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7/31/2019 Istanbul Talk Tokarev
1/59
BGW Processes and their Basic Properties
Problem: From Extinction to Reproduction
Towards a Counter-Example
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-Watso
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7/31/2019 Istanbul Talk Tokarev
2/59
BGW Processes and their Basic Properties
Problem: From Extinction to Reproduction
Towards a Counter-Example
Classification and Probability of Extinction
Daniel Tokarev From Extinction to Reproduction in Bienayme-Galton-Watso
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7/31/2019 Istanbul Talk Tokarev
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BGW Processes and their Basic Properties
Problem: From Extinction to Reproduction
Towards a Counter-Example
Classification and Probability of Extinction
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 past
The information about the process is encoded in
probability 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-Watso
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7/31/2019 Istanbul Talk Tokarev
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BGW Processes and their Basic Properties
Problem: From Extinction to Reproduction
Towards a Counter-Example
Classification and Probability of Extinction
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 past
The information about the process is encoded in
probability 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-Watso
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7/31/2019 Istanbul Talk Tokarev
5/59
BGW Processes and their Basic Properties
Problem: From Extinction to Reproduction
Towards a Counter-Example
Classification and Probability of Extinction
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 past
The information about the process is encoded in
probability 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-Watso
BGW P d h i B i P i
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BGW Processes and their Basic Properties
Problem: From Extinction to Reproduction
Towards a Counter-Example
Classification and Probability of Extinction
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 past
The information about the process is encoded in
probability 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-Watso
BGW P d th i B i P ti
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7/31/2019 Istanbul Talk Tokarev
7/59
BGW Processes and their Basic Properties
Problem: From Extinction to Reproduction
Towards a Counter-Example
Classification and Probability of Extinction
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 past
The information about the process is encoded in
probability 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-Watso
BGW Processes and their Basic Properties
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7/31/2019 Istanbul Talk Tokarev
8/59
BGW Processes and their Basic Properties
Problem: From Extinction to Reproduction
Towards a Counter-Example
Classification and Probability of Extinction
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 past
The information about the process is encoded in
probability 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-Watso
BGW Processes and their Basic Properties
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7/31/2019 Istanbul Talk Tokarev
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BGW Processes and their Basic Properties
Problem: From Extinction to Reproduction
Towards a Counter-Example
Classification and Probability of Extinction
Classification and Extinction Time
BGW processes are broadly divided into three types
Subcritical ( < 1), critical ( = 1) - extinction certain and
supercritical ( > 1) - extinction uncertainSince the iterated function fn(s) is the PGF of Z(n) inparticular fn(0) is the Pr of extinction after nsteps andtaking the limit as n , gives the Pr of eventualextinction
Daniel Tokarev From Extinction to Reproduction in Bienayme-Galton-Watso
BGW Processes and their Basic Properties
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BGW Processes and their Basic Properties
Problem: From Extinction to Reproduction
Towards a Counter-Example
Classification and Probability of Extinction
Classification and Extinction Time
BGW processes are broadly divided into three types
Subcritical ( < 1), critical ( = 1) - extinction certain and
supercritical ( > 1) - extinction uncertainSince the iterated function fn(s) is the PGF of Z(n) inparticular fn(0) is the Pr of extinction after nsteps andtaking the limit as n , gives the Pr of eventualextinction
Daniel Tokarev From Extinction to Reproduction in Bienayme-Galton-Watso
BGW Processes and their Basic Properties
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BGW Processes and their Basic Properties
Problem: From Extinction to Reproduction
Towards a Counter-Example
Classification and Probability of Extinction
Classification and Extinction Time
BGW processes are broadly divided into three types
Subcritical ( < 1), critical ( = 1) - extinction certain and
