d}z_{m})$ $\displaystyle=$
$\displaystyle\vartheta_{m}^{2}+(A_{m}^{m}-B_{m})(\mathcal{F}^{(m+1)}g)(\vartheta_{m}),$
(13.28)
$\displaystyle\int_{[0,1]\times[0,1]^{\mathbb{N}_{0}}}(\mathcal{F}^{(m)}g)(x_{m})\;{\rm
d}\Gamma_{(\vartheta_{m},y_{m})}^{\mathcal{F}^{(m)}g,E_{m},c_{m},K_{m},e_{m}}({\rm
d}z_{m})$ $\displaystyle=$
$\displaystyle(\mathcal{F}^{(m+1)}g)(\vartheta_{m}).$

###### Proof.

The proof follows the same line of argument as the proof of Proposition 13.1.
∎

####  13.2 Iterate moment relations

Recall the kernels defined in (4.20), the iterates of the kernels defined in
(5.1) and $\bar{\vartheta}^{(n)}$. Recall that
$Q^{(n)}(\bar{\vartheta}^{(n)},{\rm d}z_{0})$ is the probability density to
see the population of a single colony in state $z_{0}$ given that the
$(n+1)$-block averages equal $\bar{\vartheta}^{(n)}$.

######  Proposition 13.3.

[Iterated moment relations: single components] For $n\in\mathbb{N}_{0}$,

(13.29)
$\displaystyle\int_{[0,1]\times[0,1]^{\mathbb{N}_{0}}}x_{0}\;Q^{(n)}(\bar{\vartheta}^{(n)},{\rm
d}z_{0})$ $\displaystyle=$ $\displaystyle\vartheta_{n},$ (13.30)
$\displaystyle\int_{[0,1]\times[0,1]^{\mathbb{N}_{0}}}y_{0,0}\;Q^{(n)}(\bar{\vartheta}^{(n)},{\rm
d}z_{0})$ $\displaystyle=$ $\displaystyle\vartheta_{n},$ (13.31)
$\displaystyle\int_{[0,1]\times[0,1]^{\mathbb{N}_{0}}}x_{0}^{2}\;Q^{(n)}(\bar{\vartheta}^{(n)},{\rm
d}z_{0})$ $\displaystyle=$
$\displaystyle\;\vartheta_{n}^{2}+A_{0}^{n}\;(\mathcal{F}^{(n+1)}g)(\vartheta_{n}),$
(13.32)
$\displaystyle\int_{[0,1]\times[0,1]^{\mathbb{N}_{0}}}y_{0,0}^{2}\;Q^{(n)}(\bar{\vartheta}^{(n)},{\rm
d}z_{0})$ $\displaystyle=$
$\displaystyle\;\vartheta_{n}^{2}+(A_{0}^{n}-B_{0})(\mathcal{F}^{(n+1)}g)(\vartheta_{n}),$
(13.33)
$\displaystyle\int_{[0,1]\times[0,1]^{\mathbb{N}_{0}}}x_{0}y_{0,0}\;Q^{(n)}(\bar{\vartheta}^{(n)},{\rm
d}z_{0})$ $\displaystyle=$
$\displaystyle\int_{[0,1]\times[0,1]^{\mathbb{N}_{0}}}y^{2}_{0,0}\;Q^{(n)}(\vartheta_{n},{\rm
d}z_{0}),$ (13.34)
$\displaystyle\int_{[0,1]\times[0,1]^{\mathbb{N}_{0}}}g(x)\;Q^{(n)}(\bar{\vartheta}^{(n)},{\rm
d}z_{0})$ $\displaystyle=$
$\displaystyle(\mathcal{F}^{(n+1)}g)(\vartheta_{n}).$

###### Proof.

We prove the claim for $x_{0}^{2}$ only. The other relations follow in a
similar way. The proof proceeds by induction. The result for $n=0$ follows
directly from Proposition 13.1. Assume the result holds true for $n=n$, for
$n=n+1$ we write

(13.35)
$\displaystyle\int_{[0,1]\times[0,1]^{\mathbb{N}_{0}}}x_{0}^{2}\;Q^{(n+1)}(\bar{\vartheta}^{(n+1)},{\rm
d}z_{0})$
$\displaystyle\quad=\int_{[0,1]\times[0,1]^{\mathbb{N}_{0}}}x_{0}^{2}\,(Q^{[n+1]}\circ
Q^{(n)})(\bar{\vartheta}^{(n+1)},{\rm d}z_{0})$
$\displaystyle\quad=\int_{[0,1]\times[0,1]^{\mathbb{N}_{0}}}\int_{[0,1]\times[0,1]^{\mathbb{N}_{0}}}x_{0}^{2}\,Q^{[n+1]}(\bar{\vartheta}^{(n+1)},{\rm
d}z_{n+1})\,Q^{(n)}(z_{n+1},{\rm d}z_{0})$
$\displaystyle\quad=\int_{[0,1]\times[0,1]^{\mathbb{N}_{0}}}\left[\int_{[0,1]\times[0,1]^{\mathbb{N}_{0}}}x_{0}^{2}\,Q^{(n)}(z_{n+1},{\rm
d}z_{0})\right]\,Q^{[n+1]}(\bar{\vartheta}^{(n+1)},{\rm d}z_{n+1})$
$\displaystyle\quad=\int_{[0,1]\times[0,1]^{\mathbb{N}_{0}}}\Big{[}x_{n+1}^{2}+A_{0}^{n}\;(\mathcal{F}^{(n+1)}g)(x_{n+1})\Big{]}\Gamma_{\bar{\vartheta}^{(n+1)}}({\rm
d}z_{n+1})$
$\displaystyle\quad=\vartheta_{n+1}^{2}+A_{0}^{n}\;(\mathcal{F}^{(n+2)}g)(\vartheta_{n+1})+A_{n+1}^{n+1}\;(\mathcal{F}^{(n+2)}g)(\vartheta_{n+1})$
$\displaystyle\quad=\theta_{n+1}^{2}+A_{0}^{n+1}\;(\mathcal{F}^{(n+2)}g)(\vartheta_{n+1}).$

The first and second equality use the definition in (5.1), the third equality
uses Fubini, the fourth equality is the induction step, the fifth equality
uses Proposition 13.2, in particular, (13.26) and (13.28). ∎

Similar iterate moment relations hold for blocks. Define, for
$m,n\in\mathbb{N}_{0}$ with $n\geq m$,

(13.36) ${}Q_{m}^{(n)}=Q^{[n]}\circ\cdots\circ Q^{[m]}.$

######  Proposition 13.4.

[Iterated moment relations: blocks of components] For $n,m\in\mathbb{N}_{0}$
with $n\geq m$,

