Does Extinction distribution determine the offspring distribution in Simple Branching?

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From Extinction to Reproduction inBienaymeGaltonWatson processes
Daniel Tokarev
Monash University
11 July, 2012
Daniel Tokarev From Extinction to Reproduction in BienaymeGaltonWatson processes

Daniel Tokarev From Extinction to Reproduction in BienaymeGaltonWatson processes

BGW Processes
Let be some a random variable supported onnonnegative integers with pmf {pi} (reproductiondistribution)
Let Z0 = 1 and Zn+1 =Z (n)
i=0 i, n, where i, n are iid like and also independent of the pastThe information about the process is encoded inprobability generating function
f (s) =
i=0
pisi .
Recall that E = f (1) := , E( 1) = f (1) and thefunctional iterates fn(s), n = 1,2, . . . are the probabilitygenerating functions of the process at time n, while f (s)k ,k integer is a pgf of a process started with k individuals.
Daniel Tokarev From Extinction to Reproduction in BienaymeGaltonWatson processes

BGW Processes
Let be some a random variable supported onnonnegative integers with pmf {pi} (reproductiondistribution)
Let Z0 = 1 and Zn+1 =Z (n)
i=0 i, n, where i, n are iid like and also independent of the pastThe information about the process is encoded inprobability generating function
f (s) =
i=0
pisi .
Recall that E = f (1) := , E( 1) = f (1) and thefunctional iterates fn(s), n = 1,2, . . . are the probabilitygenerating functions of the process at time n, while f (s)k ,k integer is a pgf of a process started with k individuals.
Daniel Tokarev From Extinction to Reproduction in BienaymeGaltonWatson processes

BGW Processes
Let be some a random variable supported onnonnegative integers with pmf {pi} (reproductiondistribution)
Let Z0 = 1 and Zn+1 =Z (n)
i=0 i, n, where i, n are iid like and also independent of the pastThe information about the process is encoded inprobability generating function
f (s) =
i=0
pisi .
Recall that E = f (1) := , E( 1) = f (1) and thefunctional iterates fn(s), n = 1,2, . . . are the probabilitygenerating functions of the process at time n, while f (s)k ,k integer is a pgf of a process started with k individuals.
Daniel Tokarev From Extinction to Reproduction in BienaymeGaltonWatson processes

BGW Processes
Let be some a random variable supported onnonnegative integers with pmf {pi} (reproductiondistribution)
Let Z0 = 1 and Zn+1 =Z (n)
i=0 i, n, where i, n are iid like and also independent of the pastThe information about the process is encoded inprobability generating function
f (s) =
i=0
pisi .
Recall that E = f (1) := , E( 1) = f (1) and thefunctional iterates fn(s), n = 1,2, . . . are the probabilitygenerating functions of the process at time n, while f (s)k ,k integer is a pgf of a process started with k individuals.
Daniel Tokarev From Extinction to Reproduction in BienaymeGaltonWatson processes

BGW Processes
Let be some a random variable supported onnonnegative integers with pmf {pi} (reproductiondistribution)
Let Z0 = 1 and Zn+1 =Z (n)
i=0 i, n, where i, n are iid like and also independent of the pastThe information about the process is encoded inprobability generating function
f (s) =
i=0
pisi .
Recall that E = f (1) := , E( 1) = f (1) and thefunctional iterates fn(s), n = 1,2, . . . are the probabilitygenerating functions of the process at time n, while f (s)k ,k integer is a pgf of a process started with k individuals.
Daniel Tokarev From Extinction to Reproduction in BienaymeGaltonWatson processes

BGW Processes
Let be some a random variable supported onnonnegative integers with pmf {pi} (reproductiondistribution)
Let Z0 = 1 and Zn+1 =Z (n)
i=0 i, n, where i, n are iid like and also independent of the pastThe information about the process is encoded inprobability generating function
f (s) =
i=0
pisi .
Recall that E = f (1) := , E( 1) = f (1) and thefunctional iterates fn(s), n = 1,2, . . . are the probabilitygenerating functions of the process at time n, while f (s)k ,k integer is a pgf of a process started with k individuals.
Daniel Tokarev From Extinction to Reproduction in BienaymeGaltonWatson processes

Classification and Extinction Time
BGW processes are broadly divided into three typesSubcritical ( < 1), critical ( = 1)  extinction certain andsupercritical ( > 1)  extinction uncertainSince the iterated function fn(s) is the PGF of Z (n) inparticular fn(0) is the Pr of extinction after n steps andtaking the limit as n, gives the Pr of eventualextinction
Daniel Tokarev From Extinction to Reproduction in BienaymeGaltonWatson processes

Classification and Extinction Time
BGW processes are broadly divided into three typesSubcritical ( < 1), critical ( = 1)  extinction certain andsupercritical ( > 1)  extinction uncertainSince the iterated function fn(s) is the PGF of Z (n) inparticular fn(0) is the Pr of extinction after n steps andtaking the limit as n, gives the Pr of eventualextinction
Daniel Tokarev From Extinction to Reproduction in BienaymeGaltonWatson processes