supercritical ( > 1) - extinction uncertainSince the iterated function fn(s) is the PGF of Z(n) inparticular fn(0) is the Pr of extinction after nsteps andtaking the limit as n , gives the Pr of eventualextinction
Daniel Tokarev From Extinction to Reproduction in Bienayme-Galton-Watso
BGW Processes and their Basic Properties
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BGW Processes and their Basic Properties
Problem: From Extinction to Reproduction
Towards a Counter-Example
Classification and Probability of Extinction
Classification and Extinction Time
BGW processes are broadly divided into three types
Subcritical ( < 1), critical ( = 1) - extinction certain and
supercritical ( > 1) - extinction uncertainSince the iterated function fn(s) is the PGF of Z(n) inparticular fn(0) is the Pr of extinction after nsteps andtaking the limit as n , gives the Pr of eventualextinction
Daniel Tokarev From Extinction to Reproduction in Bienayme-Galton-Watso
BGW Processes and their Basic Properties
http://find/ -
7/31/2019 Istanbul Talk Tokarev
13/59
p
Problem: From Extinction to Reproduction
Towards a Counter-Example
Classification and Probability of Extinction
Classification and Extinction Time
BGW processes are broadly divided into three types
Subcritical ( < 1), critical ( = 1) - extinction certain and
supercritical ( > 1) - extinction uncertainSince the iterated function fn(s) is the PGF of Z(n) inparticular fn(0) is the Pr of extinction after nsteps andtaking the limit as n , gives the Pr of eventualextinction
Daniel Tokarev From Extinction to Reproduction in Bienayme-Galton-Watso
BGW Processes and their Basic Properties
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p
Problem: From Extinction to Reproduction
Towards a Counter-Example
Classification and Probability of Extinction
Classification and Extinction Time
BGW processes are broadly divided into three types
Subcritical ( < 1), critical ( = 1) - extinction certain and
supercritical ( > 1) - extinction uncertainSince the iterated function fn(s) is the PGF of Z(n) inparticular fn(0) is the Pr of extinction after nsteps andtaking the limit as n , gives the Pr of eventualextinction
Daniel Tokarev From Extinction to Reproduction in Bienayme-Galton-Watso
BGW Processes and their Basic Properties
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15/59
Problem: From Extinction to Reproduction
Towards a Counter-Example
Classification and Probability of Extinction
Classification and Extinction Time
BGW processes are broadly divided into three types
Subcritical ( < 1), critical ( = 1) - extinction certain andsupercritical ( > 1) - extinction uncertain
Since the iterated function fn(s) is the PGF of Z(n) in
particular fn(0) is the Pr of extinction after nsteps 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
0.2 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-Watso
BGW Processes and their Basic Properties
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Problem: From Extinction to Reproduction
Towards a Counter-Example
Classification and Probability of Extinction
Classification and Extinction Time
BGW processes are broadly divided into three typesSubcritical ( < 1), critical ( = 1) - extinction certain andsupercritical ( > 1) - extinction uncertain
Since the iterated function fn(s) is the PGF of Z(n) inparticular fn(0) is the Pr of extinction after nsteps and
taking 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-Watso
BGW Processes and their Basic PropertiesProblem defined
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Problem: From Extinction to Reproduction
Towards a Counter-Example
Problem defined
Moving away from analyticity
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 the
corresponding {pi} and {qi}?
Specifically if = 0, will it follow that {pi} and {qi} are thesame?
Must be true, otherwise two distinct PGFsintersect in
infinitely many points! Or is it?
Easy to construct two PGFs that share artibrarily many
iterates:
Daniel Tokarev From Extinction to Reproduction in Bienayme-Galton-Watso
BGW Processes and their Basic Properties
P bl F E ti ti t R d tiProblem defined
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18/59
Problem: From Extinction to Reproduction
Towards a Counter-Example
Problem defined
Moving away from analyticity
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 the
corresponding {pi} and {qi}?
Specifically if = 0, will it follow that {pi} and {qi} are thesame?
Must be true, otherwise two distinct PGFsintersect in
infinitely many points! Or is it?
Easy to construct two PGFs that share artibrarily many
iterates:
Daniel Tokarev From Extinction to Reproduction in Bienayme-Galton-Watso
BGW Processes and their Basic Properties
Problem From Extinction to ReproductionProblem defined
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7/31/2019 Istanbul Talk Tokarev
19/59
Problem: From Extinction to Reproduction
Towards a Counter-ExampleMoving away from analyticity
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 the
corresponding {pi} and {qi}?
Specifically if = 0, will it follow that {pi} and {qi} are thesame?
Must be true, otherwise two distinct PGFsintersect in
infinitely many points! Or is it?
