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From Extinction to Reproduction in Bienayme-Galton-Watson processes Daniel Tokarev Monash University 11 July, 2012 Daniel Tokarev From Extinction to Reproduction in Bienayme-Galton-Watson
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• From Extinction to Reproduction inBienayme-Galton-Watson processes

Daniel Tokarev

Monash University

11 July, 2012

Daniel Tokarev From Extinction to Reproduction in Bienayme-Galton-Watson processes

• Daniel Tokarev From Extinction to Reproduction in Bienayme-Galton-Watson processes

• BGW Processes

Let be some a random variable supported onnon-negative 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 Bienayme-Galton-Watson processes

• BGW Processes

Let be some a random variable supported onnon-negative 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 Bienayme-Galton-Watson processes

• BGW Processes

Let be some a random variable supported onnon-negative 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 Bienayme-Galton-Watson processes

• BGW Processes

Let be some a random variable supported onnon-negative 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 Bienayme-Galton-Watson processes

• BGW Processes

Let be some a random variable supported onnon-negative 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 Bienayme-Galton-Watson processes

• BGW Processes

Let be some a random variable supported onnon-negative 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 Bienayme-Galton-Watson 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 Bienayme-Galton-Watson 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 Bienayme-Galton-Watson 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 Bienayme-Galton-Watson 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 Bienayme-Galton-Watson 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 Bienayme-Galton-Watson 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 Bienayme-Galton-Watson 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 Bienayme-Galton-Watson 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 Bienayme-Galton-Watson 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 Bienayme-Galton-Watson 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 Bienayme-Galton-Watson 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 Bienayme-Galton-Watson 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 Bienayme-Galton-Watson 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 Bienayme-Galton-Watson 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 Bienayme-Galton-Watson 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 Bienayme-Galton-Watson 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 Bienayme-Galton-Watson 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 Bienayme-Galton-Watson 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 Bienayme-Galton-Watson 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 uniquenon-lattice 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 Bienayme-Galton-Watson 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 uniquenon-lattice 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 Bienayme-Galton-Watson 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 uniquenon-lattice 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 Bienayme-Galton-Watson 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 uniquenon-lattice 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 Bienayme-Galton-Watson 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 uniquenon-lattice 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 Bienayme-Galton-Watson 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 left-sided derivatives at 1 existBut existence of moment, factorial moments and left-sidedderivative 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 Bienayme-Galton-Watson 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 left-sided derivatives at 1 existBut existence of moment, factorial moments and left-sidedderivative 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 Bienayme-Galton-Watson 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 left-sided derivatives at 1 existBut existence of moment, factorial moments and left-sidedderivative 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 Bienayme-Galton-Watson 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 left-sided derivatives at 1 existBut existence of moment, factorial moments and left-sidedderivative 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 Bienayme-Galton-Watson 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 non-supercritical 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 Bienayme-Galton-Watson 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 non-supercritical 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 Bienayme-Galton-Watson 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 non-supercritical 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 Bienayme-Galton-Watson 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 Bienayme-Galton-Watson 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 Bienayme-Galton-Watson 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 Bienayme-Galton-Watson 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 non-negative 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 Bienayme-Galton-Watson 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 non-negative 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 Bienayme-Galton-Watson 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 non-negative 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 Bienayme-Galton-Watson 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 non-negative 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 Bienayme-Galton-Watson 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 Bienayme-Galton-Watson 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 Bienayme-Galton-Watson 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 Bienayme-Galton-Watson processes

• Towards a counter-example: 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

ai|an|

an z1 anz

.

Daniel Tokarev From Extinction to Reproduction in Bienayme-Galton-Watson processes

• Towards a counter-example: 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

ai|an|

an z1 anz

.

Daniel Tokarev From Extinction to Reproduction in Bienayme-Galton-Watson processes

• Towards a counter-example: 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

ai|an|

an z1 anz

.

Daniel Tokarev From Extinction to Reproduction in Bienayme-Galton-Watson 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 well-understood

Daniel Tokarev From Extinction to Reproduction in Bienayme-Galton-Watson 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 well-understood

Daniel Tokarev From Extinction to Reproduction in Bienayme-Galton-Watson 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 well-understood

Daniel Tokarev From Extinction to Reproduction in Bienayme-Galton-Watson 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 well-understood

Daniel Tokarev From Extinction to Reproduction in Bienayme-Galton-Watson 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 well-understood

Daniel Tokarev From Extinction to Reproduction in Bienayme-Galton-Watson 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 well-understood

Daniel Tokarev From Extinction to Reproduction in Bienayme-Galton-Watson 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 Bienayme-Galton-Watson 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 Bienayme-Galton-Watson 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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Carnicer, J. M., Pea, J. M. 1998. Characterizations of theoptimal descartes rules of signs. Math. Nachr. 189, 3348.

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 Bienayme-Galton-Watson processes

• Pakes, A. G. (1989) On the Asymptotic Behaviour of theExtinction Time of the Simple Branching Process. Adv.Appl. Prob. 21:470-471.

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Chris D. Thomas, Alison Cameron, Rhys E. Green, MichelBakkenes, Linda J. Beaumont, Yvonne C. Collingham,Barend F. N. Erasmus, Marinez Ferreira de Siqueira, AlanGrainger, Lee Hannah, Lesley Hughes, Brian Huntley,

Daniel Tokarev From Extinction to Reproduction in Bienayme-Galton-Watson processes

• Albert S. van Jaarsveld, Guy F. Midgley, Lera Miles, MiguelA. Ortega-Huerta, A. Townsend Peterson, Oliver L. Phillips& Stephen E. Williams. Extinction risk from climate change,(2004), Nature 427, 145-148.

Tokarev, D. (2008). Growth of Integral Transforms andExtinction in Critical Galton-Watson Processes, Journal ofApplied Probability, 45, 1-9.

Tokarev, D.V., and Borovkov, K.A., (2009). On theexpectations of maxima of sets of independent randomvariables. Statistics and Probability Letters, 79, 2381 2388

Tokarev D.V., From Extinction to Reproduction inBienayme-Galton-WatsonProcesses. (in preparation).Other

Daniel Tokarev From Extinction to Reproduction in Bienayme-Galton-Watson processes