Stability of point process, regular variation and ...wangyz/misc/2016Workshop/Hazra.pdf · Joint...
Transcript of Stability of point process, regular variation and ...wangyz/misc/2016Workshop/Hazra.pdf · Joint...
![Page 1: Stability of point process, regular variation and ...wangyz/misc/2016Workshop/Hazra.pdf · Joint work with Ayan Bhattacharya and Parthanil Roy Indian Statistical Institute, Kolkata](https://reader033.fdocuments.net/reader033/viewer/2022050214/5f602a16e7d6316b323e2a38/html5/thumbnails/1.jpg)
Stability of point process, regular variation andbranching random walk
Rajat Subhra HazraJoint work with Ayan Bhattacharya and Parthanil Roy
Indian Statistical Institute, Kolkata
6th May, 2016
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Table of contents
Extremes of Branching random walk
Dependent Heavy tailed Branching random walk
Stability in a nutshell
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What is Branching random walk?
I Branching random walk is a natural extension ofGalton-Watson process in a spatial sense.
I Start with one particle at origin;
I Its children who form the first generation are points of a pointprocess L on R.
I Each particle produces its own children who form secondgeneration and “positioned” (with respect to their parent)according to L.
I Each individual in the n-th generation produce independentlyof each other and everything else.
![Page 4: Stability of point process, regular variation and ...wangyz/misc/2016Workshop/Hazra.pdf · Joint work with Ayan Bhattacharya and Parthanil Roy Indian Statistical Institute, Kolkata](https://reader033.fdocuments.net/reader033/viewer/2022050214/5f602a16e7d6316b323e2a38/html5/thumbnails/4.jpg)
What is Branching random walk?
I Branching random walk is a natural extension ofGalton-Watson process in a spatial sense.
I Start with one particle at origin;
I Its children who form the first generation are points of a pointprocess L on R.
I Each particle produces its own children who form secondgeneration and “positioned” (with respect to their parent)according to L.
I Each individual in the n-th generation produce independentlyof each other and everything else.
![Page 5: Stability of point process, regular variation and ...wangyz/misc/2016Workshop/Hazra.pdf · Joint work with Ayan Bhattacharya and Parthanil Roy Indian Statistical Institute, Kolkata](https://reader033.fdocuments.net/reader033/viewer/2022050214/5f602a16e7d6316b323e2a38/html5/thumbnails/5.jpg)
What is Branching random walk?
I Branching random walk is a natural extension ofGalton-Watson process in a spatial sense.
I Start with one particle at origin;
I Its children who form the first generation are points of a pointprocess L on R.
I Each particle produces its own children who form secondgeneration and “positioned” (with respect to their parent)according to L.
I Each individual in the n-th generation produce independentlyof each other and everything else.
![Page 6: Stability of point process, regular variation and ...wangyz/misc/2016Workshop/Hazra.pdf · Joint work with Ayan Bhattacharya and Parthanil Roy Indian Statistical Institute, Kolkata](https://reader033.fdocuments.net/reader033/viewer/2022050214/5f602a16e7d6316b323e2a38/html5/thumbnails/6.jpg)
What is Branching random walk?
I Branching random walk is a natural extension ofGalton-Watson process in a spatial sense.
I Start with one particle at origin;
I Its children who form the first generation are points of a pointprocess L on R.
I Each particle produces its own children who form secondgeneration and “positioned” (with respect to their parent)according to L.
I Each individual in the n-th generation produce independentlyof each other and everything else.
![Page 7: Stability of point process, regular variation and ...wangyz/misc/2016Workshop/Hazra.pdf · Joint work with Ayan Bhattacharya and Parthanil Roy Indian Statistical Institute, Kolkata](https://reader033.fdocuments.net/reader033/viewer/2022050214/5f602a16e7d6316b323e2a38/html5/thumbnails/7.jpg)
What is Branching random walk?
I Branching random walk is a natural extension ofGalton-Watson process in a spatial sense.
I Start with one particle at origin;
I Its children who form the first generation are points of a pointprocess L on R.
I Each particle produces its own children who form secondgeneration and “positioned” (with respect to their parent)according to L.
I Each individual in the n-th generation produce independentlyof each other and everything else.
![Page 8: Stability of point process, regular variation and ...wangyz/misc/2016Workshop/Hazra.pdf · Joint work with Ayan Bhattacharya and Parthanil Roy Indian Statistical Institute, Kolkata](https://reader033.fdocuments.net/reader033/viewer/2022050214/5f602a16e7d6316b323e2a38/html5/thumbnails/8.jpg)
Growth process
0
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Growth process
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Growth process
displacements
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Growth process
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Growth process
positions Sv
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Questions?
I The underlying tree is a Galton-Watson tree.
I Various assumptions on displacements and positions can beassumed.
I Questions of interest: If Sv denotes the position of a particlev then the behaviour as n→∞ of
Nn =∑|v |=n
δa−1n (Sv−bn).
I Position of the top most particle in the n-th generation andscaling limits.
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How did it begin? and state of the art!
Branching Brownian motion (BBM):
I At time 0, particle at 0 ∈ R.
I Particle moves by a Brownian motion until for exponentialtime.
I After the step, particle splits into two. Repeat independently.
I N(t) ∼ e−t number of particles in time t and positions bedenoted by S1(t), · · · , SN(t)(t).
[Picture by Matt Roberts]
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Branching Brownian motion
I Started with connections of differential equations toprobability.
I Fisher-Kolmogorov-Petrovsky-Piscounov (F-KPP)equation:
∂tu(x , t) =1
2∂2xu(x , t) + u − u2 u(0, x) = 1x<0 .
