 2
 
 
 
 
 ⋯
 
 
 
 
 ⋮
 
 
 ⋮
 
 
 ⋮
 
 
 
 ⋮
 ⋱
 
 
 
 )
 
 
 
 
 {\displaystyle 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}}}

<h3>Variations</h3>
Some authors require that a birth process start from 0 i.e. that 
 
 
 
 
 X
 
 0
 
 
 =
 0
 
 
 {\displaystyle X_{0}=0}
 
,<a class="footnote-ref" id="fnref:5" href="#fn:5">5</a> while others allow the initial value to be given by a <a href="/facts/Probability_distribution/EpsKKVRu">probability distribution</a> on the natural numbers.<a class="footnote-ref" id="fnref:6" href="#fn:6">6</a> The <a href="/facts/State_space/H1vjb3LL">state space</a> can include infinity, in the case of an explosive birth process.<a class="footnote-ref" id="fnref:7" href="#fn:7">7</a> The birth rates are also called intensities.<a class="footnote-ref" id="fnref:8" href="#fn:8">8</a>

<h2 id="properties">Properties</h2>
As for CTMCs, a birth process has the <a href="/facts/Markov_property/g1Fs9GFS">Markov property</a>. The CTMC definitions for communicating classes, irreducibility and so on apply to birth processes. By the conditions for recurrence and transience of a <a href="/facts/Birth%25E2%2580%2593death_process/ud6IWMxu">birth–death process</a>,<a class="footnote-ref" id="fnref:9" href="#fn:9">9</a> any birth process is transient. The transition matrices 
 
 
 
 (
 (
 
 p
 
 i
 ,
 j
 
 
 (
 t
 )
 
 )
 
 i
 ,
 j
 ∈
 
 N
 
 
 
 )
 ,
 t
 ≥
 0
 )
 
 
 {\displaystyle ((p_{i,j}(t))_{i,j\in \mathbb {N} }),t\geq 0)}
 
 of a birth process satisfy the <a href="/facts/Kolmogorov_equations_(Markov_jump_process)/hElT4GCI">Kolmogorov forward and backward equations</a>.
The backwards equations are:<a class="footnote-ref" id="fnref:10" href="#fn:10">10</a>

p
 
 i
 ,
 j
 
 ′
 
 (
 t
 )
 =
 
 λ
 
 i
 
 
 (
 
 p
 
 i
 +
 1
 ,
 j
 
 
 (
 t
 )
 −
 
 p
 
 i
 ,
 j
 
 
 (
 t
 )
 )
 
 
 {\displaystyle p'_{i,j}(t)=\lambda _{i}(p_{i+1,j}(t)-p_{i,j}(t))}
 
 (for 
 
 
 
 i
 ,
 j
 ∈
 
 N
 
 
 
 {\displaystyle i,j\in \mathbb {N} }
 
)
The forward equations are:<a class="footnote-ref" id="fnref:11" href="#fn:11">11</a>

p
          
            i
            ,
            i
          
          ′
        
        (
        t
        )
        =
        −
        
          λ
          
            i
          
        
        
          p
          
            i
            ,
            i
          
        
        (
        t
        )
      
    
    {\displaystyle p'_{i,i}(t)=-\lambda _{i}p_{i,i}(t)}
  
 (for 
  
    
      
        i
        ∈
        
          N
        
      
    
    {\displaystyle i\in \mathbb {N} }
  
)

p
 
 i
 ,
 j
 
 ′
 
 (
 t
 )
 =
 
 λ
 
 j
 −
 1
 
 
 
 p
 
 i
 ,
 j
 −
 1
 
 
 (
 t
 )
 −
 
 λ
 
 j
 
 
 
 p
 
 i
 ,
 j
 
 
 (
 t
 )
 
 
 {\displaystyle p'_{i,j}(t)=\lambda _{j-1}p_{i,j-1}(t)-\lambda _{j}p_{i,j}(t)}
 
 (for 
 
 
 
 j
 ≥
 i
 +
 1
 
 
 {\displaystyle j\geq i+1}
 
)
From the forward equations it follows that:<a class="footnote-ref" id="fnref:12" href="#fn:12">12</a>

p
          
            i
            ,
            i
          
        
        (
        t
        )
        =
        
          e
          
            −
            
              λ
              
                i
              
            
            t
          
        
      