Classification and Extinction Time
BGW processes are broadly divided into three typesSubcritical ( < 1), critical ( = 1)  extinction certain andsupercritical ( > 1)  extinction uncertainSince the iterated function fn(s) is the PGF of Z (n) inparticular fn(0) is the Pr of extinction after n steps andtaking the limit as n, gives the Pr of eventualextinction
Daniel Tokarev From Extinction to Reproduction in BienaymeGaltonWatson processes

Classification and Extinction Time
BGW processes are broadly divided into three typesSubcritical ( < 1), critical ( = 1)  extinction certain andsupercritical ( > 1)  extinction uncertainSince the iterated function fn(s) is the PGF of Z (n) inparticular fn(0) is the Pr of extinction after n steps andtaking the limit as n, gives the Pr of eventualextinction
Daniel Tokarev From Extinction to Reproduction in BienaymeGaltonWatson processes

Classification and Extinction Time
BGW processes are broadly divided into three typesSubcritical ( < 1), critical ( = 1)  extinction certain andsupercritical ( > 1)  extinction uncertainSince the iterated function fn(s) is the PGF of Z (n) inparticular fn(0) is the Pr of extinction after n steps andtaking the limit as n, gives the Pr of eventualextinction
Daniel Tokarev From Extinction to Reproduction in BienaymeGaltonWatson processes

Classification and Extinction Time
BGW processes are broadly divided into three typesSubcritical ( < 1), critical ( = 1)  extinction certain andsupercritical ( > 1)  extinction uncertainSince the iterated function fn(s) is the PGF of Z (n) inparticular fn(0) is the Pr of extinction after n steps andtaking the limit as n, gives the Pr of eventualextinction
Daniel Tokarev From Extinction to Reproduction in BienaymeGaltonWatson processes

Classification and Extinction Time
BGW processes are broadly divided into three typesSubcritical ( < 1), critical ( = 1)  extinction certain andsupercritical ( > 1)  extinction uncertainSince the iterated function fn(s) is the PGF of Z (n) inparticular fn(0) is the Pr of extinction after n steps andtaking the limit as n, gives the Pr of eventualextinction
subcritical.pdf
0.2 0.4 0.6 0.8 1.0
0.2
0.4
0.6
0.8
1.0
Daniel Tokarev From Extinction to Reproduction in BienaymeGaltonWatson processes

Classification and Extinction Time
BGW processes are broadly divided into three typesSubcritical ( < 1), critical ( = 1)  extinction certain andsupercritical ( > 1)  extinction uncertainSince the iterated function fn(s) is the PGF of Z (n) inparticular fn(0) is the Pr of extinction after n steps andtaking the limit as n, gives the Pr of eventualextinction
0.2 0.4 0.6 0.8 1.0
0.2
0.4
0.6
0.8
1.0
0.05 0.10 0.15 0.20 0.25 0.30 0.35
0.05
0.10
0.15
0.20
0.25
0.30
0.35
Daniel Tokarev From Extinction to Reproduction in BienaymeGaltonWatson processes

Extinction, iterates and PGFs
Suppose two individuals have extinction time distributionsno more than apart (wrt some sensible norm):
E.g. {fn(0)} {gn(0)} := supn1{fn(0) gn(0)} .
Can we then deduce a similar statement about thecorresponding {pi} and {qi}?Specifically if = 0, will it follow that {pi} and {qi} are thesame?Must be true, otherwise two distinct PGFsintersect ininfinitely many points! Or is it?Easy to construct two PGFs that share artibrarily manyiterates:
Daniel Tokarev From Extinction to Reproduction in BienaymeGaltonWatson processes

Extinction, iterates and PGFs
Suppose two individuals have extinction time distributionsno more than apart (wrt some sensible norm):
E.g. {fn(0)} {gn(0)} := supn1{fn(0) gn(0)} .
Can we then deduce a similar statement about thecorresponding {pi} and {qi}?Specifically if = 0, will it follow that {pi} and {qi} are thesame?Must be true, otherwise two distinct PGFsintersect ininfinitely many points! Or is it?Easy to construct two PGFs that share artibrarily manyiterates:
Daniel Tokarev From Extinction to Reproduction in BienaymeGaltonWatson processes

Extinction, iterates and PGFs
Suppose two individuals have extinction time distributionsno more than apart (wrt some sensible norm):
E.g. {fn(0)} {gn(0)} := supn1{fn(0) gn(0)} .
Can we then deduce a similar statement about thecorresponding {pi} and {qi}?Specifically if = 0, will it follow that {pi} and {qi} are thesame?Must be true, otherwise two distinct PGFsintersect ininfinitely many points! Or is it?Easy to construct two PGFs that share artibrarily manyiterates:
Daniel Tokarev From Extinction to Reproduction in BienaymeGaltonWatson processes