I If u(t, x) = P(max1≤i≤N(t)Si (t) > x) then McKean (1975)showed that it satisfies F-KPP.
I Bramson (1978) showed
u(t, x + m(t))→ w(x) m(t) =√
2t − 3
2√
2log t.
![Page 16: Stability of point process, regular variation and ...wangyz/misc/2016Workshop/Hazra.pdf · Joint work with Ayan Bhattacharya and Parthanil Roy Indian Statistical Institute, Kolkata](https://reader033.fdocuments.net/reader033/viewer/2022050214/5f602a16e7d6316b323e2a38/html5/thumbnails/16.jpg)
Branching Brownian motion
I Started with connections of differential equations toprobability.
I Fisher-Kolmogorov-Petrovsky-Piscounov (F-KPP)equation:
∂tu(x , t) =1
2∂2xu(x , t) + u − u2 u(0, x) = 1x<0 .
I If u(t, x) = P(max1≤i≤N(t)Si (t) > x) then McKean (1975)showed that it satisfies F-KPP.
I Bramson (1978) showed
u(t, x + m(t))→ w(x) m(t) =√
2t − 3
2√
2log t.
![Page 17: Stability of point process, regular variation and ...wangyz/misc/2016Workshop/Hazra.pdf · Joint work with Ayan Bhattacharya and Parthanil Roy Indian Statistical Institute, Kolkata](https://reader033.fdocuments.net/reader033/viewer/2022050214/5f602a16e7d6316b323e2a38/html5/thumbnails/17.jpg)
Branching Brownian motion
I Started with connections of differential equations toprobability.
I Fisher-Kolmogorov-Petrovsky-Piscounov (F-KPP)equation:
∂tu(x , t) =1
2∂2xu(x , t) + u − u2 u(0, x) = 1x<0 .
I If u(t, x) = P(max1≤i≤N(t)Si (t) > x) then McKean (1975)showed that it satisfies F-KPP.
I Bramson (1978) showed
u(t, x + m(t))→ w(x) m(t) =√
2t − 3
2√
2log t.
![Page 18: Stability of point process, regular variation and ...wangyz/misc/2016Workshop/Hazra.pdf · Joint work with Ayan Bhattacharya and Parthanil Roy Indian Statistical Institute, Kolkata](https://reader033.fdocuments.net/reader033/viewer/2022050214/5f602a16e7d6316b323e2a38/html5/thumbnails/18.jpg)
Branching Brownian motion
I Started with connections of differential equations toprobability.
I Fisher-Kolmogorov-Petrovsky-Piscounov (F-KPP)equation:
∂tu(x , t) =1
2∂2xu(x , t) + u − u2 u(0, x) = 1x<0 .
I If u(t, x) = P(max1≤i≤N(t)Si (t) > x) then McKean (1975)showed that it satisfies F-KPP.
I Bramson (1978) showed
u(t, x + m(t))→ w(x) m(t) =√
2t − 3
2√
2log t.
![Page 19: Stability of point process, regular variation and ...wangyz/misc/2016Workshop/Hazra.pdf · Joint work with Ayan Bhattacharya and Parthanil Roy Indian Statistical Institute, Kolkata](https://reader033.fdocuments.net/reader033/viewer/2022050214/5f602a16e7d6316b323e2a38/html5/thumbnails/19.jpg)
Branching Brownian motion
I w(x) satisfies the following equation:
1
2∂2xw +
√2∂xw + w2 − w = 0
I Remarkable result of Lalley-Sellke (1987) showed
w(x) = E[e−cZe
−√
2x]
where Z is a limit of a ”derivative” martingale.
I Arguin-Bovier-Kistler (2013),Aidekon-Brunet-Berestycki-Shi (2013) showed the pointprocess
Lt =∑
1≤i≤N(t)
δSi (t)−m(t) → L, where L is superposable.
![Page 20: Stability of point process, regular variation and ...wangyz/misc/2016Workshop/Hazra.pdf · Joint work with Ayan Bhattacharya and Parthanil Roy Indian Statistical Institute, Kolkata](https://reader033.fdocuments.net/reader033/viewer/2022050214/5f602a16e7d6316b323e2a38/html5/thumbnails/20.jpg)
Branching Brownian motion
I w(x) satisfies the following equation:
1
2∂2xw +
√2∂xw + w2 − w = 0
I Remarkable result of Lalley-Sellke (1987) showed
w(x) = E[e−cZe
−√2x]
where Z is a limit of a ”derivative” martingale.
I Arguin-Bovier-Kistler (2013),Aidekon-Brunet-Berestycki-Shi (2013) showed the pointprocess
Lt =∑
1≤i≤N(t)
δSi (t)−m(t) → L, where L is superposable.
![Page 21: Stability of point process, regular variation and ...wangyz/misc/2016Workshop/Hazra.pdf · Joint work with Ayan Bhattacharya and Parthanil Roy Indian Statistical Institute, Kolkata](https://reader033.fdocuments.net/reader033/viewer/2022050214/5f602a16e7d6316b323e2a38/html5/thumbnails/21.jpg)
Branching Brownian motion
I w(x) satisfies the following equation:
1
2∂2xw +
√2∂xw + w2 − w = 0
I Remarkable result of Lalley-Sellke (1987) showed
w(x) = E[e−cZe
−√2x]
where Z is a limit of a ”derivative” martingale.
I Arguin-Bovier-Kistler (2013),Aidekon-Brunet-Berestycki-Shi (2013) showed the pointprocess
Lt =∑
1≤i≤N(t)
δSi (t)−m(t) → L, where L is superposable.