(13.37)
$\displaystyle\int_{[0,1]\times[0,1]^{\mathbb{N}_{0}}}x_{m}\;Q_{m}^{(n)}(\bar{\vartheta}^{(n)},{\rm
d}z_{m})$ $\displaystyle=$ $\displaystyle\vartheta_{n},$ (13.38)
$\displaystyle\int_{[0,1]\times[0,1]^{\mathbb{N}_{0}}}y_{m,m}\;Q_{m}^{(n)}(\bar{\vartheta}^{(n)},{\rm
d}z_{m})$ $\displaystyle=$ $\displaystyle\vartheta_{n},$ (13.39)
$\displaystyle\int_{[0,1]\times[0,1]^{\mathbb{N}_{0}}}x_{m}^{2}\;Q_{m}^{(n)}(\bar{\vartheta}^{(n)},{\rm
d}z_{m})$ $\displaystyle=$
$\displaystyle\;\vartheta_{n}^{2}+A_{m}^{n}\;(\mathcal{F}^{(n+1)}g)(\vartheta_{n}),$
(13.40)
$\displaystyle\int_{[0,1]\times[0,1]^{\mathbb{N}_{0}}}y_{m,m}^{2}\;Q_{m}^{(n)}(\bar{\vartheta}^{(n)},{\rm
d}z_{m})$ $\displaystyle=$
$\displaystyle\;\vartheta_{n}^{2}+(A_{m}^{n}-B_{m})(\mathcal{F}^{(n+1)}g)(\vartheta_{n}),$
(13.41)
$\displaystyle\int_{[0,1]\times[0,1]^{\mathbb{N}_{0}}}x_{m}y_{m,m}\;Q_{m}^{(n)}(\bar{\vartheta}^{(n)},{\rm
d}z_{m})$ $\displaystyle=$
$\displaystyle\int_{[0,1]\times[0,1]^{\mathbb{N}_{0}}}y^{2}_{m,m}\;Q_{m}^{(n)}(\bar{\vartheta}^{(n)},{\rm
d}z_{m}),$ (13.42)
$\displaystyle\int_{[0,1]\times[0,1]^{\mathbb{N}_{0}}}(\mathcal{F}^{(m)}g)(x_{m})\;Q_{m}^{(n)}(\bar{\vartheta}^{(n)},{\rm
d}z_{m})$ $\displaystyle=$
$\displaystyle(\mathcal{F}^{(n+1)}g)(\vartheta_{n}).$

###### Proof.

Follow a similar induction argument as in the proof of Proposition 13.3. ∎

####  13.3 Clustering

To prove Theorem 5.1, we proceed as in [BCGdH95]. The following clustering
property holds for the kernels associated with single colonies.

######  Proposition 13.5.

[Clustering: single colonies] Assume

(13.43) $\lim_{n\to\infty}\vartheta_{n}=\theta.$

Then

(13.44)
$\displaystyle\lim_{n\to\infty}Q^{(n)}\big{(}\bar{\vartheta}^{(n)},\\{(x_{0},y_{(0,0)})=(0,0)\\}\big{)}=1-\theta,$
$\displaystyle\lim_{n\to\infty}Q^{(n)}\big{(}\bar{\vartheta}^{(n)},\\{(x_{0},y_{(0,0)})=(1,1)\\}\big{)}=\theta,$

if and only if

(13.45) $\lim_{n\to\infty}A_{0}^{n}=\infty.$

Consequently,

(13.46)
${}\lim_{n\to\infty}Q^{(n)}\big{(}\bar{\vartheta}^{(n)},\\{(x_{0},y_{0,0})\\}\notin\\{(0,0),(1,1)\\}\big{)}=0.$

###### Proof.

The proof exploits the iterated moment relations. First assume (13.45)

1. By Proposition 13.3

(13.47)
$\int_{[0,1]\times[0,1]^{\mathbb{N}_{0}}}x_{0}(1-x_{0})\,Q^{(n)}(\bar{\vartheta}^{(n)},{\rm
d}z_{0})=\vartheta_{n}(1-\vartheta_{n})-A_{0}^{n}(\mathcal{F}^{(n+1)}g)(\vartheta_{n}).$

Because $x_{0}(1-x_{0})\geq 0$ for $x\in[0,1]$, we have

(13.48) $\vartheta_{n}(1-\vartheta_{n})\geq
A_{0}^{n}(\mathcal{F}^{(n+1)}g)(\vartheta_{n})\qquad\forall\,n\in\mathbb{N}.$

Since $\lim_{n\to\infty}A_{0}^{n}=\infty$, it follows that
$\lim_{n\to\infty}(\mathcal{F}^{(n+1)}g)(\vartheta_{n})=0$. On the other hand,

(13.49)
$(\mathcal{F}^{(n+1)}g)(\vartheta_{n})=\int_{[0,1]\times[0,1]^{\mathbb{N}_{0}}}g(x_{0})\,Q^{(n)}(\bar{\vartheta}^{(n)},{\rm
d}z_{0})$

and, because $g(x)>0$ for $x\in(0,1)$, $Q^{(n)}(\vartheta_{n},{\rm d}z_{0})$
puts all its mass on $x_{0}=0$ and $x_{0}=1$ in the limit as $n\to\infty$. Let

(13.50) ${}Q^{(n)}\big{(}\bar{\vartheta}^{(n)},\\{x_{0}=0\text{ or
}x_{0}=1\\}\big{)}=\int_{[0,1]\times[0,1]^{\mathbb{N}_{0}}}\mathbf{1}_{\\{x_{0}=0\text{
or }x_{0}=1\\}}(z_{0})\,Q^{(n)}(\bar{\vartheta}^{(n)},{\rm d}z_{0}),$

then

(13.51)
${}\lim_{n\to\infty}Q^{(n)}\big{(}\bar{\vartheta}^{(n)},\\{x_{0}=0\text{ or
}x_{1}=1\\}\big{)}=1.$

The first moment of $x_{0}$ converges to (recall (4.4))

(13.52)
$\lim_{n\to\infty}\int_{[0,1]\times[0,1]^{\mathbb{N}_{0}}}x_{0}\,Q^{(n)}(\bar{\vartheta}^{(n)},{\rm
d}z_{0})=\lim_{n\to\infty}\vartheta_{n}=\theta.$

Hence

(13.53)
$\displaystyle\lim_{n\to\infty}Q^{(n)}(\bar{\vartheta}^{(n)},\\{x_{0}=0\\})=1-\theta,$
$\displaystyle\lim_{n\to\infty}Q^{(n)}(\bar{\vartheta}^{(n)},\\{x_{0}=1\\})=\theta,$

and

(13.54)
$\displaystyle\lim_{n\to\infty}\int_{[0,1]\times[0,1]^{\mathbb{N}_{0}}}x_{0}^{2}\,Q^{(n)}(\bar{\vartheta}^{(n)},{\rm
d}z_{0})$
$\displaystyle\quad=\lim_{n\to\infty}\int_{[0,1]\times[0,1]^{\mathbb{N}_{0}}}x_{0}^{2}\Big{(}\mathbf{1}_{\\{1\\}}(x_{0})+\mathbf{1}_{\\{0\\}}(x_{0})+\mathbf{1}_{\\{(0,1)\\}}(x_{0})\Big{)}\,Q^{(n)}(\bar{\vartheta}^{(n)},{\rm
d}z_{0})=\theta.$

On the other hand,

(13.55)
$\lim_{n\to\infty}\int_{[0,1]\times[0,1]^{\mathbb{N}_{0}}}x_{0}^{2}\;Q^{(n)}(\bar{\vartheta}^{(n)},{\rm
d}z_{0})=\theta^{2}+\lim_{n\to\infty}A_{0}^{n}(\mathcal{F}^{(n+1)}g)(\theta),$

and so, combining (13.54)–(13.55), we obtain

(13.56)
$\lim_{n\to\infty}A_{0}^{n}(\mathcal{F}^{(n+1)}g)(\theta)=\theta(1-\theta).$