Extinction, iterates and PGFs
Suppose two individuals have extinction time distributionsno more than apart (wrt some sensible norm):
E.g. {fn(0)} {gn(0)} := supn1{fn(0) gn(0)} .
Can we then deduce a similar statement about thecorresponding {pi} and {qi}?Specifically if = 0, will it follow that {pi} and {qi} are thesame?Must be true, otherwise two distinct PGFsintersect ininfinitely many points! Or is it?Easy to construct two PGFs that share artibrarily manyiterates:
Daniel Tokarev From Extinction to Reproduction in BienaymeGaltonWatson processes

Extinction, iterates and PGFs
Suppose two individuals have extinction time distributionsno more than apart (wrt some sensible norm):
E.g. {fn(0)} {gn(0)} := supn1{fn(0) gn(0)} .
Can we then deduce a similar statement about thecorresponding {pi} and {qi}?Specifically if = 0, will it follow that {pi} and {qi} are thesame?Must be true, otherwise two distinct PGFsintersect ininfinitely many points! Or is it?Easy to construct two PGFs that share artibrarily manyiterates:
Daniel Tokarev From Extinction to Reproduction in BienaymeGaltonWatson processes

Extinction, iterates and PGFs
Suppose two individuals have extinction time distributionsno more than apart (wrt some sensible norm):
E.g. {fn(0)} {gn(0)} := supn1{fn(0) gn(0)} .
Can we then deduce a similar statement about thecorresponding {pi} and {qi}?Specifically if = 0, will it follow that {pi} and {qi} are thesame?Must be true, otherwise two distinct PGFsintersect ininfinitely many points! Or is it?Easy to construct two PGFs that share artibrarily manyiterates:
Daniel Tokarev From Extinction to Reproduction in BienaymeGaltonWatson processes

Finitely many points in common
Let f (s) = ex1 and denote its extinction pmf by{fn(0)} =: {tn}. For some integer j , and small > 0 leth(s) := s
ji=1(s ti).
Then for sufficiently small = (j), f (s) + h(s) will be aPGF with the same first j iterates as f (s). More generally
TheoremLet 0 a1 < a2 < < an = 1 be a finite ordered sequencewith f (ai) =: bi , i = 1, . . . ,n and bn = 1. Further letk := #{j 0 : pj > 0} , so that f (s) =
ki=1 pji s
ji , whereji s are the indices of strictly positive probabilities pj . Thereexists a distribution {qi} on Z+ with {qi} 6= {pi}, such that for itsPGF g(s) =
i0 qis
i , g(ai) = f (ai) = bi , for i = 1, . . . ,n if andonly if n k.
Daniel Tokarev From Extinction to Reproduction in BienaymeGaltonWatson processes

Finitely many points in common
Let f (s) = ex1 and denote its extinction pmf by{fn(0)} =: {tn}. For some integer j , and small > 0 leth(s) := s
ji=1(s ti).
Then for sufficiently small = (j), f (s) + h(s) will be aPGF with the same first j iterates as f (s). More generally
TheoremLet 0 a1 < a2 < < an = 1 be a finite ordered sequencewith f (ai) =: bi , i = 1, . . . ,n and bn = 1. Further letk := #{j 0 : pj > 0} , so that f (s) =
ki=1 pji s
ji , whereji s are the indices of strictly positive probabilities pj . Thereexists a distribution {qi} on Z+ with {qi} 6= {pi}, such that for itsPGF g(s) =
i0 qis
i , g(ai) = f (ai) = bi , for i = 1, . . . ,n if andonly if n k.
Daniel Tokarev From Extinction to Reproduction in BienaymeGaltonWatson processes

Finitely many points in common
Let f (s) = ex1 and denote its extinction pmf by{fn(0)} =: {tn}. For some integer j , and small > 0 leth(s) := s
ji=1(s ti).
Then for sufficiently small = (j), f (s) + h(s) will be aPGF with the same first j iterates as f (s). More generally
TheoremLet 0 a1 < a2 < < an = 1 be a finite ordered sequencewith f (ai) =: bi , i = 1, . . . ,n and bn = 1. Further letk := #{j 0 : pj > 0} , so that f (s) =
ki=1 pji s
ji , whereji s are the indices of strictly positive probabilities pj . Thereexists a distribution {qi} on Z+ with {qi} 6= {pi}, such that for itsPGF g(s) =
i0 qis
i , g(ai) = f (ai) = bi , for i = 1, . . . ,n if andonly if n k.
Daniel Tokarev From Extinction to Reproduction in BienaymeGaltonWatson processes