Easy to construct two PGFs that share artibrarily many
iterates:
Daniel Tokarev From Extinction to Reproduction in Bienayme-Galton-Watso
BGW Processes and their Basic Properties
Problem: From Extinction to ReproductionProblem defined
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7/31/2019 Istanbul Talk Tokarev
20/59
Problem: From Extinction to Reproduction
Towards a Counter-ExampleMoving away from analyticity
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 the
corresponding {pi} and {qi}?
Specifically if = 0, will it follow that {pi} and {qi} are thesame?
Must be true, otherwise two distinct PGFsintersect in
infinitely many points! Or is it?
Easy to construct two PGFs that share artibrarily many
iterates:
Daniel Tokarev From Extinction to Reproduction in Bienayme-Galton-Watso
BGW Processes and their Basic Properties
Problem: From Extinction to ReproductionProblem defined
http://find/ -
7/31/2019 Istanbul Talk Tokarev
21/59
Problem: From Extinction to Reproduction
Towards a Counter-ExampleMoving away from analyticity
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 the
corresponding {pi} and {qi}?
Specifically if = 0, will it follow that {pi} and {qi} are thesame?
Must be true, otherwise two distinct PGFsintersect in
infinitely many points! Or is it?
Easy to construct two PGFs that share artibrarily many
iterates:
Daniel Tokarev From Extinction to Reproduction in Bienayme-Galton-Watso
BGW Processes and their Basic Properties
Problem: From Extinction to ReproductionProblem defined
http://find/ -
7/31/2019 Istanbul Talk Tokarev
22/59
Problem: From Extinction to Reproduction
Towards a Counter-ExampleMoving away from analyticity
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 the
corresponding {pi} and {qi}?
Specifically if = 0, will it follow that {pi} and {qi} are thesame?
Must be true, otherwise two distinct PGFsintersect in
infinitely many points! Or is it?
Easy to construct two PGFs that share artibrarily many
iterates:
Daniel Tokarev From Extinction to Reproduction in Bienayme-Galton-Watso
BGW Processes and their Basic PropertiesProblem: From Extinction to Reproduction
Problem defined
M i f l i i
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Problem: From Extinction to Reproduction
Towards a Counter-ExampleMoving away from analyticity
Finitely many points in common
Let f(s) = ex
1
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
Theorem
Let0 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) =
k
i=1pj
i
sji, wherejis are the indices of strictly positive probabilities pj. Thereexists a distribution{qi} onZ+ with{qi} = {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-Watso
BGW Processes and their Basic PropertiesProblem: From Extinction to Reproduction
Problem defined
M i f l ti it
http://find/ -
7/31/2019 Istanbul Talk Tokarev
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p
Towards a Counter-ExampleMoving away from analyticity
Finitely many points in common
Let f(s) = ex
1
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
Theorem
Let0 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) =
k
i=1pj
i
sji, wherejis are the indices of strictly positive probabilities pj. Thereexists a distribution{qi} onZ+ with{qi} = {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-Watso
BGW Processes and their Basic PropertiesProblem: From Extinction to Reproduction
Problem defined
Moving away from analyticity
http://find/ -
7/31/2019 Istanbul Talk Tokarev
25/59
p
Towards a Counter-ExampleMoving away from analyticity
Finitely many points in common
Let f(s) = ex
1
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
Theorem
Let0 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) =
k
i=1pj
i
sji, wherejis are the indices of strictly positive probabilities pj. Thereexists a distribution{qi} onZ+ with{qi} = {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-Watso
BGW Processes and their Basic PropertiesProblem: From Extinction to Reproduction
Problem defined
Moving away from analyticity
http://find/ -
7/31/2019 Istanbul Talk Tokarev
26/59
Towards a Counter-ExampleMoving away from analyticity
Finitely many points in common
Let f(s) = ex
1
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
Theorem
Let0 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) =
k
i=1pj
i
sji, wherejis are the indices of strictly positive probabilities pj. Thereexists a distribution{qi} onZ+ with{qi} = {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-Watso
BGW Processes and their Basic PropertiesProblem: From Extinction to Reproduction
Problem defined
Moving away from analyticity
http://find/ -
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Towards a Counter-ExampleMoving away from analyticity
Supercritical case
Recall that in supercritical case the iterates accumulate toa point inside the unit interval
Hence by Identity principle, we cannot have zeroes
accumulating to a point inside the region of analyticity.