2. We know also that

(13.57)
$\displaystyle\lim_{n\to\infty}\int_{[0,1]\times[0,1]^{\mathbb{N}_{0}}}x_{0}y_{0,0}\,Q^{(n)}(\bar{\vartheta}^{(n)},{\rm
d}z_{0})$
$\displaystyle\qquad=\lim_{n\to\infty}\;\vartheta_{n}^{2}+\left(A_{0}^{n}-\frac{E_{0}^{2}}{E_{0}c_{0}+e_{0}+E_{0}K_{0}e_{0}}\right)(\mathcal{F}^{(n+1)}g)(\vartheta_{n})$
$\displaystyle\qquad=\theta^{2}+\theta(1-\theta)=\theta$

and

(13.58)
$\displaystyle\lim_{n\to\infty}\int_{[0,1]\times[0,1]^{\mathbb{N}_{0}}}xy_{0}\,Q^{(n)}(\bar{\vartheta}^{(n)},{\rm
d}z_{0})$
$\displaystyle\quad=\lim_{n\to\infty}\int_{[0,1]\times[0,1]^{\mathbb{N}_{0}}}x_{0}y_{0,0}\,\Big{(}\mathbf{1}_{\\{1\\}}(x_{0})+\mathbf{1}_{\\{0\\}}(x_{0})+\mathbf{1}_{\\{(0,1)\\}}(x_{0})\Big{)}\,Q^{(n)}(\bar{\vartheta}^{(n)},{\rm
d}z_{0})$
$\displaystyle\quad=\lim_{n\to\infty}\int_{[0,1]\times[0,1]^{\mathbb{N}_{0}}}y_{0,0}\,\mathbf{1}_{\\{1\\}}(x_{0})\,Q^{(n)}(\bar{\vartheta}^{(n)},{\rm
d}z_{0}).$

Therefore

(13.59)
$\displaystyle\lim_{n\to\infty}\int_{[0,1]\times[0,1]^{\mathbb{N}_{0}}}y_{0,0}\,\mathbf{1}_{\\{1\\}}(x_{0})\,Q^{(n)}(\bar{\vartheta}^{(n)},{\rm
d}z_{0})=\lim_{n\to\infty}\vartheta_{n}=\theta,$

and hence

(13.60)
$\displaystyle\lim_{n\to\infty}\int_{[0,1]\times[0,1]^{\mathbb{N}_{0}}}(1-y_{(0,0)})\,\mathbf{1}_{\\{1\\}}(x_{0})\,Q^{(n)}(\bar{\vartheta}^{(n)},{\rm
d}z_{0})=\theta-\theta=0.$

Since $1-y_{(0,0)}\geq 0$, we conclude that if $x_{0}=1$, then
$Q^{(n)}(\bar{\vartheta}^{(n)},{\rm d}z_{0})$ puts all its mass on $y_{0,0}=1$
in the limit as $n\to\infty$. Hence

(13.61)
${}\lim_{n\to\infty}Q^{(n)}\big{(}\bar{\vartheta}^{(n)},\\{(x_{0},y_{0,0})=(1,1)\\}\big{)}=\theta.$

From Proposition 13.4 it also follows that

(13.62)
$\displaystyle\lim_{n\to\infty}\int_{[0,1]\times[0,1]^{\mathbb{N}_{0}}}(1-x_{0})(1-y_{0,0})\,Q^{(n)}(\bar{\vartheta}^{(n)},{\rm
d}z_{0})$
$\displaystyle\quad=1-\theta-\theta+\theta^{2}+\lim_{n\to\infty}\left(A_{0}^{n}-\frac{E_{0}^{2}}{E_{0}c_{0}+e_{0}+E_{0}K_{0}e_{0}}\right)(\mathcal{F}^{(n+1)}g)(\vartheta_{n})$
$\displaystyle\quad=1-\theta.$

On the other hand,

(13.63)
$\displaystyle\lim_{n\to\infty}\int_{[0,1]\times[0,1]^{\mathbb{N}_{0}}}(1-x_{0})(1-y_{0,0})\,Q^{(n)}(\bar{\vartheta}^{(n)},{\rm
d}z_{0})$
$\displaystyle=\lim_{n\to\infty}\int_{[0,1]\times[0,1]^{\mathbb{N}_{0}}}(1-x_{0})(1-y_{0,0})\,\Big{(}\mathbf{1}_{\\{1\\}}(x_{0})+\mathbf{1}_{\\{0\\}}(x_{0})+\mathbf{1}_{\\{(0,1)\\}}(x_{0})\Big{)}$
$\displaystyle\qquad\qquad\qquad\times Q^{(n)}(\bar{\vartheta}^{(n)},{\rm
d}z_{0})$
$\displaystyle=\lim_{n\to\infty}\int_{[0,1]\times[0,1]^{\mathbb{N}_{0}}}(1-y_{0,0})\,\mathbf{1}_{\\{0\\}}(x_{0})\,Q^{(n)}(\bar{\vartheta}^{(n)},{\rm
d}z_{0})$ $\displaystyle=1-\theta.$

Since $y\in[0,1]$ and

(13.64)
$\displaystyle\lim_{n\to\infty}\int_{[0,1]\times[0,1]^{\mathbb{N}_{0}}}\mathbf{1}_{\\{0\\}}(x_{0})\,Q^{(n)}(\bar{\vartheta}^{(n)},{\rm
d}z_{0})=1-\theta,$

it follows that

(13.65)
$\displaystyle\lim_{n\to\infty}\int_{[0,1]\times[0,1]^{\mathbb{N}_{0}}}y_{0,0}\mathbf{1}_{\\{0\\}}(x_{0})\,Q^{(n)}(\bar{\vartheta}^{(n)},{\rm
d}z_{0})=0.$

This implies that if $x_{0}=0$, then $Q^{(n)}(\bar{\vartheta}^{(n)},{\rm
d}z_{0})$ puts all its mass on $y_{0,0}=0$ in the limit as $n\to\infty$. Hence

(13.66)
$\displaystyle\lim_{n\to\infty}Q^{(n)}\big{(}\bar{\vartheta}^{(n)},\\{(x_{0},y_{0,0})=(0,0)\\}\big{)}=1-\theta,$
$\displaystyle\lim_{n\to\infty}Q^{(n)}\big{(}\bar{\vartheta}^{(n)},\\{(x_{0},y_{0,0})=(1,1)\\}\big{)}=\theta.$

Now assume (LABEL:ifc). Then (13.54) still holds. On the other hand, also
(13.55) still holds by Proposition 13.3. Therefore we obtain (13.56). On the
other hand, by (LABEL:ifc)

(13.67)
$\lim_{n\to\infty}\mathcal{F}^{(n+1)}g=\lim_{n\to\infty}\int_{[0,1]\times[0,1]^{\mathbb{N}_{0}}}g(x_{0})Q^{(n)}(\bar{\vartheta}^{(n)},{\rm
d}z_{0})=0.$

Hence (13.45) holds. ∎

A similar clustering property holds for the kernels associated with blocks.

######  Proposition 13.6.

[Clustering: blocks] Assume

(13.68) $\lim_{n\to\infty}\vartheta_{n}=\theta.$

Then

(13.69)
$\displaystyle\lim_{n\to\infty}Q_{m}^{(n)}\big{(}\bar{\vartheta}^{(n)},\\{(x_{m},y_{(m,m)})=(0,0)\\}\big{)}=1-\theta,$
$\displaystyle\lim_{n\to\infty}Q_{m}^{(n)}\big{(}\bar{\vartheta}^{(n)},\\{(x_{m},y_{(m,m)})=(1,1)\\}\big{)}=\theta,$

if and only if

(13.70) $\lim_{n\to\infty}A_{m}^{n}=\infty.$

Consequently,

(13.71)
${}\lim_{n\to\infty}Q^{(n)}_{m}\big{(}\bar{\vartheta}^{(n)},\\{(x_{m},y_{m,m})\\}\notin\\{(0,0),(1,1)\\}\big{)}=0.$

###### Proof.

We can proceed exactly as in the proof of Proposition 13.5. ∎

Finally, we can prove Theorem 5.1.