![Page 22: Stability of point process, regular variation and ...wangyz/misc/2016Workshop/Hazra.pdf · Joint work with Ayan Bhattacharya and Parthanil Roy Indian Statistical Institute, Kolkata](https://reader033.fdocuments.net/reader033/viewer/2022050214/5f602a16e7d6316b323e2a38/html5/thumbnails/22.jpg)
Branching Random Walk
I Strong law for topmost particle: Hammerseley, Kingman,Biggins (70’s)
I Addario-Berry, Reed (2009): Order of expected maxima.
I Bramson-Zeitouni (2009) : Tightness for recenteredmaxima.
I Aidekon (2013) ( weak law for minimum position, same asBBM). Relies on works of Biggins and Kyprianou (2004) onconvergence of derivative martingale in boundary case.
I Madaule (2015) : Point process convergence of the positionin n-th generation (seen from the tip).
I Non-boundary, heavy tails: Durrett (1979,1983),Bhattacharya, H., Roy (2015, 2016), Bhattacharya,Maulik, Palmowski, Roy (2016+).
![Page 23: Stability of point process, regular variation and ...wangyz/misc/2016Workshop/Hazra.pdf · Joint work with Ayan Bhattacharya and Parthanil Roy Indian Statistical Institute, Kolkata](https://reader033.fdocuments.net/reader033/viewer/2022050214/5f602a16e7d6316b323e2a38/html5/thumbnails/23.jpg)
Branching Random Walk
I Strong law for topmost particle: Hammerseley, Kingman,Biggins (70’s)
I Addario-Berry, Reed (2009): Order of expected maxima.
I Bramson-Zeitouni (2009) : Tightness for recenteredmaxima.
I Aidekon (2013) ( weak law for minimum position, same asBBM). Relies on works of Biggins and Kyprianou (2004) onconvergence of derivative martingale in boundary case.
I Madaule (2015) : Point process convergence of the positionin n-th generation (seen from the tip).
I Non-boundary, heavy tails: Durrett (1979,1983),Bhattacharya, H., Roy (2015, 2016), Bhattacharya,Maulik, Palmowski, Roy (2016+).
![Page 24: Stability of point process, regular variation and ...wangyz/misc/2016Workshop/Hazra.pdf · Joint work with Ayan Bhattacharya and Parthanil Roy Indian Statistical Institute, Kolkata](https://reader033.fdocuments.net/reader033/viewer/2022050214/5f602a16e7d6316b323e2a38/html5/thumbnails/24.jpg)
Branching Random Walk
I Strong law for topmost particle: Hammerseley, Kingman,Biggins (70’s)
I Addario-Berry, Reed (2009): Order of expected maxima.
I Bramson-Zeitouni (2009) : Tightness for recenteredmaxima.
I Aidekon (2013) ( weak law for minimum position, same asBBM). Relies on works of Biggins and Kyprianou (2004) onconvergence of derivative martingale in boundary case.
I Madaule (2015) : Point process convergence of the positionin n-th generation (seen from the tip).
I Non-boundary, heavy tails: Durrett (1979,1983),Bhattacharya, H., Roy (2015, 2016), Bhattacharya,Maulik, Palmowski, Roy (2016+).
![Page 25: Stability of point process, regular variation and ...wangyz/misc/2016Workshop/Hazra.pdf · Joint work with Ayan Bhattacharya and Parthanil Roy Indian Statistical Institute, Kolkata](https://reader033.fdocuments.net/reader033/viewer/2022050214/5f602a16e7d6316b323e2a38/html5/thumbnails/25.jpg)
Branching Random Walk
I Strong law for topmost particle: Hammerseley, Kingman,Biggins (70’s)
I Addario-Berry, Reed (2009): Order of expected maxima.
I Bramson-Zeitouni (2009) : Tightness for recenteredmaxima.
I Aidekon (2013) ( weak law for minimum position, same asBBM). Relies on works of Biggins and Kyprianou (2004) onconvergence of derivative martingale in boundary case.
I Madaule (2015) : Point process convergence of the positionin n-th generation (seen from the tip).
I Non-boundary, heavy tails: Durrett (1979,1983),Bhattacharya, H., Roy (2015, 2016), Bhattacharya,Maulik, Palmowski, Roy (2016+).
![Page 26: Stability of point process, regular variation and ...wangyz/misc/2016Workshop/Hazra.pdf · Joint work with Ayan Bhattacharya and Parthanil Roy Indian Statistical Institute, Kolkata](https://reader033.fdocuments.net/reader033/viewer/2022050214/5f602a16e7d6316b323e2a38/html5/thumbnails/26.jpg)
Branching Random Walk
I Strong law for topmost particle: Hammerseley, Kingman,Biggins (70’s)
I Addario-Berry, Reed (2009): Order of expected maxima.
I Bramson-Zeitouni (2009) : Tightness for recenteredmaxima.
I Aidekon (2013) ( weak law for minimum position, same asBBM). Relies on works of Biggins and Kyprianou (2004) onconvergence of derivative martingale in boundary case.
I Madaule (2015) : Point process convergence of the positionin n-th generation (seen from the tip).
I Non-boundary, heavy tails: Durrett (1979,1983),Bhattacharya, H., Roy (2015, 2016), Bhattacharya,Maulik, Palmowski, Roy (2016+).