    
    {\displaystyle p_{i,i}(t)=e^{-\lambda _{i}t}}
  
 (for 
  
    
      
        i
        ∈
        
          N
        
      
    
    {\displaystyle i\in \mathbb {N} }
  
)

p
 
 i
 ,
 j
 
 
 (
 t
 )
 =
 
 λ
 
 j
 −
 1
 
 
 
 e
 
 −
 
 λ
 
 j
 
 
 t
 
 
 
 ∫
 
 0
 
 
 t
 
 
 
 e
 
 
 λ
 
 j
 
 
 s
 
 
 
 p
 
 i
 ,
 j
 −
 1
 
 
 (
 s
 )
 
 
 d
 
 s
 
 
 {\displaystyle 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}
 
 (for 
 
 
 
 j
 ≥
 i
 +
 1
 
 
 {\displaystyle j\geq i+1}
 
)
Unlike a Poisson process, a birth process may have infinitely many births in a finite amount of time. We define 
 
 
 
 
 T
 
 ∞
 
 
 =
 sup
 {
 
 T
 
 n
 
 
 :
 n
 ∈
 
 N
 
 }
 
 
 {\displaystyle T_{\infty }=\sup\{T_{n}:n\in \mathbb {N} \}}
 
 and say that a birth process explodes if 
 
 
 
 
 T
 
 ∞
 
 
 
 
 {\displaystyle T_{\infty }}
 
 is finite. If 
 
 
 
 
 ∑
 
 n
 =
 0
 
 
 ∞
 
 
 
 
 1
 
 λ
 
 n
 
 
 
 
 <
 ∞
 
 
 {\displaystyle \sum _{n=0}^{\infty }{\frac {1}{\lambda _{n}}}<\infty }
 
 then the process is explosive with probability 1; otherwise, it is non-explosive with probability 1 ("honest").<a class="footnote-ref" id="fnref:13" href="#fn:13">13</a><a class="footnote-ref" id="fnref:14" href="#fn:14">14</a>

<h2 id="examples">Examples</h2>

A <a href="/facts/Poisson_process/8uphjAUc">Poisson process</a> is a birth process where the birth rates are constant i.e. 
 
 
 
 
 λ
 
 n
 
 
 =
 λ
 
 
 {\displaystyle \lambda _{n}=\lambda }
 
 for some 
 
 
 
 λ
 >
 0
 
 
 {\displaystyle \lambda >0}
 
.<a class="footnote-ref" id="fnref:15" href="#fn:15">15</a>

<h3>Simple birth process</h3>

A simple birth process is a birth process with rates 
 
 
 
 
 λ
 
 n
 
 
 =
 n
 λ
 
 
 {\displaystyle \lambda _{n}=n\lambda }
 
.<a class="footnote-ref" id="fnref:16" href="#fn:16">16</a> It models a population in which each individual gives birth repeatedly and independently at rate 
 
 
 
 λ
 
 
 {\displaystyle \lambda }
 
. <a href="/facts/Udny_Yule/3MHsP37s">Udny Yule</a> studied the processes, so they may be known as Yule processes.<a class="footnote-ref" id="fnref:17" href="#fn:17">17</a>
The number of births in time 
 
 
 
 t
 
 
 {\displaystyle t}
 
 from a simple birth process of population 
 
 
 
 n
 
 
 {\displaystyle n}
 
 is given by:<a class="footnote-ref" id="fnref:18" href="#fn:18">18</a>

p
          
            n
            ,
            n
            +
            m
          
        
        (
        t
        )
        =
        
          
            
              (
            
            
              n
              m
            
            
              )
            
          
        
        (
        λ
        t
        
          )
          
            m
          
        
        (
        1
        −
        λ
        t
        
          )
          
            n
            −
            m
          
        
        +
        o
        (
        h
        )
      
    
    {\displaystyle p_{n,n+m}(t)={\binom {n}{m}}(\lambda t)^{m}(1-\lambda t)^{n-m}+o(h)}

In exact form, the number of births is the <a href="/facts/Negative_binomial_distribution/znzj365Y">negative binomial distribution</a> with parameters 
 
 
 
 n
 
 
 {\displaystyle n}
 
 and 
 
 
 
 
 e
 
 −
 λ
 t
 
 
 