###### Proof.

Note that (13.45) implies (13.70). Recall that single colonies of deep seed-
banks that have already interacted and reached their quasi-equilibrium equal
the block average of the level on which they interact (see Theorem 4.4). It
follows that, for $m\in\mathbb{N}_{0}$,

(13.72)
$\displaystyle\lim_{n\to\infty}Q^{(n)}\big{(}\bar{\vartheta}^{(n)},{(x_{0},y_{0,m})=(1,1)}\big{)}=\theta,$
$\displaystyle\lim_{n\to\infty}Q^{(n)}\big{(}\bar{\vartheta}^{(n)},{(x_{0},y_{0,m})=(0,0)}\big{)}=1-\theta.$

Therefore, for $N\in\mathbb{N}_{0}$,

(13.73)
$\displaystyle\lim_{n\to\infty}Q^{(n)}\left(\bar{\vartheta}^{(n)},\bigcap_{m=0}^{N}\big{\\{}(x_{0},y_{0,m})=(1,1)\text{
or }(x_{0},y_{0,m})=(0,0)\big{\\}}\right)$
$\displaystyle\qquad=1-\lim_{n\to\infty}Q^{(n)}\left(\bar{\vartheta}^{(n)},\bigcup_{m=0}^{N}\big{\\{}(x_{0},y_{0,m})\in[0,1]^{2}\backslash\\{(0,0),(1,1)\\}\big{\\}}\right)$
$\displaystyle\qquad\geq
1-\lim_{n\to\infty}\sum_{m=0}^{N}Q^{(n)}\Big{(}\bar{\vartheta}^{(n)},\big{\\{}(x_{0},y_{0,m})\in[0,1]^{2}\backslash\\{(0,0),(1,1)\\}\big{\\}}\Big{)}=1-0=1.$

Note that

(13.74)
$\displaystyle\lim_{n\to\infty}Q^{(n)}\left(\bar{\vartheta}^{(n)},\bigcap_{m=0}^{N}\big{\\{}(x_{0},y_{0,m})=(1,1)\text{
or }(x_{0},y_{0,m})=(0,0)\big{\\}}\right)$
$\displaystyle\qquad=\lim_{n\to\infty}Q^{(n)}\Big{(}\bar{\vartheta}^{(n)},\big{\\{}(x_{0},(y_{0,m})_{0\leq
m\leq N})=(0,0^{N+1})$ $\displaystyle\qquad\qquad\qquad\qquad\qquad\text{ or
}(x_{0},(y_{0,m})_{0\leq m\leq N})=(1,1^{N+1})\big{\\}}\Big{)}=1.$

On the other hand,

(13.75)
$\displaystyle\lim_{n\to\infty}Q^{(n)}\Big{(}\bar{\vartheta}^{(n)},\\{(x_{0},(y_{0,m})_{0\leq
m\leq N})=(1,1^{N+1})\\}\Big{)}$
$\displaystyle\qquad\leq\lim_{n\to\infty}Q^{(n)}\big{(}\bar{\vartheta}^{(n)},\\{(x_{0},y_{0,0})=(1,1)\\}\big{)}=\theta$

and

(13.76)
$\displaystyle\lim_{n\to\infty}Q^{(n)}\Big{(}\bar{\vartheta}^{(n)},\\{(x_{0},(y_{0,m})_{0\leq
m\leq N})=(0,0^{N+1})\\}\Big{)}$
$\displaystyle\qquad\leq\lim_{n\to\infty}Q^{(n)}\big{(}\bar{\vartheta}^{(n)},\\{(x_{0},y_{0,0})=(0,0)\\}\big{)}=1-\theta.$

Hence we conclude that

(13.77)
${}\lim_{n\to\infty}Q^{(n)}\Big{(}\bar{\vartheta}^{(n)},\\{(x_{0},(y_{0,m})_{0\leq
m\leq N})=(1,1^{N+1})\\}\Big{)}=\theta$

and

(13.78)
${}\lim_{n\to\infty}Q^{(n)}\Big{(}\bar{\vartheta}^{(n)},\\{(x_{0},(y_{0,m})_{0\leq
m\leq N})=(0,0^{N+1})\\}\Big{)}=1-\theta.$

We can do the same for all finite-dimensional distributions. Since
$[0,1]\times[0,1]^{\mathbb{N}_{0}}$ is compact, the process
$z_{0}=(x_{0},(y_{0,m})_{m\in\mathbb{N}_{0}})$ is tight. Therefore, by
(13.77)–(13.78) we find for every converging subsequence

(13.79)
$\lim_{k\to\infty}Q^{(n_{k})}\Big{(}\bar{\vartheta}^{(n_{k})},\cdot\Big{)}=(1-\theta)\,\delta_{(0,0^{\mathbb{N}_{0}})}+\theta\,\delta_{(1,1^{\mathbb{N}_{0}})}.$

We conclude that

(13.80) $\lim_{n\to\infty}Q^{(n)}(\bar{\vartheta}^{(n)},{\rm
d}z_{0})=(1-\theta)\,\delta_{(0,0^{\mathbb{N}_{0}})}+\theta\,\delta_{(1,1^{\mathbb{N}_{0}})},$

which is the claim in (5.4). ∎

####  13.4 Dichotomy finite versus infinite seed-bank

In this section we prove Theorem 5.3.

###### Proof.

We investigate for what choices of the sequences
$\underline{c},\underline{K},\underline{e}$ defined in (2.5) and
(LABEL:defKem) we meet the _clustering criterion_
$\lim_{n\to\infty}A_{n}\to\infty$ in (5.5). Recall from (4.5) and (5.3) that

(13.81)
$A_{n}=\frac{1}{2}\sum_{k=0}^{n-1}\frac{E_{k}}{c_{k}}\frac{(E_{k}c_{k}+e_{k})}{(E_{k}c_{k}+e_{k})+E_{k}K_{k}e_{k}},\qquad
E_{k}=\frac{1}{1+\sum_{m=0}^{k-1}K_{m}}.$

We distinguish between three regimes as $k\to\infty$:

  1. 1.

$E_{k}c_{k}+e_{k}\gg E_{k}K_{k}e_{k}$.

  2. 2.

$E_{k}c_{k}+e_{k}\asymp E_{k}K_{k}e_{k}$.

  3. 3.

$E_{k}c_{k}+e_{k}\ll E_{k}K_{k}e_{k}$.

These regimes correspond to the following scaling for $A_{n}$ as $n\to\infty$:

  1. 1.

$A_{n}\sim\frac{1}{2}\sum_{k=0}^{n-1}\frac{E_{k}}{c_{k}}$.

  2. 2.

$A_{n}\asymp\sum_{k=0}^{n-1}\frac{E_{k}}{c_{k}}$.

  3. 3.

$A_{n}\sim\frac{1}{2}\sum_{k=0}^{n-1}\frac{E_{k}c_{k}+e_{k}}{c_{k}K_{k}e_{k}}$.

Recall from that (2.14) that

(13.82) $\rho=\sum_{m\in\mathbb{N}_{0}}K_{m}.$

Different behaviour shows up for finite seed-bank ($\rho<\infty$) and infinite
seed-bank ($\rho=\infty$).

###### (I) $\rho<\infty$.

Note that $k\mapsto E_{k}$ is non-increasing and converges to $1/(1+\rho)>0$.
Since $E_{k}\leq 1$, we have $\lim_{k\to\infty}E_{k}K_{k}=0$ and hence we are
in regime 1. Therefore

(13.83) ${}A_{n}\sim\frac{1}{2(1+\rho)}\sum_{k=0}^{n-1}\frac{1}{c_{k}},\qquad
n\to\infty,$

and clustering occurs if and only if
$\sum_{k\in\mathbb{N}_{0}}\frac{1}{c_{k}}=\infty$, which is the same criterion
as for the system without seed-bank.