Finitely many points in common
Let f (s) = ex1 and denote its extinction pmf by{fn(0)} =: {tn}. For some integer j , and small > 0 leth(s) := s
ji=1(s ti).
Then for sufficiently small = (j), f (s) + h(s) will be aPGF with the same first j iterates as f (s). More generally
TheoremLet 0 a1 < a2 < < an = 1 be a finite ordered sequencewith f (ai) =: bi , i = 1, . . . ,n and bn = 1. Further letk := #{j 0 : pj > 0} , so that f (s) =
ki=1 pji s
ji , whereji s are the indices of strictly positive probabilities pj . Thereexists a distribution {qi} on Z+ with {qi} 6= {pi}, such that for itsPGF g(s) =
i0 qis
i , g(ai) = f (ai) = bi , for i = 1, . . . ,n if andonly if n k.
Daniel Tokarev From Extinction to Reproduction in BienaymeGaltonWatson processes

Supercritical case
Recall that in supercritical case the iterates accumulate toa point inside the unit intervalHence by Identity principle, we cannot have zeroesaccumulating to a point inside the region of analyticity.More generally
Theorem
For any sequence of extinction probabilities {f ki (0)} of a mortalsupercritical BGW process with Z (0) = r , there is a uniquenonlattice offspring distribution {pi}.
Indeed if f (s) a PGF of a lattice RV on N, letg(s) = f(s1/), then the BGW process corresponding to fand starting with Z0 = will have the same extinction distas that corresponding to g.
Daniel Tokarev From Extinction to Reproduction in BienaymeGaltonWatson processes

Supercritical case
Recall that in supercritical case the iterates accumulate toa point inside the unit intervalHence by Identity principle, we cannot have zeroesaccumulating to a point inside the region of analyticity.More generally
Theorem
For any sequence of extinction probabilities {f ki (0)} of a mortalsupercritical BGW process with Z (0) = r , there is a uniquenonlattice offspring distribution {pi}.
Indeed if f (s) a PGF of a lattice RV on N, letg(s) = f(s1/), then the BGW process corresponding to fand starting with Z0 = will have the same extinction distas that corresponding to g.
Daniel Tokarev From Extinction to Reproduction in BienaymeGaltonWatson processes

Supercritical case
Recall that in supercritical case the iterates accumulate toa point inside the unit intervalHence by Identity principle, we cannot have zeroesaccumulating to a point inside the region of analyticity.More generally
Theorem
For any sequence of extinction probabilities {f ki (0)} of a mortalsupercritical BGW process with Z (0) = r , there is a uniquenonlattice offspring distribution {pi}.
Indeed if f (s) a PGF of a lattice RV on N, letg(s) = f(s1/), then the BGW process corresponding to fand starting with Z0 = will have the same extinction distas that corresponding to g.
Daniel Tokarev From Extinction to Reproduction in BienaymeGaltonWatson processes

Supercritical case
Recall that in supercritical case the iterates accumulate toa point inside the unit intervalHence by Identity principle, we cannot have zeroesaccumulating to a point inside the region of analyticity.More generally
Theorem
For any sequence of extinction probabilities {f ki (0)} of a mortalsupercritical BGW process with Z (0) = r , there is a uniquenonlattice offspring distribution {pi}.
Indeed if f (s) a PGF of a lattice RV on N, letg(s) = f(s1/), then the BGW process corresponding to fand starting with Z0 = will have the same extinction distas that corresponding to g.
Daniel Tokarev From Extinction to Reproduction in BienaymeGaltonWatson processes

Supercritical case
Recall that in supercritical case the iterates accumulate toa point inside the unit intervalHence by Identity principle, we cannot have zeroesaccumulating to a point inside the region of analyticity.More generally
Theorem
For any sequence of extinction probabilities {f ki (0)} of a mortalsupercritical BGW process with Z (0) = r , there is a uniquenonlattice offspring distribution {pi}.
Indeed if f (s) a PGF of a lattice RV on N, letg(s) = f(s1/), then the BGW process corresponding to fand starting with Z0 = will have the same extinction distas that corresponding to g.
Daniel Tokarev From Extinction to Reproduction in BienaymeGaltonWatson processes

When moments exist
So if the PGF is analytic at accumulation point of theiterates at 0 (call it q), the question is settledFor the case q = 1, what if all moments exist? Then allfactorial moments exist, ie leftsided derivatives at 1 existBut existence of moment, factorial moments and leftsidedderivative does not imply that the PGF is analytic at 1, eglet pi = c2
k , c = 1/
2
k , easy to check that allmoments
cpk2
k exist but the PGF f (s) =
pisi
cannot be continued beyond 1 since
(1 + a)k2
k =for all a > 0.
Daniel Tokarev From Extinction to Reproduction in BienaymeGaltonWatson processes

When moments exist
So if the PGF is analytic at accumulation point of theiterates at 0 (call it q), the question is settledFor the case q = 1, what if all moments exist? Then allfactorial moments exist, ie leftsided derivatives at 1 existBut existence of moment, factorial moments and leftsidedderivative does not imply that the PGF is analytic at 1, eglet pi = c2
k , c = 1/
2
k , easy to check that allmoments
cpk2
k exist but the PGF f (s) =
pisi
cannot be continued beyond 1 since
(1 + a)k2
k =for all a > 0.
Daniel Tokarev From Extinction to Reproduction in BienaymeGaltonWatson processes