![Page 27: Stability of point process, regular variation and ...wangyz/misc/2016Workshop/Hazra.pdf · Joint work with Ayan Bhattacharya and Parthanil Roy Indian Statistical Institute, Kolkata](https://reader033.fdocuments.net/reader033/viewer/2022050214/5f602a16e7d6316b323e2a38/html5/thumbnails/27.jpg)
Branching Random Walk
I Strong law for topmost particle: Hammerseley, Kingman,Biggins (70’s)
I Addario-Berry, Reed (2009): Order of expected maxima.
I Bramson-Zeitouni (2009) : Tightness for recenteredmaxima.
I Aidekon (2013) ( weak law for minimum position, same asBBM). Relies on works of Biggins and Kyprianou (2004) onconvergence of derivative martingale in boundary case.
I Madaule (2015) : Point process convergence of the positionin n-th generation (seen from the tip).
I Non-boundary, heavy tails: Durrett (1979,1983),Bhattacharya, H., Roy (2015, 2016), Bhattacharya,Maulik, Palmowski, Roy (2016+).
![Page 28: Stability of point process, regular variation and ...wangyz/misc/2016Workshop/Hazra.pdf · Joint work with Ayan Bhattacharya and Parthanil Roy Indian Statistical Institute, Kolkata](https://reader033.fdocuments.net/reader033/viewer/2022050214/5f602a16e7d6316b323e2a38/html5/thumbnails/28.jpg)
Assumptions on Branching Mechanism
I Underlying tree is a Galton-Watson tree.
I Zn denotes the number of particles at n-th generation andµ := E(Z1) ∈ (1,∞).
I We shall assume that P(Z1 = 0) = 0 (no leaves).
I Using martingale convergence theorem,
Zn
µn→W (≥ 0) almost surely.
I Kesten-Stigum condition :
E(Z1 logZ1) <∞⇔ P(W > 0) = 1.
![Page 29: Stability of point process, regular variation and ...wangyz/misc/2016Workshop/Hazra.pdf · Joint work with Ayan Bhattacharya and Parthanil Roy Indian Statistical Institute, Kolkata](https://reader033.fdocuments.net/reader033/viewer/2022050214/5f602a16e7d6316b323e2a38/html5/thumbnails/29.jpg)
Assumptions on Displacement Random Variables
I Each particle produces an independent copy of
L =
Z1∑i=1
δXi
where Z1 ⊥ (X1,X2, . . .) is a K := [0,∞)∞-valued randomvariables such that
I each Xi ∼ F ∈ RV−α(α > 0);
I (X1,X2, . . .) ∈ RV−α(K \ 0∞, λ)
![Page 30: Stability of point process, regular variation and ...wangyz/misc/2016Workshop/Hazra.pdf · Joint work with Ayan Bhattacharya and Parthanil Roy Indian Statistical Institute, Kolkata](https://reader033.fdocuments.net/reader033/viewer/2022050214/5f602a16e7d6316b323e2a38/html5/thumbnails/30.jpg)
Assumptions on Displacement Random Variables
I Each particle produces an independent copy of
L =
Z1∑i=1
δXi
where Z1 ⊥ (X1,X2, . . .) is a K := [0,∞)∞-valued randomvariables such that
I each Xi ∼ F ∈ RV−α(α > 0);
I (X1,X2, . . .) ∈ RV−α(K \ 0∞, λ)
![Page 31: Stability of point process, regular variation and ...wangyz/misc/2016Workshop/Hazra.pdf · Joint work with Ayan Bhattacharya and Parthanil Roy Indian Statistical Institute, Kolkata](https://reader033.fdocuments.net/reader033/viewer/2022050214/5f602a16e7d6316b323e2a38/html5/thumbnails/31.jpg)
Assumptions on Displacement Random Variables
I Each particle produces an independent copy of
L =
Z1∑i=1
δXi
where Z1 ⊥ (X1,X2, . . .) is a K := [0,∞)∞-valued randomvariables such that
I each Xi ∼ F ∈ RV−α(α > 0);
I (X1,X2, . . .) ∈ RV−α(K \ 0∞, λ)
![Page 32: Stability of point process, regular variation and ...wangyz/misc/2016Workshop/Hazra.pdf · Joint work with Ayan Bhattacharya and Parthanil Roy Indian Statistical Institute, Kolkata](https://reader033.fdocuments.net/reader033/viewer/2022050214/5f602a16e7d6316b323e2a38/html5/thumbnails/32.jpg)
Assumptions on Displacement Random Variables
I Each particle produces an independent copy of
L =
Z1∑i=1
δXi
where Z1 ⊥ (X1,X2, . . .) is a K := [0,∞)∞-valued randomvariables such that
I each Xi ∼ F ∈ RV−α(α > 0);
I (X1,X2, . . .) ∈ RV−α(K \ 0∞, λ)
![Page 33: Stability of point process, regular variation and ...wangyz/misc/2016Workshop/Hazra.pdf · Joint work with Ayan Bhattacharya and Parthanil Roy Indian Statistical Institute, Kolkata](https://reader033.fdocuments.net/reader033/viewer/2022050214/5f602a16e7d6316b323e2a38/html5/thumbnails/33.jpg)
Assumptions on Displacement Random Variables
I We say (X1,X2, . . .) ∈ RV−α(K \ 0∞, λ)
if there exists asequence {cn} such that
µnP(c−1n (X1,X2, . . .) ∈ A)→ λ(A).
for all A ⊂ K0 = K \ {0∞} such that 0∞ /∈ A and λ(∂A) = 0and λ(·) is a measure on K0 such that for all ε > 0,λ(K \ B(0∞, ε)) <∞.
I This convergence is introduced by Hult and Lindskog(2006)and have been extended by Das, Mitra, Resnick (2013),Lindskog, Resnick and Roy(2014).