###### (II) $\rho=\infty$.

We focus on the settings in (3.10) and (3.11), which fall in regimes 1 and 2.

Asymptotically polynomial. Suppose that

(13.84) ${}K_{k}\sim Ak^{-\alpha},\quad
k\to\infty,\,A\in(0,\infty),\,\alpha\in(-\infty,1).$

Then

(13.85) $E_{k}\sim\frac{1-\alpha}{A}k^{-(1-\alpha)},\qquad
E_{k}K_{k}\sim\frac{1-\alpha}{A}k^{-1},\qquad k\to\infty.$

Hence we are in regime 1. Suppose that

(13.86) ${}c_{k}\sim Fk^{-\phi},\quad
k\to\infty,\,F\in(0,\infty),\,\phi\in\mathbb{R}.$

Then

(13.87)
$A_{n}\sim\frac{1-\alpha}{2AF}\sum_{k=1}^{n-1}k^{-1+\alpha+\phi},\qquad
n\to\infty,$

and clustering occurs if and only if $-\phi\leq\alpha<1$. In this case

(13.88) $\displaystyle-\phi<\alpha\colon\quad
A_{n}\sim\frac{1-\alpha}{2AF(\alpha+\phi)}\,n^{\alpha+\phi},$
$\displaystyle-\phi=\alpha\colon\quad A_{n}\sim\frac{1-\alpha}{2AF}\,\log n.$

The case $\alpha=1$ can be included. Then (13.85) becomes

(13.89) $E_{k}\sim\frac{1}{A\log k},\qquad E_{k}K_{k}\sim\frac{1}{k\log
k},\qquad k\to\infty,$

so that we are again in regime 1. Now (13.87) becomes

(13.90) $A_{n}\sim\tfrac{1}{2AF}\sum_{k=1}^{n-1}\frac{k^{\phi}}{\log k},\qquad
n\to\infty,$

and clustering occurs if and only if $-\phi\leq 1$. In this case

(13.91) $\displaystyle-\phi<1\colon\quad
A_{n}\sim\frac{1}{2AF(1+\phi)}\,\frac{n^{1+\phi}}{\log n},$
$\displaystyle-\phi=1\colon\quad A_{n}\sim\frac{1}{2AF}\,\log\log n.$

Pure exponential. Suppose that

(13.92) ${}K_{k}=K^{k},\quad k\in\mathbb{N}_{0},\,K\in(1,\infty).$

Then

(13.93)
$E_{k}=\frac{1}{1+\sum_{m=0}^{k-1}K^{m}}=\frac{1}{1+\frac{K^{k}-1}{K-1}}=\frac{K-1}{K^{k}+K-2}.$

Suppose that

(13.94) $e_{k}=e^{k},\quad c_{k}=c^{k},\quad
k\in\mathbb{N}_{0},\,e,c\in(0,\infty).$

Then

(13.95) ${}E_{k}c_{k}+e_{k}=\frac{K-1}{K^{k}+K-2}c^{k}+e^{k},\qquad
E_{k}K_{k}e_{k}=\frac{K-1}{K^{k}+K-2}K^{k}e^{k},$

and so

(13.96) $E_{k}c_{k}+e_{k}\sim(K-1)\left(\frac{c}{K}\right)^{k}+e^{k},\qquad
E_{k}K_{k}e_{k}\sim(K-1)e^{k},\qquad k\to\infty.$

For $c\leq Ke$ we are in regime 2, and hence

(13.97)
$A_{n}\sim\frac{1}{2}\sum_{k=0}^{n-1}\frac{K-1}{(Kc)^{k}}\,\frac{(K-1)(\frac{c}{K})^{k}+e^{k}}{(K-1)(\frac{c}{K})^{k}+Ke^{k}},\qquad
n\to\infty,$

which simplifies to

(13.98) $\displaystyle c<Ke\colon\quad
A_{n}\sim\frac{1}{2K}\sum_{k=0}^{n-1}\frac{K-1}{(Kc)^{k}},$ $\displaystyle
c=Ke\colon\quad
A_{n}\sim\frac{K-1}{2(2K-1)}\sum_{k=0}^{n-1}\frac{K-1}{(Kc)^{k}}.$

Clustering occurs if and only if $Kc\leq 1$. In this case

(13.99) $\displaystyle
Kc<1\colon\quad\sum_{k=0}^{n-1}\frac{1}{(Kc)^{k}}\sim\frac{1}{1-Kc}\,(Kc)^{-(n-1)},$
$\displaystyle Kc=1\colon\quad\sum_{k=0}^{n-1}\frac{1}{(Kc)^{k}}\sim n.$

For $c>Ke$, on the other hand, we are in regime 1, and hence

(13.100) ${}A_{n}\sim\frac{1}{2}\sum_{k=0}^{n-1}\frac{K-1}{(Kc)^{k}},\qquad
n\to\infty,$

for which we can again use (13.99).

The case $K=1$ can be included. Then $E_{k}=1/k$ and (13.96) becomes

(13.101) $E_{k}c_{k}+e_{k}\sim\frac{1}{k}c^{k}+e^{k},\qquad
E_{k}K_{k}e_{k}\sim\frac{1}{k}e^{k},\qquad k\to\infty,$

we are again in regime 1. Hence

(13.102) $A_{n}\sim\frac{1}{2}\sum_{k=0}^{n-1}\frac{1}{kc^{k}},\qquad
n\to\infty,$

and clustering occurs if and only if $c\leq 1$. In that case

(13.103) $\displaystyle c<1\colon\quad
A_{n}\sim\frac{1}{2(1-c)}\,\frac{1}{n}\,c^{-(n-1)},$ $\displaystyle
c=1\colon\quad A_{n}\sim\frac{1}{2}\,\log n.$

∎

In the above computations, only regimes 1 and 2 arise. Regime 3 arises, for
instance, when

(13.104) $\lim_{k\to\infty}\frac{1}{k}\log
K_{k}=\infty,\qquad\lim_{k\to\infty}K_{k}e_{k}/c_{k}=\infty.$

Indeed, the first condition implies that $E_{k}\sim 1/K_{k-1}$ and
$E_{k}K_{k}\gg 1$, while the second implies that $E_{k}K_{k}e_{k}\gg
E_{k}c_{k}$. There are two subcases:

(13.105) $\displaystyle K_{k-1}e_{k}\ll c_{k}\colon\quad
A_{n}\sim\tfrac{1}{2}\sum_{k=0}^{n-1}\frac{1}{K_{k}K_{k-1}e_{k}},$
$\displaystyle K_{k-1}e_{k}\gg c_{k}\colon\quad
A_{n}\sim\tfrac{1}{2}\sum_{k=0}^{n-1}\frac{1}{K_{k}c_{k}}.$

By picking, for instance, $e_{k}=1/K_{k}K_{k-1}$, we find that
$A_{n}\sim\tfrac{1}{2}n$ in the first subcase and $A_{n}\gg n$ in the second
subcase. By picking, alternatively, $c_{k}=1/K_{k}$, we find that $A_{n}\gg n$
in the first subcase and $A_{n}\sim\tfrac{1}{2}n$ in the second subcase.

###  Appendix A Computation of scaling coefficients

In Appendices A.1–A.2 we spell out a technical computation for the tail of the
wake-up time defined in (2.39)–(2.40) in the two parameter regimes given by
(3.10)–(3.11). In Appendix A.3 we carry out a computation that is needed in
Section 6.1.