When moments exist
So if the PGF is analytic at accumulation point of theiterates at 0 (call it q), the question is settledFor the case q = 1, what if all moments exist? Then allfactorial moments exist, ie leftsided derivatives at 1 existBut existence of moment, factorial moments and leftsidedderivative does not imply that the PGF is analytic at 1, eglet pi = c2
k , c = 1/
2
k , easy to check that allmoments
cpk2
k exist but the PGF f (s) =
pisi
cannot be continued beyond 1 since
(1 + a)k2
k =for all a > 0.
Daniel Tokarev From Extinction to Reproduction in BienaymeGaltonWatson processes

When moments exist
So if the PGF is analytic at accumulation point of theiterates at 0 (call it q), the question is settledFor the case q = 1, what if all moments exist? Then allfactorial moments exist, ie leftsided derivatives at 1 existBut existence of moment, factorial moments and leftsidedderivative does not imply that the PGF is analytic at 1, eglet pi = c2
k , c = 1/
2
k , easy to check that allmoments
cpk2
k exist but the PGF f (s) =
pisi
cannot be continued beyond 1 since
(1 + a)k2
k =for all a > 0.
Daniel Tokarev From Extinction to Reproduction in BienaymeGaltonWatson processes

When moments exist continued
So the previous result does not guarantee that the iteratesuniquely determine reproduction distribution {pi}Divided differences come to the rescue and give us more!
TheoremLet {Zn} be either a supercritical or a nonsupercritical BGWprocess for which the moment generating function exists. Then{fi(0)} =: qi uniquely characterises {pi} which can bedetermined from the Taylor expansion of f around q given by
f (s) = q +
i=1 (qn, . . . ,qn+i)(s q)i , where (qi) := qi+1
and (qi , . . . ,qi+j) :=(qi+1,...,qi+j )(qi ,...,qi+j1)
qi+jqi
Daniel Tokarev From Extinction to Reproduction in BienaymeGaltonWatson processes

When moments exist continued
So the previous result does not guarantee that the iteratesuniquely determine reproduction distribution {pi}Divided differences come to the rescue and give us more!
TheoremLet {Zn} be either a supercritical or a nonsupercritical BGWprocess for which the moment generating function exists. Then{fi(0)} =: qi uniquely characterises {pi} which can bedetermined from the Taylor expansion of f around q given by
f (s) = q +
i=1 (qn, . . . ,qn+i)(s q)i , where (qi) := qi+1
and (qi , . . . ,qi+j) :=(qi+1,...,qi+j )(qi ,...,qi+j1)
qi+jqi
Daniel Tokarev From Extinction to Reproduction in BienaymeGaltonWatson processes

When moments exist continued
So the previous result does not guarantee that the iteratesuniquely determine reproduction distribution {pi}Divided differences come to the rescue and give us more!
TheoremLet {Zn} be either a supercritical or a nonsupercritical BGWprocess for which the moment generating function exists. Then{fi(0)} =: qi uniquely characterises {pi} which can bedetermined from the Taylor expansion of f around q given by
f (s) = q +
i=1 (qn, . . . ,qn+i)(s q)i , where (qi) := qi+1
and (qi , . . . ,qi+j) :=(qi+1,...,qi+j )(qi ,...,qi+j1)
qi+jqi
Daniel Tokarev From Extinction to Reproduction in BienaymeGaltonWatson processes

Approximation theory to the rescue!
We will need the following key result  evolution ofWeierstrass Approximation Theorem through to MntzsTheorem  Full Mntzs Theorem (Schwartz, Siegel):
TheoremLet {i}i=0 be a sequence of distinct positive real numbers
including 0, = Span{n
i=0
aixi ai R}, and C[0,1] is the
space of continuous functions on [0,1]. Then
= C[0,1] iff i
2i + 1=.
Daniel Tokarev From Extinction to Reproduction in BienaymeGaltonWatson processes

Approximation theory to the rescue!
We will need the following key result  evolution ofWeierstrass Approximation Theorem through to MntzsTheorem  Full Mntzs Theorem (Schwartz, Siegel):
TheoremLet {i}i=0 be a sequence of distinct positive real numbers
including 0, = Span{n
i=0
aixi ai R}, and C[0,1] is the
space of continuous functions on [0,1]. Then
= C[0,1] iff i
2i + 1=.
Daniel Tokarev From Extinction to Reproduction in BienaymeGaltonWatson processes

Approximation theory to the rescue!
We will need the following key result  evolution ofWeierstrass Approximation Theorem through to MntzsTheorem  Full Mntzs Theorem (Schwartz, Siegel):
TheoremLet {i}i=0 be a sequence of distinct positive real numbers
including 0, = Span{n
i=0
aixi ai R}, and C[0,1] is the
space of continuous functions on [0,1]. Then
= C[0,1] iff i
2i + 1=.
Daniel Tokarev From Extinction to Reproduction in BienaymeGaltonWatson processes