More generally
Theorem
For any sequence of extinction probabilities{fki (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-Watso
BGW Processes and their Basic PropertiesProblem: From Extinction to Reproduction
Problem defined
Moving away from analyticity
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Towards a Counter-ExampleMoving away from analyticity
Supercritical case
Recall that in supercritical case the iterates accumulate toa point inside the unit interval
Hence by Identity principle, we cannot have zeroes
accumulating to a point inside the region of analyticity.
More generally
Theorem
For any sequence of extinction probabilities{fki (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-Watso
BGW Processes and their Basic PropertiesProblem: From Extinction to Reproduction
T d C t E l
Problem defined
Moving away from analyticity
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Towards a Counter-ExampleMoving away from analyticity
Supercritical case
Recall that in supercritical case the iterates accumulate toa point inside the unit interval
Hence by Identity principle, we cannot have zeroes
accumulating to a point inside the region of analyticity.
More generally
Theorem
For any sequence of extinction probabilities{fki (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-Watso
BGW Processes and their Basic PropertiesProblem: From Extinction to Reproduction
T d C t E l
Problem defined
Moving away from analyticity
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Towards a Counter-Exampleg y y y
Supercritical case
Recall that in supercritical case the iterates accumulate toa point inside the unit interval
Hence by Identity principle, we cannot have zeroes
accumulating to a point inside the region of analyticity.
More generally
Theorem
For any sequence of extinction probabilities{fki (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-Watso
BGW Processes and their Basic PropertiesProblem: From Extinction to Reproduction
Towards a Counter Example
Problem defined
Moving away from analyticity
http://find/ -
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31/59
Towards a Counter-Exampleg y y y
Supercritical case
Recall that in supercritical case the iterates accumulate toa point inside the unit interval
Hence by Identity principle, we cannot have zeroes
accumulating to a point inside the region of analyticity.
More generally
Theorem
For any sequence of extinction probabilities{fki (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-Watso
BGW Processes and their Basic PropertiesProblem: From Extinction to Reproduction
Towards a Counter-Example
Problem defined
Moving away from analyticity
http://find/ -
7/31/2019 Istanbul Talk Tokarev
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Towards a Counter-Example
When moments exist
So if the PGF is analytic at accumulation point of the
iterates at 0 (call it q), the question is settled
For the case q= 1, what if all moments exist? Then all
factorial moments exist, ie left-sided derivatives at 1 existBut existence of moment, factorial moments and left-sided
derivative does not imply that the PGF is analytic at 1, eg
let pi = c2
k, c= 1/
2
k, easy to check that all
moments cpk
2
k exist but the PGF f(s) = p
isi
cannot be continued beyond 1 since
(1 + a)k2k =
for all a> 0.
Daniel Tokarev From Extinction to Reproduction in Bienayme-Galton-Watso
BGW Processes and their Basic PropertiesProblem: From Extinction to Reproduction
Towards a Counter-Example
Problem defined
Moving away from analyticity
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7/31/2019 Istanbul Talk Tokarev
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Towards a Counter-Example
When moments exist
So if the PGF is analytic at accumulation point of the
iterates at 0 (call it q), the question is settled
For the case q= 1, what if all moments exist? Then all
factorial moments exist, ie left-sided derivatives at 1 existBut existence of moment, factorial moments and left-sided
derivative does not imply that the PGF is analytic at 1, eg
let pi = c2
k, c= 1/
2
k, easy to check that all
moments cpk
2
k exist but the PGF f(s) = p
isi
cannot be continued beyond 1 since
(1 + a)k2k =
for all a> 0.