 
 {\displaystyle e^{-\lambda t}}
 
. For the special case 
 
 
 
 n
 =
 1
 
 
 {\displaystyle n=1}
 
, this is the <a href="/facts/Geometric_distribution/c8wjfaoV">geometric distribution</a> with success rate 
 
 
 
 
 e
 
 −
 λ
 t
 
 
 
 
 {\displaystyle e^{-\lambda t}}
 
.<a class="footnote-ref" id="fnref:19" href="#fn:19">19</a>
The <a href="/facts/Expected_value/1XV0JKL8">expectation</a> of the process grows exponentially; specifically, if 
 
 
 
 
 X
 
 0
 
 
 =
 1
 
 
 {\displaystyle X_{0}=1}
 
 then 
 
 
 
 
 E
 
 (
 
 X
 
 t
 
 
 )
 =
 
 e
 
 λ
 t
 
 
 
 
 {\displaystyle \mathbb {E} (X_{t})=e^{\lambda t}}
 
.<a class="footnote-ref" id="fnref:20" href="#fn:20">20</a>
A simple birth process with immigration is a modification of this process with rates 
 
 
 
 
 λ
 
 n
 
 
 =
 n
 λ
 +
 ν
 
 
 {\displaystyle \lambda _{n}=n\lambda +\nu }
 
. This models a population with births by each population member in addition to a constant rate of immigration into the system.<a class="footnote-ref" id="fnref:21" href="#fn:21">21</a>