Identity theorem for completely monotone functions
Recall that h(s) is completely monotone if h C[0,)]and for all n N, s R+, (1)nh(n)(s) 0.Given a family of functionsM with common domain D, wesay that a function is uniquely characterised by its valueson {i} D if for any two f, g M with f(i) = g(i) for alli = 1,2, . . ., implies f(s) = g(s) for all s D. We have
TheoremGiven a sequence of distinct nonnegative real numbers{i} 3 0, a completely monotone function is uniquelycharacterised by its values on {i} iff i
2i + 1=. (1)
Daniel Tokarev From Extinction to Reproduction in BienaymeGaltonWatson processes

Identity theorem for completely monotone functions
Recall that h(s) is completely monotone if h C[0,)]and for all n N, s R+, (1)nh(n)(s) 0.Given a family of functionsM with common domain D, wesay that a function is uniquely characterised by its valueson {i} D if for any two f, g M with f(i) = g(i) for alli = 1,2, . . ., implies f(s) = g(s) for all s D. We have
TheoremGiven a sequence of distinct nonnegative real numbers{i} 3 0, a completely monotone function is uniquelycharacterised by its values on {i} iff i
2i + 1=. (1)
Daniel Tokarev From Extinction to Reproduction in BienaymeGaltonWatson processes

Identity theorem for completely monotone functions
Recall that h(s) is completely monotone if h C[0,)]and for all n N, s R+, (1)nh(n)(s) 0.Given a family of functionsM with common domain D, wesay that a function is uniquely characterised by its valueson {i} D if for any two f, g M with f(i) = g(i) for alli = 1,2, . . ., implies f(s) = g(s) for all s D. We have
TheoremGiven a sequence of distinct nonnegative real numbers{i} 3 0, a completely monotone function is uniquelycharacterised by its values on {i} iff i
2i + 1=. (1)
Daniel Tokarev From Extinction to Reproduction in BienaymeGaltonWatson processes

Identity theorem for completely monotone functions
Recall that h(s) is completely monotone if h C[0,)]and for all n N, s R+, (1)nh(n)(s) 0.Given a family of functionsM with common domain D, wesay that a function is uniquely characterised by its valueson {i} D if for any two f, g M with f(i) = g(i) for alli = 1,2, . . ., implies f(s) = g(s) for all s D. We have
TheoremGiven a sequence of distinct nonnegative real numbers{i} 3 0, a completely monotone function is uniquelycharacterised by its values on {i} iff i
2i + 1=. (1)
Daniel Tokarev From Extinction to Reproduction in BienaymeGaltonWatson processes

And back to PGFs!
Observe that if f is a PGF and h is completely monotone,then f (g) is completely monotone, from this we easilyobtain
TheoremLet {qn}n=j , for some j N, l be a tail of a distribution ofextinction time of a BGW process {Zn}, with Z0 = r . Let Tdenote the RV time to extinction of {Zn}. Then {qn}n=j uniquelydetermines the reproduction distribution {pi} and r if
i=j
(1 qr ) = or equivalently ET =.
Daniel Tokarev From Extinction to Reproduction in BienaymeGaltonWatson processes

And back to PGFs!
Observe that if f is a PGF and h is completely monotone,then f (g) is completely monotone, from this we easilyobtain
TheoremLet {qn}n=j , for some j N, l be a tail of a distribution ofextinction time of a BGW process {Zn}, with Z0 = r . Let Tdenote the RV time to extinction of {Zn}. Then {qn}n=j uniquelydetermines the reproduction distribution {pi} and r if
i=j
(1 qr ) = or equivalently ET =.
Daniel Tokarev From Extinction to Reproduction in BienaymeGaltonWatson processes

And back to PGFs!
Observe that if f is a PGF and h is completely monotone,then f (g) is completely monotone, from this we easilyobtain
TheoremLet {qn}n=j , for some j N, l be a tail of a distribution ofextinction time of a BGW process {Zn}, with Z0 = r . Let Tdenote the RV time to extinction of {Zn}. Then {qn}n=j uniquelydetermines the reproduction distribution {pi} and r if
i=j
(1 qr ) = or equivalently ET =.
Daniel Tokarev From Extinction to Reproduction in BienaymeGaltonWatson processes

Towards a counterexample: Blaschke Products
Generalisation of Weierstrass products to functionsanalytic on the open unit disk
TheoremGiven a set of points {an} on the unit disk, there exists afunction analytic on the unit disk with zeros at {an} and uniqueup to a zero free analytic factor iff
i=1
(1 ai )
in which case it is given by
B(z) =
i=1
aian
an z1 anz
.
Daniel Tokarev From Extinction to Reproduction in BienaymeGaltonWatson processes