Daniel Tokarev From Extinction to Reproduction in Bienayme-Galton-Watso
BGW Processes and their Basic PropertiesProblem: From Extinction to Reproduction
Towards a Counter-Example
Problem defined
Moving away from analyticity
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Towards a Counter Example
When moments exist
So if the PGF is analytic at accumulation point of the
iterates at 0 (call it q), the question is settled
For the case q= 1, what if all moments exist? Then all
factorial moments exist, ie left-sided derivatives at 1 existBut existence of moment, factorial moments and left-sided
derivative does not imply that the PGF is analytic at 1, eg
let pi = c2
k, c= 1/
2
k, easy to check that all
moments cpk
2
k exist but the PGF f(s) = p
isi
cannot be continued beyond 1 since
(1 + a)k2k =
for all a> 0.
Daniel Tokarev From Extinction to Reproduction in Bienayme-Galton-Watso
BGW Processes and their Basic PropertiesProblem: From Extinction to Reproduction
Towards a Counter-Example
Problem defined
Moving away from analyticity
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o a ds a Cou te a p e
When moments exist
So if the PGF is analytic at accumulation point of the
iterates at 0 (call it q), the question is settled
For the case q= 1, what if all moments exist? Then all
factorial moments exist, ie left-sided derivatives at 1 existBut existence of moment, factorial moments and left-sided
derivative does not imply that the PGF is analytic at 1, eg
let pi = c2
k, c= 1/
2
k, easy to check that all
moments cpk
2
k exist but the PGF f(s) = p
isi
cannot be continued beyond 1 since
(1 + a)k2k =
for all a> 0.
Daniel Tokarev From Extinction to Reproduction in Bienayme-Galton-Watso
BGW Processes and their Basic PropertiesProblem: From Extinction to Reproduction
Towards a Counter-Example
Problem defined
Moving away from analyticity
http://find/ -
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p
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!
Theorem
Let{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-Watso
BGW Processes and their Basic PropertiesProblem: From Extinction to Reproduction
Towards a Counter-Example
Problem defined
Moving away from analyticity
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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!
Theorem
Let{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-Watso
BGW Processes and their Basic PropertiesProblem: From Extinction to Reproduction
Towards a Counter-Example
Problem defined
Moving away from analyticity
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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!
Theorem
Let{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-Watso
BGW Processes and their Basic PropertiesProblem: From Extinction to Reproduction
Towards a Counter-Example
Problem defined
Moving away from analyticity
http://find/ -
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Approximation theory to the rescue!
We will need the following key result - evolution of
Weierstrass Approximation Theorem through to Mntzs
Theorem - Full Mntzs Theorem (Schwartz, Siegel):
Theorem
Let{i}i=0 be a sequence of distinct positive real numbers
including0, = 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-Watso
BGW Processes and their Basic PropertiesProblem: From Extinction to Reproduction
Towards a Counter-Example
Problem defined
Moving away from analyticity
http://find/ -
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Approximation theory to the rescue!
We will need the following key result - evolution of
Weierstrass Approximation Theorem through to Mntzs
Theorem - Full Mntzs Theorem (Schwartz, Siegel):
Theorem
Let{i}i=0 be a sequence of distinct positive real numbers
including0, = 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-Watso
BGW Processes and their Basic PropertiesProblem: From Extinction to Reproduction
Towards a Counter-Example
Problem defined
Moving away from analyticity
http://find/ -
7/31/2019 Istanbul Talk Tokarev
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Approximation theory to the rescue!