![Page 34: Stability of point process, regular variation and ...wangyz/misc/2016Workshop/Hazra.pdf · Joint work with Ayan Bhattacharya and Parthanil Roy Indian Statistical Institute, Kolkata](https://reader033.fdocuments.net/reader033/viewer/2022050214/5f602a16e7d6316b323e2a38/html5/thumbnails/34.jpg)
Assumptions on Displacement Random Variables
I We say (X1,X2, . . .) ∈ RV−α(K \ 0∞, λ) if there exists asequence {cn} such that
µnP(c−1n (X1,X2, . . .) ∈ A)→ λ(A).
for all A ⊂ K0 = K \ {0∞} such that 0∞ /∈ A and λ(∂A) = 0and λ(·) is a measure on K0 such that for all ε > 0,λ(K \ B(0∞, ε)) <∞.
I This convergence is introduced by Hult and Lindskog(2006)and have been extended by Das, Mitra, Resnick (2013),Lindskog, Resnick and Roy(2014).
![Page 35: Stability of point process, regular variation and ...wangyz/misc/2016Workshop/Hazra.pdf · Joint work with Ayan Bhattacharya and Parthanil Roy Indian Statistical Institute, Kolkata](https://reader033.fdocuments.net/reader033/viewer/2022050214/5f602a16e7d6316b323e2a38/html5/thumbnails/35.jpg)
Assumptions on Displacement Random Variables
I We say (X1,X2, . . .) ∈ RV−α(K \ 0∞, λ) if there exists asequence {cn} such that
µnP(c−1n (X1,X2, . . .) ∈ A)→ λ(A).
for all A ⊂ K0 = K \ {0∞} such that 0∞ /∈ A and λ(∂A) = 0
and λ(·) is a measure on K0 such that for all ε > 0,λ(K \ B(0∞, ε)) <∞.
I This convergence is introduced by Hult and Lindskog(2006)and have been extended by Das, Mitra, Resnick (2013),Lindskog, Resnick and Roy(2014).
![Page 36: Stability of point process, regular variation and ...wangyz/misc/2016Workshop/Hazra.pdf · Joint work with Ayan Bhattacharya and Parthanil Roy Indian Statistical Institute, Kolkata](https://reader033.fdocuments.net/reader033/viewer/2022050214/5f602a16e7d6316b323e2a38/html5/thumbnails/36.jpg)
Assumptions on Displacement Random Variables
I We say (X1,X2, . . .) ∈ RV−α(K \ 0∞, λ) if there exists asequence {cn} such that
µnP(c−1n (X1,X2, . . .) ∈ A)→ λ(A).
for all A ⊂ K0 = K \ {0∞} such that 0∞ /∈ A and λ(∂A) = 0and λ(·) is a measure on K0 such that for all ε > 0,λ(K \ B(0∞, ε)) <∞.
I This convergence is introduced by Hult and Lindskog(2006)and have been extended by Das, Mitra, Resnick (2013),Lindskog, Resnick and Roy(2014).
![Page 37: Stability of point process, regular variation and ...wangyz/misc/2016Workshop/Hazra.pdf · Joint work with Ayan Bhattacharya and Parthanil Roy Indian Statistical Institute, Kolkata](https://reader033.fdocuments.net/reader033/viewer/2022050214/5f602a16e7d6316b323e2a38/html5/thumbnails/37.jpg)
First main result
Let us denote the random point process of the positions of theparticles by
Nn =∑|v |=n
δc−1n Sv
where cn ≈ µn/α.
Theorem (Bhattacharya, H. and Roy (2016))
Under our assumptions, the random point configuration convergesin distribution to the Cox cluster process N∗ where
N∗d=∞∑l=1
Ul∑k=1
T(k)l δ
W 1/αj(k)l
![Page 38: Stability of point process, regular variation and ...wangyz/misc/2016Workshop/Hazra.pdf · Joint work with Ayan Bhattacharya and Parthanil Roy Indian Statistical Institute, Kolkata](https://reader033.fdocuments.net/reader033/viewer/2022050214/5f602a16e7d6316b323e2a38/html5/thumbnails/38.jpg)
Description of N∗: One Large Bunch Phenomenon
U = 3
T (1) = 2 T (2) = 3 T (3) = 5
N∗d=∞∑l=1
Ul∑k=1
T(k)l δ
j(k)l W
1α, where
∞∑l=1
δ(j(1)l , j
(2)l , ...
) ∼ PRM(K0, λ)
![Page 39: Stability of point process, regular variation and ...wangyz/misc/2016Workshop/Hazra.pdf · Joint work with Ayan Bhattacharya and Parthanil Roy Indian Statistical Institute, Kolkata](https://reader033.fdocuments.net/reader033/viewer/2022050214/5f602a16e7d6316b323e2a38/html5/thumbnails/39.jpg)
Maxima
I Let Mn denotes the maximal position of the nth generationparticles.
Theorem (Bhattacharya, H. and Roy (2016))
Under the assumptions, for every x > 0,
limn→∞
P(Mn > cnx) = E[e−Wκλx
−α]
where κλ > 0 is a constant.
I This is an extension of main result of Durrett(1983).
I Extensions of point process result to multi-type in forthcoming article by Bhattacharya, Maulik, Palmowski, Roy(2016+)
![Page 40: Stability of point process, regular variation and ...wangyz/misc/2016Workshop/Hazra.pdf · Joint work with Ayan Bhattacharya and Parthanil Roy Indian Statistical Institute, Kolkata](https://reader033.fdocuments.net/reader033/viewer/2022050214/5f602a16e7d6316b323e2a38/html5/thumbnails/40.jpg)
Maxima
I Let Mn denotes the maximal position of the nth generationparticles.