<h2 id="notes">Notes</h2>

<ul><li><a href="/facts/Geoffrey_Grimmett/GymuLIPK">Grimmett, G. R.</a>; Stirzaker, D. R. (1992). Probability and Random Processes (second ed.). Oxford University Press. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0198572220.</li>
<li><a href="/facts/Samuel_Karlin/zr0uUIzg">Karlin, Samuel</a>; McGregor, James (1957). <a href="https://www.ams.org/journals/tran/1957-086-02/S0002-9947-1957-0094854-8/S0002-9947-1957-0094854-8.pdf">"The classification of birth and death processes"</a> (PDF). Transactions of the American Mathematical Society. 86 (2): 366–400.</li>
<li>Norris, J.R. (1997). Markov Chains. Cambridge University Press. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 9780511810633.</li>
<li>Ross, Sheldon M. (2010). Introduction to Probability Models (tenth ed.). Academic Press. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 9780123756862.</li>
<li>Upton, G.; Cook, I. (2014). A Dictionary of Statistics (third ed.). <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 9780191758317.</li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1">Upton & Cook (2014), birth-and-death process. - Upton, G.; Cook, I. (2014). A Dictionary of Statistics (third ed.). ISBN 9780191758317. <a href="#fnref:1" class="footnote-back-ref">↩</a></li>
<li id="fn:2">Norris (1997), p. 81. - Norris, J.R. (1997). Markov Chains. Cambridge University Press. ISBN 9780511810633. <a href="#fnref:2" class="footnote-back-ref">↩</a></li>
<li id="fn:3">Grimmett & Stirzaker (1992), p. 232. - Grimmett, G. R.; Stirzaker, D. R. (1992). Probability and Random Processes (second ed.). Oxford University Press. ISBN 0198572220. <a href="#fnref:3" class="footnote-back-ref">↩</a></li>
<li id="fn:4">Norris (1997), p. 81–82. - Norris, J.R. (1997). Markov Chains. Cambridge University Press. ISBN 9780511810633. <a href="#fnref:4" class="footnote-back-ref">↩</a></li>
<li id="fn:5">Grimmett & Stirzaker (1992), p. 232. - Grimmett, G. R.; Stirzaker, D. R. (1992). Probability and Random Processes (second ed.). Oxford University Press. ISBN 0198572220. <a href="#fnref:5" class="footnote-back-ref">↩</a></li>
<li id="fn:6">Norris (1997), p. 81. - Norris, J.R. (1997). Markov Chains. Cambridge University Press. ISBN 9780511810633. <a href="#fnref:6" class="footnote-back-ref">↩</a></li>
<li id="fn:7">Norris (1997), p. 81. - Norris, J.R. (1997). Markov Chains. Cambridge University Press. ISBN 9780511810633. <a href="#fnref:7" class="footnote-back-ref">↩</a></li>
<li id="fn:8">Grimmett & Stirzaker (1992), p. 232. - Grimmett, G. R.; Stirzaker, D. R. (1992). Probability and Random Processes (second ed.). Oxford University Press. ISBN 0198572220. <a href="#fnref:8" class="footnote-back-ref">↩</a></li>
<li id="fn:9">Karlin & McGregor (1957). - Karlin, Samuel; McGregor, James (1957). "The classification of birth and death processes" (PDF). Transactions of the American Mathematical Society. 86 (2): 366–400. <a href="https://www.ams.org/journals/tran/1957-086-02/S0002-9947-1957-0094854-8/S0002-9947-1957-0094854-8.pdf" target="_blank">https://www.ams.org/journals/tran/1957-086-02/S0002-9947-1957-0094854-8/S0002-9947-1957-0094854-8.pdf</a> <a href="#fnref:9" class="footnote-back-ref">↩</a></li>
<li id="fn:10">Ross (2010), p. 386. - Ross, Sheldon M. (2010). Introduction to Probability Models (tenth ed.). Academic Press. ISBN 9780123756862. <a href="#fnref:10" class="footnote-back-ref">↩</a></li>
<li id="fn:11">Ross (2010), p. 389. - Ross, Sheldon M. (2010). Introduction to Probability Models (tenth ed.). Academic Press. ISBN 9780123756862. <a href="#fnref:11" class="footnote-back-ref">↩</a></li>
<li id="fn:12">Ross (2010), p. 389. - Ross, Sheldon M. (2010). Introduction to Probability Models (tenth ed.). Academic Press. ISBN 9780123756862. <a href="#fnref:12" class="footnote-back-ref">↩</a></li>
<li id="fn:13">Norris (1997), p. 83. - Norris, J.R. (1997). Markov Chains. Cambridge University Press. ISBN 9780511810633. <a href="#fnref:13" class="footnote-back-ref">↩</a></li>
<li id="fn:14">Grimmett & Stirzaker (1992), p. 234. - Grimmett, G. R.; Stirzaker, D. R. (1992). Probability and Random Processes (second ed.). Oxford University Press. ISBN 0198572220. <a href="#fnref:14" class="footnote-back-ref">↩</a></li>
<li id="fn:15">Grimmett & Stirzaker (1992), p. 232. - Grimmett, G. R.; Stirzaker, D. R. (1992). Probability and Random Processes (second ed.). Oxford University Press. ISBN 0198572220. <a href="#fnref:15" class="footnote-back-ref">↩</a></li>
<li id="fn:16">Norris (1997), p. 82. - Norris, J.R. (1997). Markov Chains. Cambridge University Press. ISBN 9780511810633. <a href="#fnref:16" class="footnote-back-ref">↩</a></li>
<li id="fn:17">Ross (2010), p. 375. - Ross, Sheldon M. (2010). Introduction to Probability Models (tenth ed.). Academic Press. ISBN 9780123756862. <a href="#fnref:17" class="footnote-back-ref">↩</a></li>
<li id="fn:18">Grimmett & Stirzaker (1992), p. 232. - Grimmett, G. R.; Stirzaker, D. R. (1992). Probability and Random Processes (second ed.). Oxford University Press. ISBN 0198572220. <a href="#fnref:18" class="footnote-back-ref">↩</a></li>
<li id="fn:19">Ross (2010), p. 383. - Ross, Sheldon M. (2010). Introduction to Probability Models (tenth ed.). Academic Press. ISBN 9780123756862. <a href="#fnref:19" class="footnote-back-ref">↩</a></li>
<li id="fn:20">Norris (1997), p. 82. - Norris, J.R. (1997). Markov Chains. Cambridge University Press. ISBN 9780511810633. <a href="#fnref:20" class="footnote-back-ref">↩</a></li>
<li id="fn:21">Grimmett & Stirzaker (1992), p. 232. - Grimmett, G. R.; Stirzaker, D. R. (1992). Probability and Random Processes (second ed.). Oxford University Press. ISBN 0198572220. <a href="#fnref:21" class="footnote-back-ref">↩</a></li>
</ol>

Birth process open-in-new

Birth process