We will need the following key result - evolution of
Weierstrass Approximation Theorem through to Mntzs
Theorem - Full Mntzs Theorem (Schwartz, Siegel):
Theorem
Let{i}i=0 be a sequence of distinct positive real numbers
including0, = 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-Watso
BGW Processes and their Basic PropertiesProblem: From Extinction to Reproduction
Towards a Counter-Example
Problem defined
Moving away from analyticity
http://find/ -
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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 functions M 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 all
i = 1, 2, . . ., implies f(s) = g(s) for all s D. We have
Theorem
Given a sequence of distinct non-negative real numbers{i} 0, a completely monotone function is uniquelycharacterised by its values on{i} iff
i2i + 1
= . (1)
Daniel Tokarev From Extinction to Reproduction in Bienayme-Galton-Watso
BGW Processes and their Basic PropertiesProblem: From Extinction to Reproduction
Towards a Counter-Example
Problem defined
Moving away from analyticity
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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 functions M 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 all
i = 1, 2, . . ., implies f(s) = g(s) for all s D. We have
Theorem
Given a sequence of distinct non-negative real numbers{i} 0, a completely monotone function is uniquelycharacterised by its values on{i} iff
i2i + 1
= . (1)
Daniel Tokarev From Extinction to Reproduction in Bienayme-Galton-Watso
BGW Processes and their Basic PropertiesProblem: From Extinction to Reproduction
Towards a Counter-Example
Problem defined
Moving away from analyticity
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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 functions M 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 all
i = 1, 2, . . ., implies f(s) = g(s) for all s D. We have
Theorem
Given a sequence of distinct non-negative real numbers{i} 0, a completely monotone function is uniquelycharacterised by its values on{i} iff
i2i + 1
= . (1)
Daniel Tokarev From Extinction to Reproduction in Bienayme-Galton-Watso
BGW Processes and their Basic PropertiesProblem: From Extinction to Reproduction
Towards a Counter-Example
Problem defined
Moving away from analyticity
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7/31/2019 Istanbul Talk Tokarev
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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 functions M 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 all
i = 1, 2, . . ., implies f(s) = g(s) for all s D. We have
Theorem
Given a sequence of distinct non-negative real numbers{i} 0, a completely monotone function is uniquelycharacterised by its values on{i} iff
i2i + 1
= . (1)
Daniel Tokarev From Extinction to Reproduction in Bienayme-Galton-Watso
BGW Processes and their Basic PropertiesProblem: From Extinction to Reproduction
Towards a Counter-Example
Problem defined
Moving away from analyticity
G
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And back to PGFs!
Observe that if f is a PGF and his completely monotone,then f(g) is completely monotone, from this we easilyobtain
Theorem
Let{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-Watso
BGW Processes and their Basic PropertiesProblem: From Extinction to Reproduction
Towards a Counter-Example
Problem definedMoving away from analyticity
A d b k PGF !
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And back to PGFs!
Observe that if f is a PGF and his completely monotone,then f(g) is completely monotone, from this we easilyobtain
Theorem
Let{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-Watso
BGW Processes and their Basic PropertiesProblem: From Extinction to Reproduction
Towards a Counter-Example
Problem definedMoving away from analyticity
A d b k t PGF !
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And back to PGFs!
Observe that if f is a PGF and his completely monotone,then f(g) is completely monotone, from this we easilyobtain
Theorem
Let{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-Watso
BGW Processes and their Basic PropertiesProblem: From Extinction to Reproduction
Towards a Counter-Example
T d t l Bl hk P d t
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Towards a counter-example: Blaschke Products
Generalisation of Weierstrass products to functions
analytic on the open unit disk
Theorem
Given a set of points{an} on the unit disk, there exists afunction analytic on the unit disk with zeros at{an} and unique
up 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-Watso
BGW Processes and their Basic PropertiesProblem: From Extinction to Reproduction
Towards a Counter-Example
T d t l Bl hk P d t
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Towards a counter-example: Blaschke Products
Generalisation of Weierstrass products to functions
analytic on the open unit disk
Theorem
Given a set of points{an} on the unit disk, there exists afunction analytic on the unit disk with zeros at{an} and unique
up 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-Watso
BGW Processes and their Basic PropertiesProblem: From Extinction to Reproduction
Towards a Counter-Example
Towards a counter example: Blaschke Products
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Towards a counter-example: Blaschke Products