Theorem (Bhattacharya, H. and Roy (2016))
Under the assumptions, for every x > 0,
limn→∞
P(Mn > cnx) = E[e−Wκλx
−α]
where κλ > 0 is a constant.
I This is an extension of main result of Durrett(1983).
I Extensions of point process result to multi-type in forthcoming article by Bhattacharya, Maulik, Palmowski, Roy(2016+)
![Page 41: Stability of point process, regular variation and ...wangyz/misc/2016Workshop/Hazra.pdf · Joint work with Ayan Bhattacharya and Parthanil Roy Indian Statistical Institute, Kolkata](https://reader033.fdocuments.net/reader033/viewer/2022050214/5f602a16e7d6316b323e2a38/html5/thumbnails/41.jpg)
Maxima
I Let Mn denotes the maximal position of the nth generationparticles.
Theorem (Bhattacharya, H. and Roy (2016))
Under the assumptions, for every x > 0,
limn→∞
P(Mn > cnx) = E[e−Wκλx
−α]
where κλ > 0 is a constant.
I This is an extension of main result of Durrett(1983).
I Extensions of point process result to multi-type in forthcoming article by Bhattacharya, Maulik, Palmowski, Roy(2016+)
![Page 42: Stability of point process, regular variation and ...wangyz/misc/2016Workshop/Hazra.pdf · Joint work with Ayan Bhattacharya and Parthanil Roy Indian Statistical Institute, Kolkata](https://reader033.fdocuments.net/reader033/viewer/2022050214/5f602a16e7d6316b323e2a38/html5/thumbnails/42.jpg)
Maxima
I Let Mn denotes the maximal position of the nth generationparticles.
Theorem (Bhattacharya, H. and Roy (2016))
Under the assumptions, for every x > 0,
limn→∞
P(Mn > cnx) = E[e−Wκλx
−α]
where κλ > 0 is a constant.
I This is an extension of main result of Durrett(1983).
I Extensions of point process result to multi-type in forthcoming article by Bhattacharya, Maulik, Palmowski, Roy(2016+)
![Page 43: Stability of point process, regular variation and ...wangyz/misc/2016Workshop/Hazra.pdf · Joint work with Ayan Bhattacharya and Parthanil Roy Indian Statistical Institute, Kolkata](https://reader033.fdocuments.net/reader033/viewer/2022050214/5f602a16e7d6316b323e2a38/html5/thumbnails/43.jpg)
Stable Point process
I (Scalar Multiplication) For every a ∈ (0,∞), define
a ◦ P =∞∑i=1
δaui .
I (Superposition) The superposition of two point measures L1and L2 will be denoted by L1 + L2.
Definition (Davydov, Molchanov and Zuyev (2008,2011))
A point process N is called a strictly α-stable (α > 0) pointprocess if for all a1, a2 ∈ (0,∞)
a1 ◦ N1 + a2 ◦ N2d= (aα1 + aα2 )1/α ◦ N
where N1 and N2 are two independent copies of N.
![Page 44: Stability of point process, regular variation and ...wangyz/misc/2016Workshop/Hazra.pdf · Joint work with Ayan Bhattacharya and Parthanil Roy Indian Statistical Institute, Kolkata](https://reader033.fdocuments.net/reader033/viewer/2022050214/5f602a16e7d6316b323e2a38/html5/thumbnails/44.jpg)
Stable Point process
I (Scalar Multiplication) For every a ∈ (0,∞), define
a ◦ P =∞∑i=1
δaui .
I (Superposition) The superposition of two point measures L1and L2 will be denoted by L1 + L2.
Definition (Davydov, Molchanov and Zuyev (2008,2011))
A point process N is called a strictly α-stable (α > 0) pointprocess if for all a1, a2 ∈ (0,∞)
a1 ◦ N1 + a2 ◦ N2d= (aα1 + aα2 )1/α ◦ N
where N1 and N2 are two independent copies of N.
![Page 45: Stability of point process, regular variation and ...wangyz/misc/2016Workshop/Hazra.pdf · Joint work with Ayan Bhattacharya and Parthanil Roy Indian Statistical Institute, Kolkata](https://reader033.fdocuments.net/reader033/viewer/2022050214/5f602a16e7d6316b323e2a38/html5/thumbnails/45.jpg)
Stable Point process
I (Scalar Multiplication) For every a ∈ (0,∞), define
a ◦ P =∞∑i=1
δaui .
I (Superposition) The superposition of two point measures L1and L2 will be denoted by L1 + L2.
Definition (Davydov, Molchanov and Zuyev (2008,2011))
A point process N is called a strictly α-stable (α > 0) pointprocess if for all a1, a2 ∈ (0,∞)
a1 ◦ N1 + a2 ◦ N2d= (aα1 + aα2 )1/α ◦ N
where N1 and N2 are two independent copies of N.
![Page 46: Stability of point process, regular variation and ...wangyz/misc/2016Workshop/Hazra.pdf · Joint work with Ayan Bhattacharya and Parthanil Roy Indian Statistical Institute, Kolkata](https://reader033.fdocuments.net/reader033/viewer/2022050214/5f602a16e7d6316b323e2a38/html5/thumbnails/46.jpg)
Stable Point process
I (Scalar Multiplication) For every a ∈ (0,∞), define
a ◦ P =∞∑i=1
δaui .
I (Superposition) The superposition of two point measures L1and L2 will be denoted by L1 + L2.