####  A.1 Regularly varying coefficients

In (2.39), note that for large $t$ in the sum over $m$ only small values of
$e_{m}/N^{m}$ contribute, which means large $m$. Hence, by the Euler-MacLaurin
approximation formula, we have

(A.1)
$P(\tau>t)=\frac{1}{\chi}\sum_{m\in\mathbb{N}_{0}}K_{m}\frac{e_{m}}{N^{m}}\,{\rm
e}^{-(e_{m}/N^{m})t}\sim\frac{1}{\chi}\int_{c}^{\infty}{\rm
d}m\,K_{m}\frac{e_{m}}{N^{m}}\,{\rm e}^{-(e_{m}/N^{m})t},$

where $c$ is a constant that identifies from which value of $m$ onward terms
contribute significantly. Make the change of variable $s=\frac{e_{m}}{N^{m}}$.
Since $e_{m}\sim Bm^{-\beta}$ and $K_{m}\sim Am^{-\alpha}$ as $m\to\infty$, we
have

(A.2) $s\sim Bm^{-\beta}N^{-m}$

and hence

(A.3) $\displaystyle\log s$ $\displaystyle\sim\log B-\beta\log m-m\log N,$
$\displaystyle\log\frac{1}{s}$ $\displaystyle=m\log N\left(-\frac{B}{m\log
N}+\frac{\beta\log m}{m\log N}+1\right)=[1+o(1)]\,m\log N,$

which gives

(A.4) $m=[1+o(1)]\,\frac{\log(\frac{1}{s})}{\log N}.$

Thus,

(A.5) $\frac{1}{s}\frac{{\rm d}s}{{\rm d}m}=-\log
N-\frac{\beta}{m}=-[1+o(1)]\,\log N,$

which implies

(A.6) $\frac{{\rm d}s}{{\rm d}m}=-[1+o(1)]\,s\log N,$

so that $s(m)$ is asymptotically decreasing in $m$, and

(A.7) $\frac{{\rm d}m}{{\rm d}s}=-[1+o(1)]\,\left(s\log N\right)^{-1}.$

Note that if $c\leq m<\infty$, then asymptotically
$0<m^{-\beta}N^{-m}<c^{-\beta}N^{-c}=C_{2}$. Doing the substitution, we get

(A.8) $\displaystyle\mathbb{P}(\tau>t)\sim\frac{1}{\chi}\int_{0}^{C_{2}}{\rm
d}s\,K_{m}s\left(s\log N\right)^{-1}\,{\rm
e}^{-st}\sim\frac{1}{\chi}\int_{0}^{C_{2}}{\rm d}s\,Am^{-\alpha}\left(\log
N\right)^{-1}\,{\rm e}^{-st}$
$\displaystyle\sim\frac{1}{\chi}\int_{0}^{C_{2}}{\rm
d}s\,A\left(\frac{\log(\frac{1}{s})}{\log N}\right)^{-\alpha}\left(\log
N\right)^{-1}\,{\rm e}^{-st}\sim\frac{A}{\chi}\left(\frac{1}{\log
N}\right)^{-\alpha+1}\int_{0}^{C_{2}}{\rm
d}s\,\log\left(\tfrac{1}{s}\right)^{-\alpha}\,{\rm e}^{-st}.$

Next, put $st=u$, so $s=\frac{u}{t}$ and $\frac{{\rm d}s}{{\rm
d}u}=\frac{1}{t}$ and $0<u<tC_{2}$. Then

(A.9) $\mathbb{P}(\tau>t)\sim\frac{A}{\chi}\left(\frac{1}{\log
N}\right)^{-\alpha+1}\frac{1}{t}\int_{0}^{C_{2}t}{\rm
d}u\,\log\left(\tfrac{t}{u}\right)^{-\alpha}\,{\rm e}^{-u}.$

We will show that

(A.10) $\frac{A}{\chi}\left(\frac{1}{\log
N}\right)^{-\alpha+1}\frac{1}{t}\int_{0}^{C_{2}t}{\rm
d}u\,\log\left(\tfrac{t}{u}\right)^{-\alpha}\,{\rm
e}^{-u}\asymp\frac{A}{\chi}\left(\frac{1}{\log
N}\right)^{-\alpha+1}\frac{1}{t}\int_{0}^{C_{2}t}{\rm d}u\,\log
t^{-\alpha}\,{\rm e}^{-u}.$

For $\alpha=0$ this claim is immediate. For $\alpha\in(-\infty,0)$, note that
$\log\left(\frac{t}{u}\right)^{-\alpha}$ is a decreasing function on
$(0,C_{2}t)$. Therefore we can reason as follows:

(A.11) $\displaystyle\int_{0}^{C_{2}t}{\rm
d}u\,\log\left(\tfrac{t}{u}\right)^{-\alpha}{\rm e}^{-u}$
$\displaystyle=\int_{0}^{1}{\rm
d}u\,\log\left(\tfrac{t}{u}\right)^{-\alpha}{\rm e}^{-u}+\int_{1}^{C_{2}t}{\rm
d}u\,\log\left(\tfrac{t}{u}\right)^{-\alpha}{\rm e}^{-u}$
$\displaystyle\leq\int_{0}^{1}{\rm
d}u\,\log\left(\tfrac{t}{u}\right)^{-\alpha}+\int_{1}^{C_{2}t}{\rm
d}u\,\log{t}^{-\alpha}{\rm e}^{-u}$ $\displaystyle\leq
2^{-\alpha}\int_{0}^{\frac{1}{t}}{\rm
d}u\,\log\left(\tfrac{1}{u}\right)^{-\alpha}+2^{-\alpha}\int_{\frac{1}{t}}^{1}{\rm
d}u\,\log t^{-\alpha}+\log t^{-\alpha}\left[1-{\rm e}^{-1}\right]$
$\displaystyle\leq 2^{-\alpha}\Gamma(-\alpha+1)+2^{-\alpha}\log
t^{-\alpha}\left[1-\tfrac{1}{t}\right]+\log t^{-\alpha}\left[1-{\rm
e}^{-1}\right]$ $\displaystyle=\log
t^{-\alpha}\left[2^{-\alpha}\tfrac{\Gamma(-\alpha+1)}{\log
t^{-\alpha}}+2^{-\alpha}\left[1-\tfrac{1}{t}\right]+\left[1-{\rm
e}^{-1}\right]\right]\asymp\log t^{-\alpha}.$

For the lower bound, note that

(A.12) $\displaystyle\int_{0}^{C_{2}t}{\rm
d}u\,\log\left(\tfrac{t}{u}\right)^{-\alpha}{\rm e}^{-u}$
$\displaystyle\geq\log\left(t\right)^{-\alpha}\int_{0}^{1}{\rm d}u\,{\rm
e}^{-u}+\log(\tfrac{1}{C_{2}})^{-\alpha}\int_{1}^{C_{2}t}{\rm d}u\,{\rm
e}^{-u}$ $\displaystyle=\log
t^{-\alpha}\left[1-e^{-1}+\frac{\log(\tfrac{1}{C_{2}})^{-\alpha}}{\log
t^{-\alpha}}\right]\asymp\log t^{-\alpha}.$

For $\alpha\in(0,1]$, note that the function
$\log\left(\frac{t}{u}\right)^{-\alpha}$ is increasing in $u$. For the lower
bound estimate

(A.13) $\displaystyle\int_{0}^{C_{2}t}{\rm
d}u\,\log\left(\tfrac{t}{u}\right)^{-\alpha}{\rm e}^{-u}$
$\displaystyle\geq\lim_{u\to 0}\log\left(\tfrac{t}{u}\right)^{-\alpha}[1-{\rm
e}^{-1}]+\log t^{-\alpha}[{\rm e}^{-1}-e^{-C_{2}t}]$ $\displaystyle=\log
t^{-\alpha}\left[0+{\rm e}^{-1}-{\rm e}^{-C_{2}t}\right]\asymp\log
t^{-\alpha}.$