Generalisation of Weierstrass products to functions
analytic on the open unit disk
Theorem
Given a set of points{an} on the unit disk, there exists afunction analytic on the unit disk with zeros at{an} and unique
up 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-Watso
BGW Processes and their Basic PropertiesProblem: From Extinction to Reproduction
Towards a Counter-Example
The trouble with the negatives
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The trouble with the negatives
If f and gagree 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), and
We need to find E(s) that would make the coefficient of the
product E(s)B(s) decay faster than i2
The trouble is that we dont understand the pattern of signs
in bis - real Blaschke products are not well-understood
Daniel Tokarev From Extinction to Reproduction in Bienayme-Galton-Watso
BGW Processes and their Basic PropertiesProblem: From Extinction to Reproduction
Towards a Counter-Example
The trouble with the negatives
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7/31/2019 Istanbul Talk Tokarev
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The trouble with the negatives
If f and gagree 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), and
We need to find E(s) that would make the coefficient of the
product E(s)B(s) decay faster than i2
The trouble is that we dont understand the pattern of signs
in bis - real Blaschke products are not well-understood
Daniel Tokarev From Extinction to Reproduction in Bienayme-Galton-Watso
BGW Processes and their Basic PropertiesProblem: From Extinction to Reproduction
Towards a Counter-Example
The trouble with the negatives
http://find/http://goback/ -
7/31/2019 Istanbul Talk Tokarev
54/59
The trouble with the negatives
If f and gagree 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), and
We need to find E(s) that would make the coefficient of the
product E(s)B(s) decay faster than i2
The trouble is that we dont understand the pattern of signs
in bis - real Blaschke products are not well-understood
Daniel Tokarev From Extinction to Reproduction in Bienayme-Galton-Watso
BGW Processes and their Basic Properties
Problem: From Extinction to Reproduction
Towards a Counter-Example
The trouble with the negatives
http://find/http://goback/ -
7/31/2019 Istanbul Talk Tokarev
55/59
The trouble with the negatives
If f and gagree 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), and
We need to find E(s) that would make the coefficient of the
product E(s)B(s) decay faster than i2
The trouble is that we dont understand the pattern of signs
in bis - real Blaschke products are not well-understood
Daniel Tokarev From Extinction to Reproduction in Bienayme-Galton-Watso
BGW Processes and their Basic Properties
Problem: From Extinction to Reproduction
Towards a Counter-Example
The trouble with the negatives
http://find/http://goback/ -
7/31/2019 Istanbul Talk Tokarev
56/59
The trouble with the negatives
If f and gagree 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), and
We need to find E(s) that would make the coefficient of the
product E(s)B(s) decay faster than i2
The trouble is that we dont understand the pattern of signs
in bis - real Blaschke products are not well-understood
Daniel Tokarev From Extinction to Reproduction in Bienayme-Galton-Watso
BGW Processes and their Basic Properties
Problem: From Extinction to Reproduction
Towards a Counter-Example
The trouble with the negatives
http://find/http://goback/ -
7/31/2019 Istanbul Talk Tokarev
57/59
The trouble with the negatives
If f and gagree 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), and
We need to find E(s) that would make the coefficient of the
product E(s)B(s) decay faster than i2
The trouble is that we dont understand the pattern of signs
in bis - real Blaschke products are not well-understood
Daniel Tokarev From Extinction to Reproduction in Bienayme-Galton-Watso
BGW Processes and their Basic Properties
Problem: From Extinction to Reproduction
Towards a Counter-Example
Ath K B d N P E (1972) B hi P
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Athreya, K. B. and Ney, P.E. (1972) Branching Processes.Springer-Verlag.
Feller, W. (1971) An Introduction to Probability Theory andIts Applications, Volume II, 2nd Ed., John Wiley & Sons,Inc.
Feller, W. (1968) On Muntz Theorem and Completely
Monotone Functions. The American Mathematical Monthly,Vol. 75, No. 4 (Apr., 1968), pp. 342-350
R. Remmert, Classical topics in complex function theory,
Volume 172, GTM, Springer, 1998.
L.Schwartz, tude des Sommes DExponentielles,Hermann, Paris, 1959.
A.R. Siegel, On the Mntz-Scsz Theorem for C[0, 1],Proc. Amer. Math, Soc. 36 (1972), 161-166.
Daniel Tokarev From Extinction to Reproduction in Bienayme-Galton-Watso
BGW Processes and their Basic Properties
Problem: From Extinction to Reproduction
Towards a Counter-Example
I E Verbitskii Taylor coefficients and LP moduli of
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7/31/2019 Istanbul Talk Tokarev
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I. E. Verbitskii. Taylor coefficients and LP-moduli of
continuity of Blaschke products. Zapiski Nauchnykh
Seminarov Leningradskogo Otdeleniya MatematicheskogoInstituta im. V. A. Steklova AN SSSR, Vol. 107, pp. 27-35,
1982.
Daniel Tokarev From Extinction to Reproduction in Bienayme-Galton-Watso
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