Definition (Davydov, Molchanov and Zuyev (2008,2011))
A point process N is called a strictly α-stable (α > 0) pointprocess if for all a1, a2 ∈ (0,∞)
a1 ◦ N1 + a2 ◦ N2d= (aα1 + aα2 )1/α ◦ N
where N1 and N2 are two independent copies of N.
![Page 47: Stability of point process, regular variation and ...wangyz/misc/2016Workshop/Hazra.pdf · Joint work with Ayan Bhattacharya and Parthanil Roy Indian Statistical Institute, Kolkata](https://reader033.fdocuments.net/reader033/viewer/2022050214/5f602a16e7d6316b323e2a38/html5/thumbnails/47.jpg)
Stable Point process
I (Scalar Multiplication) For every a ∈ (0,∞), define
a ◦ P =∞∑i=1
δaui .
I (Superposition) The superposition of two point measures L1and L2 will be denoted by L1 + L2.
Definition (Davydov, Molchanov and Zuyev (2008,2011))
A point process N is called a strictly α-stable (α > 0) pointprocess if for all a1, a2 ∈ (0,∞)
a1 ◦ N1 + a2 ◦ N2d= (aα1 + aα2 )1/α ◦ N
where N1 and N2 are two independent copies of N.
![Page 48: Stability of point process, regular variation and ...wangyz/misc/2016Workshop/Hazra.pdf · Joint work with Ayan Bhattacharya and Parthanil Roy Indian Statistical Institute, Kolkata](https://reader033.fdocuments.net/reader033/viewer/2022050214/5f602a16e7d6316b323e2a38/html5/thumbnails/48.jpg)
Representation of Stable Point Processes
Theorem (Davydov, Molchanov, Zuyev (2008,2011))
N be a strictly α-stable (α > 0) point process if and only if
Nd=∞∑i=1
λi ◦ Pi
where
I {λi : i ≥ 1} are such that Λ =∑∞
i=1 δλi ∼ PRM(να) whereνα((x ,∞]) = x−α for all x > 0;
I Pi s are independent copies of the point process P and alsoindependent of Λ.
![Page 49: Stability of point process, regular variation and ...wangyz/misc/2016Workshop/Hazra.pdf · Joint work with Ayan Bhattacharya and Parthanil Roy Indian Statistical Institute, Kolkata](https://reader033.fdocuments.net/reader033/viewer/2022050214/5f602a16e7d6316b323e2a38/html5/thumbnails/49.jpg)
Representation of Stable Point Processes
Theorem (Davydov, Molchanov, Zuyev (2008,2011))
N be a strictly α-stable (α > 0) point process if and only if
Nd=∞∑i=1
λi ◦ Pi
where
I {λi : i ≥ 1} are such that Λ =∑∞
i=1 δλi ∼ PRM(να) where
να((x ,∞]) = x−α for all x > 0;
I Pi s are independent copies of the point process P and alsoindependent of Λ.
![Page 50: Stability of point process, regular variation and ...wangyz/misc/2016Workshop/Hazra.pdf · Joint work with Ayan Bhattacharya and Parthanil Roy Indian Statistical Institute, Kolkata](https://reader033.fdocuments.net/reader033/viewer/2022050214/5f602a16e7d6316b323e2a38/html5/thumbnails/50.jpg)
Representation of Stable Point Processes
Theorem (Davydov, Molchanov, Zuyev (2008,2011))
N be a strictly α-stable (α > 0) point process if and only if
Nd=∞∑i=1
λi ◦ Pi
where
I {λi : i ≥ 1} are such that Λ =∑∞
i=1 δλi ∼ PRM(να) whereνα((x ,∞]) = x−α for all x > 0;
I Pi s are independent copies of the point process P and alsoindependent of Λ.
![Page 51: Stability of point process, regular variation and ...wangyz/misc/2016Workshop/Hazra.pdf · Joint work with Ayan Bhattacharya and Parthanil Roy Indian Statistical Institute, Kolkata](https://reader033.fdocuments.net/reader033/viewer/2022050214/5f602a16e7d6316b323e2a38/html5/thumbnails/51.jpg)
Brunet-Derrida in Our Setup
BD1 The analogue of the superposability is
N∗d= W 1/α ◦ Q
where W is the martingale limit and Q is a strictly α-stablepoint process. (Randomly scaled strictly α-stable pointprocess).
BD2 The analogous representation is randomly scaledscale-decorated Poisson point process (Randomly scaled DMZrepresentation).
I We have shown that BD1 and BD2 are equivalent(heavy-tailed extension of Subag and Zeitouni (2014)).
![Page 52: Stability of point process, regular variation and ...wangyz/misc/2016Workshop/Hazra.pdf · Joint work with Ayan Bhattacharya and Parthanil Roy Indian Statistical Institute, Kolkata](https://reader033.fdocuments.net/reader033/viewer/2022050214/5f602a16e7d6316b323e2a38/html5/thumbnails/52.jpg)
Brunet-Derrida in Our Setup
BD1 The analogue of the superposability is
N∗d= W 1/α ◦ Q
where W is the martingale limit and Q is a strictly α-stablepoint process. (Randomly scaled strictly α-stable pointprocess).
BD2 The analogous representation is randomly scaledscale-decorated Poisson point process (Randomly scaled DMZrepresentation).
I We have shown that BD1 and BD2 are equivalent(heavy-tailed extension of Subag and Zeitouni (2014)).