For the upper bound estimate

(A.14) $\displaystyle\int_{0}^{C_{2}t}{\rm
d}u\,\log\left(\tfrac{t}{u}\right)^{-\alpha}{\rm e}^{-u}$
$\displaystyle\leq\log t^{-\alpha}[1-{\rm
e}^{-1}]+\log\left(\frac{t}{\sqrt{C_{2}t}}\right)^{-\alpha}\int_{1}^{\sqrt{C_{2}t}}{\rm
d}u\,e^{-u}+\log\left(\frac{1}{C_{2}}\right)^{-\alpha}\int_{\sqrt{C_{2}t}}^{C_{2}t}{\rm
d}u\,{\rm e}^{-u}$ $\displaystyle=\log t^{-\alpha}[1-{\rm
e}^{-1}]+(\tfrac{1}{2})^{-\alpha}\log\left(\frac{t}{C_{2}}\right)^{-\alpha}\left[{\rm
e}^{-1}-{\rm
e}^{-\sqrt{C_{2}t}}\right]+\log\left(\frac{1}{C_{2}}\right)^{-\alpha}\left[{\rm
e}^{-\sqrt{C_{2}t}}-{\rm e}^{-C_{2}t}\right]$ $\displaystyle=\log
t^{-\alpha}\Bigg{[}1-{\rm e}^{-1}+(\tfrac{1}{2})^{-\alpha}\left(\frac{\log
t-\log C_{2}}{\log t}\right)^{-\alpha}\left[{\rm e}^{-1}-{\rm
e}^{-\sqrt{C_{2}t}}\,\right]$
$\displaystyle\qquad\qquad\qquad+\log\left(\frac{1}{C_{2}}\right)^{-\alpha}\frac{\left[{\rm
e}^{-\sqrt{C_{2}t}}-{\rm e}^{-C_{2}t}\right]}{\log
t^{-\alpha}}\Bigg{]}\asymp\log t^{-\alpha}.$

####  A.2 Pure exponential coefficients

In order to satisfy condition in (2.12), we must assume that $Ke<N$. Since
$K\geq 1$ for $\rho=\infty$, we also have $e<N$. We again use that for large
$t$ only large $m$ contribute to the sum. Hence, again by the Euler-MacLaurin
approximation formula, we have

(A.15)
$P(\tau>t)=\frac{1}{\chi}\sum_{m\in\mathbb{N}_{0}}K_{m}\frac{e_{m}}{N^{m}}\,{\rm
e}^{-(e_{m}/N^{m})t}\sim\int_{M}^{\infty}{\rm
d}m\,K_{m}\frac{e_{m}}{N^{m}}\,{\rm e}^{-(e_{m}/N^{m})t}.\\\ $

Again we put $s=\frac{e^{m}}{N^{m}}$. Hence

(A.16) $\log s=m\log\left(\frac{e}{N}\right),\qquad m=\frac{\log
s}{\log\frac{e}{N}},\qquad\frac{{\rm d}m}{{\rm
d}s}=\frac{1}{s\log\frac{e}{N}},$

and

(A.17) $K_{m}\sim K^{m}\sim{\rm e}^{\log s\frac{\log K}{\log\frac{e}{N}}}\sim
s^{\frac{\log K}{\log\frac{e}{N}}}.$

Since $s(m)$ is decreasing in $m$, putting $C=(\frac{e}{N})^{M}$ we obtain

(A.18) $\mathbb{P}(\tau>t)\sim\int_{0}^{C}{\rm
d}s\,K_{m}\frac{s}{s\log\frac{e}{N}}\,{\rm
e}^{-st}\sim\frac{1}{\log\frac{e}{N}}\int_{0}^{C}{\rm d}s\,s^{\frac{\log
K}{\log\frac{e}{N}}}\,{\rm e}^{-st}.$

Substitute $u=st$, i.e., $\frac{u}{t}=s$, to get

(A.19)
$\displaystyle\mathbb{P}(\tau>t)\sim\frac{1}{\log\frac{e}{N}}\,t^{-1-{\frac{\log
K}{\log\frac{e}{N}}}}\int_{0}^{Ct}{\rm d}u\,u^{\frac{\log
K}{\log\frac{e}{N}}}\,{\rm e}^{-u}$
$\displaystyle\sim\frac{1}{\log\frac{e}{N}}\,t^{\frac{-\log(\frac{e}{N})-\log
K}{\log\frac{e}{N}}}\int_{0}^{Ct}{\rm d}u\,u^{\frac{\log
K}{\log\frac{e}{N}}}\,{\rm
e}^{-u}\sim\frac{1}{\log\frac{e}{N}}\,t^{-\frac{\log(\frac{N}{Ke})}{\log\frac{N}{e}}}\int_{0}^{Ct}{\rm
d}u\,u^{\frac{\log K}{\log\frac{e}{N}}}\,{\rm e}^{-u}.$

The last integral converges because $\frac{\log K}{\log(\frac{e}{N})}>-1$, and

(A.20) $\int_{0}^{Ct}{\rm d}u\,u^{\frac{\log K}{\log\frac{e}{N}}}\,{\rm
e}^{-u}\leq\int_{0}^{\infty}{\rm d}u\,u^{\frac{\log K}{\log\frac{e}{N}}}\,{\rm
e}^{-u}=\Gamma\left(\frac{\log K}{\log(\frac{e}{N})}+1\right).$

####  A.3 Slowly varying functions

Return to Section 6.1. Note that $t(s)=\varphi(s)^{-1}s^{\gamma}$. Since this
is the total time two lineages are active up to time $s$, $t(s)$ must be
smaller than $s$. By (3.7), we have

(A.21)
$\displaystyle\frac{{\varphi}(t)}{{\varphi}(s)}=\exp\left[-\int_{t(s)}^{s}\frac{{\rm
d}u}{u}\psi(u)\right].$

Since we are interested in $s\to\infty$, we may assume that $s\gg 1$ and
$t(s)>1$, and estimate

(A.22) $\displaystyle\frac{{\varphi}(t)}{{\varphi}(s)}$
$\displaystyle\leq\exp\left[\int_{t(s)}^{s}\frac{{\rm d}u}{u}\frac{C}{\log
u}\right]=\exp\big{[}C(\log\log s-\log\log t(s))\big{]}$
$\displaystyle=\exp\left[C\log\left(\frac{\log
s}{\log\left(\varphi(s)^{-1}s^{\gamma}\right)}\right)\right]=\exp\left[-C\log\left(\frac{\gamma\log
s-\log\varphi(s)}{\log s}\right)\right].$

A similar lower bound holds with the sign reversed. Using that
$\lim_{s\to\infty}\frac{\log\varphi(s)}{\log s}=0$, we get

(A.23)
$\displaystyle\gamma^{C}\leq\liminf_{s\to\infty}\frac{{\varphi}(t)}{{\varphi}(s)}\leq\limsup_{s\to\infty}\frac{{\varphi}(t)}{{\varphi}(s)}\leq\gamma^{-C}.$

Both bounds above are positive, so indeed
$\frac{{\varphi}(t)}{{\varphi}(s)}\asymp 1$.

