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Difference between revisions of "Degeneration, probability of"

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The probability of no particles being left in a [[Branching process|branching process]] at an epoch <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030880/d0308801.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030880/d0308802.png" /> be the number of particles in a branching process with one type of particles. The probability of degeneration
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<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030880/d0308803.png" /></td> </tr></table>
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does not decrease as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030880/d0308804.png" /> increases; the value
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The probability of no particles being left in a [[Branching process|branching process]] at an epoch  $  t $.  
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Let  $  \mu ( t) $
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be the number of particles in a branching process with one type of particles. The probability of degeneration
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030880/d0308805.png" /></td> </tr></table>
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$$
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{\mathsf P} _ {0} ( t)  = \
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{\mathsf P} \{ \mu ( t) = 0 \mid  \mu ( 0) = 1 \}
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$$
  
is called the probability of degeneration in infinite time or simply the probability of degeneration. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030880/d0308806.png" /> is the time elapsing from the beginning of the process to the epoch of disappearance of the last particle, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030880/d0308807.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030880/d0308808.png" />. The rate of convergence of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030880/d0308809.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030880/d03088010.png" /> as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030880/d03088011.png" /> has been studied for various models of branching processes.
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does not decrease as  $  t $
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increases; the value
  
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$$
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q  =  \lim\limits _ {t \rightarrow \infty }  {\mathsf P} _ {0} ( t)
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$$
  
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is called the probability of degeneration in infinite time or simply the probability of degeneration. If  $  \tau $
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is the time elapsing from the beginning of the process to the epoch of disappearance of the last particle, then  $  {\mathsf P} \{ \tau < t \} = {\mathsf P} _ {0} ( t) $
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and  $  {\mathsf P} \{ \tau < \infty \} = q $.
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The rate of convergence of  $  {\mathsf P} _ {0} ( t) $
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to  $  q $
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as  $  t \rightarrow \infty $
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has been studied for various models of branching processes.
  
 
====Comments====
 
====Comments====

Latest revision as of 17:32, 5 June 2020


The probability of no particles being left in a branching process at an epoch $ t $. Let $ \mu ( t) $ be the number of particles in a branching process with one type of particles. The probability of degeneration

$$ {\mathsf P} _ {0} ( t) = \ {\mathsf P} \{ \mu ( t) = 0 \mid \mu ( 0) = 1 \} $$

does not decrease as $ t $ increases; the value

$$ q = \lim\limits _ {t \rightarrow \infty } {\mathsf P} _ {0} ( t) $$

is called the probability of degeneration in infinite time or simply the probability of degeneration. If $ \tau $ is the time elapsing from the beginning of the process to the epoch of disappearance of the last particle, then $ {\mathsf P} \{ \tau < t \} = {\mathsf P} _ {0} ( t) $ and $ {\mathsf P} \{ \tau < \infty \} = q $. The rate of convergence of $ {\mathsf P} _ {0} ( t) $ to $ q $ as $ t \rightarrow \infty $ has been studied for various models of branching processes.

Comments

The probability of degeneration is more commonly called the probability of extinction (in infinite time).

References

[a1] P.E. Ney, K.B. Athreya, "Branching processes" , Springer (1972)
[a2] T.E. Harris, "The theory of branching processes" , Springer (1963)
How to Cite This Entry:
Degeneration, probability of. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Degeneration,_probability_of&oldid=46617
This article was adapted from an original article by V.P. Chistyakov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article