![Page 53: Stability of point process, regular variation and ...wangyz/misc/2016Workshop/Hazra.pdf · Joint work with Ayan Bhattacharya and Parthanil Roy Indian Statistical Institute, Kolkata](https://reader033.fdocuments.net/reader033/viewer/2022050214/5f602a16e7d6316b323e2a38/html5/thumbnails/53.jpg)
Brunet-Derrida in Our Setup
BD1 The analogue of the superposability is
N∗d= W 1/α ◦ Q
where W is the martingale limit and Q is a strictly α-stablepoint process. (Randomly scaled strictly α-stable pointprocess).
BD2 The analogous representation is randomly scaledscale-decorated Poisson point process (Randomly scaled DMZrepresentation).
I We have shown that BD1 and BD2 are equivalent(heavy-tailed extension of Subag and Zeitouni (2014)).
![Page 54: Stability of point process, regular variation and ...wangyz/misc/2016Workshop/Hazra.pdf · Joint work with Ayan Bhattacharya and Parthanil Roy Indian Statistical Institute, Kolkata](https://reader033.fdocuments.net/reader033/viewer/2022050214/5f602a16e7d6316b323e2a38/html5/thumbnails/54.jpg)
Brunet-Derrida in Our Setup
BD1 The analogue of the superposability is
N∗d= W 1/α ◦ Q
where W is the martingale limit and Q is a strictly α-stablepoint process. (Randomly scaled strictly α-stable pointprocess).
BD2 The analogous representation is randomly scaledscale-decorated Poisson point process (Randomly scaled DMZrepresentation).
I We have shown that BD1 and BD2 are equivalent
(heavy-tailed extension of Subag and Zeitouni (2014)).
![Page 55: Stability of point process, regular variation and ...wangyz/misc/2016Workshop/Hazra.pdf · Joint work with Ayan Bhattacharya and Parthanil Roy Indian Statistical Institute, Kolkata](https://reader033.fdocuments.net/reader033/viewer/2022050214/5f602a16e7d6316b323e2a38/html5/thumbnails/55.jpg)
Brunet-Derrida in Our Setup
BD1 The analogue of the superposability is
N∗d= W 1/α ◦ Q
where W is the martingale limit and Q is a strictly α-stablepoint process. (Randomly scaled strictly α-stable pointprocess).
BD2 The analogous representation is randomly scaledscale-decorated Poisson point process (Randomly scaled DMZrepresentation).
I We have shown that BD1 and BD2 are equivalent(heavy-tailed extension of Subag and Zeitouni (2014)).
![Page 56: Stability of point process, regular variation and ...wangyz/misc/2016Workshop/Hazra.pdf · Joint work with Ayan Bhattacharya and Parthanil Roy Indian Statistical Institute, Kolkata](https://reader033.fdocuments.net/reader033/viewer/2022050214/5f602a16e7d6316b323e2a38/html5/thumbnails/56.jpg)
Domain of Attraction Theorem
I Recall that M set of point measures, is a complete, separablemetric space equipped with vague metric.
I One can define regular variation for measures on M usingworks of Hult and Lindskog(2006).
Theorem (Bhattacharya, H., Roy (2015))
Let L be a point process on S . Suppose L is RV−α, that is,
nP(b−1n ◦ L ∈ ·)HL→ µα(·).
Then
b−1n ◦n∑
i=1
Li ⇒ Strictly α-stable Point Process
![Page 57: Stability of point process, regular variation and ...wangyz/misc/2016Workshop/Hazra.pdf · Joint work with Ayan Bhattacharya and Parthanil Roy Indian Statistical Institute, Kolkata](https://reader033.fdocuments.net/reader033/viewer/2022050214/5f602a16e7d6316b323e2a38/html5/thumbnails/57.jpg)
Domain of Attraction Theorem
I Recall that M set of point measures, is a complete, separablemetric space equipped with vague metric.
I One can define regular variation for measures on M usingworks of Hult and Lindskog(2006).
Theorem (Bhattacharya, H., Roy (2015))
Let L be a point process on S . Suppose L is RV−α, that is,
nP(b−1n ◦ L ∈ ·)HL→ µα(·).
Then
b−1n ◦n∑
i=1
Li ⇒ Strictly α-stable Point Process
![Page 58: Stability of point process, regular variation and ...wangyz/misc/2016Workshop/Hazra.pdf · Joint work with Ayan Bhattacharya and Parthanil Roy Indian Statistical Institute, Kolkata](https://reader033.fdocuments.net/reader033/viewer/2022050214/5f602a16e7d6316b323e2a38/html5/thumbnails/58.jpg)
Domain of Attraction Theorem
I Recall that M set of point measures, is a complete, separablemetric space equipped with vague metric.
I One can define regular variation for measures on M usingworks of Hult and Lindskog(2006).
Theorem (Bhattacharya, H., Roy (2015))
Let L be a point process on S . Suppose L is RV−α, that is,
nP(b−1n ◦ L ∈ ·)HL→ µα(·).
Then
b−1n ◦n∑
i=1
Li ⇒ Strictly α-stable Point Process
![Page 59: Stability of point process, regular variation and ...wangyz/misc/2016Workshop/Hazra.pdf · Joint work with Ayan Bhattacharya and Parthanil Roy Indian Statistical Institute, Kolkata](https://reader033.fdocuments.net/reader033/viewer/2022050214/5f602a16e7d6316b323e2a38/html5/thumbnails/59.jpg)
Thank You
I Point process convergence for branching random walkswith regularly varying steps: arXiv:1411.5646, to appear inAnnales de l’Institut Henri Poincare.
I Branching random walks, stable point processes andregular variation: arXiv:1601.01656, submitted.