Istanbul Talk Tokarev

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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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Classification and Probability of Extinction

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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.

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BGW Processes






f(s) =

i=0

pisi.


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BGW Processes






f(s) =

i=0

pisi.



BGW P d h i B i P i


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BGW Processes






f(s) =

i=0

pisi.



BGW P d th i B i P ti


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BGW Processes






f(s) =

i=0

pisi.





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BGW Processes






f(s) =

i=0

pisi.



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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


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p











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p









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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


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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


BGW Processes and their Basic PropertiesProblem defined
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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:



P bl F E ti ti t R d tiProblem defined
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Problem defined




E.g. {fn(0)} {gn(0)} := supn1

{|fn(0) gn(0)|} .







iterates:



Problem From Extinction to ReproductionProblem defined
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Towards a Counter-ExampleMoving away from analyticity



E.g. {fn(0)} {gn(0)} := supn1

{|fn(0) gn(0)|} .







iterates:



Problem: From Extinction to ReproductionProblem defined


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E.g. {fn(0)} {gn(0)} := supn1

{|fn(0) gn(0)|} .







iterates:



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E.g. {fn(0)} {gn(0)} := supn1

{|fn(0) gn(0)|} .







iterates:



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E.g. {fn(0)} {gn(0)} := supn1

{|fn(0) gn(0)|} .







iterates:


BGW Processes and their Basic PropertiesProblem: From Extinction to Reproduction

Problem defined

M i f l i i
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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.



Problem defined

M i f l ti it
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p



Let f(s) = ex

1


ji=1(s ti).


Theorem


k

i=1pj

i


i0 qis




Problem defined

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p



Let f(s) = ex

1


ji=1(s ti).


Theorem


k

i=1pj

i


i0 qis




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Let f(s) = ex

1


ji=1(s ti).


Theorem


k

i=1pj

i


i0 qis




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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.



Problem defined

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Supercritical case




More generally

Theorem





T d C t E l

Problem defined

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Supercritical case




More generally

Theorem





T d C t E l

Problem defined

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Towards a Counter-Exampleg y y y

Supercritical case




More generally

Theorem





Towards a Counter Example

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Towards a Counter-Exampleg y y y

Supercritical case




More generally

Theorem






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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.




Problem defined

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When moments exist






let pi = c2

k, c= 1/

2


moments cpk

2


isi


(1 + a)k2k =

for all a> 0.




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Towards a Counter Example

When moments exist






let pi = c2

k, c= 1/

2


moments cpk

2


isi


(1 + a)k2k =

for all a> 0.




Problem defined



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o a ds a Cou te a p e

When moments exist






let pi = c2

k, c= 1/

2


moments cpk

2


isi


(1 + a)k2k =

for all a> 0.




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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




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Theorem




qi+jqi




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Theorem




qi+jqi




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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= .




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Theorem



i=0



= C[0, 1] iff i

2i + 1= .




Problem defined

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Theorem



i=0



= C[0, 1] iff i

2i + 1= .




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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)




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Theorem


i2i + 1

= . (1)




Problem defined



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Theorem


i2i + 1

= . (1)




Problem defined



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Theorem


i2i + 1

= . (1)




Problem defined


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 = .




Problem definedMoving away from analyticity

A d b k PGF !


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And back to PGFs!


Theorem


i=j





Problem definedMoving away from analyticity

A d b k t PGF !
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And back to PGFs!


Theorem


i=j





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

.




T d t l Bl hk P d t


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Theorem



i=1

(1 |ai|)


B(z) =

i=1

ai|an|

an z1 anz

.




Towards a counter example: Blaschke Products


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Theorem



i=1

(1 |ai|)


B(z) =

i=1

ai|an|

an z1 anz

.




The trouble with the negatives


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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






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bisi with











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bisi with












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bisi with












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bisi with












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bisi with










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.





I E Verbitskii Taylor coefficients and LP moduli of


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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.


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