Towards a counterexample: Blaschke Products
Generalisation of Weierstrass products to functionsanalytic on the open unit disk
TheoremGiven a set of points {an} on the unit disk, there exists afunction analytic on the unit disk with zeros at {an} and uniqueup to a zero free analytic factor iff
i=1
(1 ai )
in which case it is given by
B(z) =
i=1
aian
an z1 anz
.
Daniel Tokarev From Extinction to Reproduction in BienaymeGaltonWatson processes

Towards a counterexample: Blaschke Products
Generalisation of Weierstrass products to functionsanalytic on the open unit disk
TheoremGiven a set of points {an} on the unit disk, there exists afunction analytic on the unit disk with zeros at {an} and uniqueup to a zero free analytic factor iff
i=1
(1 ai )
in which case it is given by
B(z) =
i=1
aian
an z1 anz
.
Daniel Tokarev From Extinction to Reproduction in BienaymeGaltonWatson processes

The Number of Positive Zeros of Transforms
If f and g agree on the iterates, f (s) g(s) = E(s)B(s)where B(s) is a Blaschke product and E(s) is a zero freefunction analytic on the unit disk.For subcritical PGFs, we know that B(s) =
bisi with
bi  1/ig(s) would have Taylor coefficients = o(i2) and sincef (s) = g(s) + E(s)B(s), andWe need to find E(s) that would make the coefficient of theproduct E(s)B(s) decay faster than i2
The trouble is that we dont understand the pattern of signsin bi s  real Blaschke products are not wellunderstood
Daniel Tokarev From Extinction to Reproduction in BienaymeGaltonWatson processes

The Number of Positive Zeros of Transforms
If f and g agree on the iterates, f (s) g(s) = E(s)B(s)where B(s) is a Blaschke product and E(s) is a zero freefunction analytic on the unit disk.For subcritical PGFs, we know that B(s) =
bisi with
bi  1/ig(s) would have Taylor coefficients = o(i2) and sincef (s) = g(s) + E(s)B(s), andWe need to find E(s) that would make the coefficient of theproduct E(s)B(s) decay faster than i2
The trouble is that we dont understand the pattern of signsin bi s  real Blaschke products are not wellunderstood
Daniel Tokarev From Extinction to Reproduction in BienaymeGaltonWatson processes

The Number of Positive Zeros of Transforms
If f and g agree on the iterates, f (s) g(s) = E(s)B(s)where B(s) is a Blaschke product and E(s) is a zero freefunction analytic on the unit disk.For subcritical PGFs, we know that B(s) =
bisi with
bi  1/ig(s) would have Taylor coefficients = o(i2) and sincef (s) = g(s) + E(s)B(s), andWe need to find E(s) that would make the coefficient of theproduct E(s)B(s) decay faster than i2
The trouble is that we dont understand the pattern of signsin bi s  real Blaschke products are not wellunderstood
Daniel Tokarev From Extinction to Reproduction in BienaymeGaltonWatson processes

The Number of Positive Zeros of Transforms
If f and g agree on the iterates, f (s) g(s) = E(s)B(s)where B(s) is a Blaschke product and E(s) is a zero freefunction analytic on the unit disk.For subcritical PGFs, we know that B(s) =
bisi with
bi  1/ig(s) would have Taylor coefficients = o(i2) and sincef (s) = g(s) + E(s)B(s), andWe need to find E(s) that would make the coefficient of theproduct E(s)B(s) decay faster than i2
The trouble is that we dont understand the pattern of signsin bi s  real Blaschke products are not wellunderstood
Daniel Tokarev From Extinction to Reproduction in BienaymeGaltonWatson processes

The Number of Positive Zeros of Transforms
If f and g agree on the iterates, f (s) g(s) = E(s)B(s)where B(s) is a Blaschke product and E(s) is a zero freefunction analytic on the unit disk.For subcritical PGFs, we know that B(s) =
bisi with
bi  1/ig(s) would have Taylor coefficients = o(i2) and sincef (s) = g(s) + E(s)B(s), andWe need to find E(s) that would make the coefficient of theproduct E(s)B(s) decay faster than i2
The trouble is that we dont understand the pattern of signsin bi s  real Blaschke products are not wellunderstood
Daniel Tokarev From Extinction to Reproduction in BienaymeGaltonWatson processes

The Number of Positive Zeros of Transforms
If f and g agree on the iterates, f (s) g(s) = E(s)B(s)where B(s) is a Blaschke product and E(s) is a zero freefunction analytic on the unit disk.For subcritical PGFs, we know that B(s) =
bisi with
bi  1/ig(s) would have Taylor coefficients = o(i2) and sincef (s) = g(s) + E(s)B(s), andWe need to find E(s) that would make the coefficient of theproduct E(s)B(s) decay faster than i2
The trouble is that we dont understand the pattern of signsin bi s  real Blaschke products are not wellunderstood
Daniel Tokarev From Extinction to Reproduction in BienaymeGaltonWatson processes

Mixing AdvantageHamza, Jagers, Sudbury & Tokarev (2009), Extr.
How do mixed and unmixed populations compare?
Theorem (Hamza, Jagers, Sudbury and Tokarev (2009) Extr.)
Assume FSS that d = n and let Mi := Emax{(1)i , . . . ,
(n)i }
1n
ni=1
Mi Emax{1, . . . , n} 1n
ni=1
Mi +n 1
nmax
i=1,...,n{Mi}
In particular, if all the unmixed expected lifetimes are the sameand equal to M, then we have
M Emax{1, . . . , n} (2 1/n)M.
Daniel Tokarev From Extinction to Reproduction in BienaymeGaltonWatson processes

Mixing AdvantageHamza, Jagers, Sudbury & Tokarev (2009), Extr.
How do mixed and unmixed populations compare?
Theorem (Hamza, Jagers, Sudbury and Tokarev (2009) Extr.)
Assume FSS that d = n and let Mi := Emax{(1)i , . . . ,
(n)i }
1n
ni=1
Mi Emax{1, . . . , n} 1n
ni=1
Mi +n 1
nmax
i=1,...,n{Mi}
In particular, if all the unmixed expected lifetimes are the sameand equal to M, then we have
M Emax{1, . . . , n} (2 1/n)M.
Daniel Tokarev From Extinction to Reproduction in BienaymeGaltonWatson processes

What does tell us about ?
When does {fn(0)} uniquely characterizes f?When f is analytic in the neighbourhood of the fixed point q
From Identity Principle (an analytic function is uniquelycharacterized by its values at a set of points accumulatinginside the domain of analyticity).
This applies when f is a PGF of a supercritical process withp0 6= 0
When E =  from a slight generalisation of Fellers proofof Mntzs TheoremNot when E

What does tell us about ?
When does {fn(0)} uniquely characterizes f?When f is analytic in the neighbourhood of the fixed point q
From Identity Principle (an analytic function is uniquelycharacterized by its values at a set of points accumulatinginside the domain of analyticity).
This applies when f is a PGF of a supercritical process withp0 6= 0
When E =  from a slight generalisation of Fellers proofof Mntzs TheoremNot when E

What does tell us about ?
When does {fn(0)} uniquely characterizes f?When f is analytic in the neighbourhood of the fixed point q
From Identity Principle (an analytic function is uniquelycharacterized by its values at a set of points accumulatinginside the domain of analyticity).
This applies when f is a PGF of a supercritical process withp0 6= 0
When E =  from a slight generalisation of Fellers proofof Mntzs TheoremNot when E

What does tell us about ?
When does {fn(0)} uniquely characterizes f?When f is analytic in the neighbourhood of the fixed point q
From Identity Principle (an analytic function is uniquelycharacterized by its values at a set of points accumulatinginside the domain of analyticity).
This applies when f is a PGF of a supercritical process withp0 6= 0
When E =  from a slight generalisation of Fellers proofof Mntzs TheoremNot when E

What does tell us about ?
When does {fn(0)} uniquely characterizes f?When f is analytic in the neighbourhood of the fixed point q
From Identity Principle (an analytic function is uniquelycharacterized by its values at a set of points accumulatinginside the domain of analyticity).
This applies when f is a PGF of a supercritical process withp0 6= 0
When E =  from a slight generalisation of Fellers proofof Mntzs TheoremNot when E

What does tell us about ?
When does {fn(0)} uniquely characterizes f?When f is analytic in the neighbourhood of the fixed point q
From Identity Principle (an analytic function is uniquelycharacterized by its values at a set of points accumulatinginside the domain of analyticity).
This applies when f is a PGF of a supercritical process withp0 6= 0
When E =  from a slight generalisation of Fellers proofof Mntzs TheoremNot when E

What does tell us about ?
When does {fn(0)} uniquely characterizes f?When f is analytic in the neighbourhood of the fixed point q
From Identity Principle (an analytic function is uniquelycharacterized by its values at a set of points accumulatinginside the domain of analyticity).
This applies when f is a PGF of a supercritical process withp0 6= 0
When E =  from a slight generalisation of Fellers proofof Mntzs TheoremNot when E

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Feller, W. (1971) An Introduction to Probability Theory andIts Applications, Volume II, 2nd Ed., John Wiley & Sons,Inc.
Hamza, K., Jagers, P., Sudbury, A., Tokarev D. (2009)Mixing advantage is less than 2. Extremes, 12, 1931.
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Lawton, J.H., May, R.M., Extinction rates, (1990).textitOxford University Press, Oxford, UK
Daniel Tokarev From Extinction to Reproduction in BienaymeGaltonWatson processes

Pakes, A. G. (1989) On the Asymptotic Behaviour of theExtinction Time of the Simple Branching Process. Adv.Appl. Prob. 21:470471.
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Daniel Tokarev From Extinction to Reproduction in BienaymeGaltonWatson processes

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Daniel Tokarev From Extinction to Reproduction in BienaymeGaltonWatson processes