###  Appendix B Meyer-Zheng topology

####  B.1 Basic facts about the Meyer-Zheng topology

In the Meyer-Zheng topology we assign to each real-valued Borel measurable
function $(w(t))_{t\geq 0}$ a probability law on
$[0,\infty]\times\bar{\mathbb{R}}$ that is called the pseudopath $\psi_{w}$.
Note that the Borel-$\sigma$ algebra on $[0,\infty]\times\bar{\mathbb{R}}$ is
generated by sets of the form $[a,b]\times B$ for $B\in\mathcal{B}$ and
$0<a<b$. For $A=[a,b]\times B$, set

(B.1) $\psi_{w}(A)=\int 1_{A}(t,w(t)){\rm e}^{-t}{\rm
d}t=\int_{a}^{b}1_{B}(w(t))e^{-t}{\rm d}t,$

i.e., $\psi_{w}$ is the image measure of the mapping $t\to(t,w(t))$ under the
measure $\lambda({\rm d}t)={\rm e}^{-t}{\rm d}t$. The set of all pseudopaths
is denoted by $\Psi$. Note that a pseudopath corresponding to $(w(t))_{t>0}$
is simply its occupation measure. The following important facts are stated in
[MZ84]:

  * •

If two paths $w_{1}$ and $w_{2}$ are the same Lebesgue a.e., then
$\psi_{w_{1}}=\psi_{w_{2}}$.

  * •

Denote by $\mathbf{D}$ the space of càdlàg paths on
$[0,\infty]\times\mathbb{R}$. The mapping
$\psi\colon\,\mathbf{D}\to\Psi,\,w\mapsto\psi_{w}$ is one-to-one on
$\mathbf{D}$ and hence gives an embedding of $\mathbf{D}$ into the compact
space $\bar{\mathcal{P}}$, the space of probability measures on
$[0,\infty]\times\bar{\mathbb{R}}$.

  * •

Note if $f$ is a function on $[0,\infty]\times\mathbb{R}$ and
$w\in\mathbf{D}$, then

(B.2) $\psi_{w}(f)=\int_{0}^{\infty}f(t,w(t))\,{\rm e}^{-t}{\rm d}t.$

Therefore we say that the sequence of pseudopaths induced by
$(w_{n})\subset\mathbf{D}$ converges to a pseudopath $w$ if, for all
continuous bounded function $f(t,w(t))$ on $[0,\infty]\times\bar{\mathbb{R}}$,

(B.3) $\lim_{n\to\infty}\int_{0}^{\infty}f(t,w_{n}(t))\,{\rm e}^{-t}{\rm
d}t=\int_{0}^{\infty}f(t,w(t))\,{\rm e}^{-t}{\rm d}t.$

Since a pseudopath is a measure, convergence of pseudopaths is convergence of
measures.

  * •

$\mathbf{D}$ endowed with the pseudopath topology is _not_ a Polish space.
$\Psi$ endowed with the pseudopath topology is a Polish space.

  * •

According to [MZ84][Lemma 1], the pseudopath topology on $\Psi$ is convergence
in Lebesgue measure on $\mathbf{D}$.

####  B.2 Pseudopaths of stochastic processes on a general metric separable
space

In [Kur91] the results of [MZ84] on state space $\mathbb{R}$ are generalised
to a general metric separable space $E$. Let $(Z(t))_{t>0}$ be a stochastic
process with state space $E$. Then we assign a random pseudopath to $(Z(t))$
as follows: for $\omega\in\Omega$ and $A=[a,b]\times B$, $0\leq a<b$ and
$B\in\mathcal{B}(E)$,

(B.4) $\psi_{(Z(t,\omega))_{t\geq 0}}(A)=\int_{a}^{b}1_{B}(Z(t,\omega))\,{\rm
e}^{-t}{\rm d}t.$

Hence $\psi_{(Z(t))_{t\geq 0}}$ is a random variable with state space $\Psi$,
i.e., $\psi_{(Z(t))_{t\geq 0}}\in\mathcal{M}(\Psi)$, the set of probability
measures on pseudopaths. Note that

(B.5) $\mathbb{E}\left[\psi_{(Z(t))_{t\geq
0}}f\right]=\mathbb{E}\left[\int_{0}^{\infty}f(t,Z(t,\omega))\,{\rm
e}^{-t}{\rm d}t\right]=\mathbb{E}\left[\int_{0}^{\infty}f(t,Z(t))\,{\rm
e}^{-t}{\rm d}t\right].$

###### Weak convergence in the Meyer-Zheng topology.

Let $(Z_{n}(t))_{t\geq 0}$ and $(Z(t))_{t\geq 0}$ be stochastic processes with
state-space $E$. We say that

(B.6) $\mathcal{L}\left[(Z_{n}(t))_{t\geq
0}\right]=\mathcal{L}\left[(Z(t))_{t\geq 0}\right]\text{ in the Meyer-Zheng
topology}$

if, for all $f\in\mathcal{C}_{b}(\Psi)$,

(B.7) $\lim_{n\to\infty}\mathbb{E}[f(\psi_{(Z^{n}(t))_{t\geq
0}})]=\mathbb{E}[f(\psi_{(Z(t))_{t\geq 0}})].$

Let $\mathcal{C}_{m}([0,\infty)\times
E)\subset\mathcal{C}_{b}([0,\infty)\times E)$ be the set of functions of the
form

(B.8) $\displaystyle\mathcal{C}_{m}([0,\infty)\times E)$
$\displaystyle=\Big{\\{}F\in\mathcal{C}_{b}([0,\infty)\times
E):F(t,x(t))=\prod_{i=1}^{m}\int_{0}^{T_{i}}f_{i}(t,x(t)){\rm d}t,$
$\displaystyle\qquad m\in\mathbb{N},\,\forall 1\leq i\leq
m,\,f_{i}\in\mathcal{C}_{b}([0,\infty)\times E),\,T_{i}>0\Big{\\}}.$

Note that $\mathcal{C}_{m}$ is an algebra. Let $M_{E}[0,\infty)$ be the space
of measurable processes from $[0,\infty)$ to $\mathbb{E}$, so
$\mathbf{D}\subset M_{E}[0,\infty)$. Note that $\mathcal{C}_{m}$ separates
points in $M_{E}[0,\infty)$. By [Kur91][Proposition 4.5], the set
$\mathcal{C}_{m}$ is separating in the set of measures on $M_{E}[0,\infty)$.
This means that if two stochastic processes $(Z_{1}(t))_{t>0}$ and
$(Z_{2}(t))_{t\geq 0}$ satisfy

(B.9) $\mathbb{E}[F(Z_{1})]=\mathbb{E}[F(Z_{2})]\qquad\forall
F\in\mathcal{C}_{m},$

then $\mathcal{L}[Z_{1}]=\mathcal{L}[Z_{2}]$.

Define

(B.10) $F(\psi)=\int{\rm
d}\psi\prod_{i=1}^{m}\int_{0}^{T_{i}}f_{i}(t,x(t))\,{\rm d}t.$

Recall that a pseudopath $\psi$ is associated with a path $w\in
M_{E}[0,\infty)$. Hence this becomes

(B.11) $F(\psi_{w})=\prod_{i=1}^{m}\int_{0}^{T_{i}}f_{i}(t,w(t))\,{\rm d}t.$

Since each pseudopath $\psi\in\Psi$ is associated with a path in
$M_{E}[0\infty)$, $\mathcal{C}_{m}$ also separates points on $\Psi$ and hence
$\mathcal{C}_{m}$ separates measures on $\Psi$. This implies that if

(B.12) $\mathbb{E}[F(\psi_{Z_{1}})]=\mathbb{E}[F(\psi_{Z_{2}})]\qquad\forall
F\in\mathcal{C}_{m},$

then $\mathcal{L}[\psi_{Z_{1}}]=\mathcal{L}[\psi_{Z_{2}}]$. Therefore
$\mathcal{L}[Z_{1}]=\mathcal{L}[Z_{2}]$ if and only if
$\mathcal{L}[\psi_{Z_{1}}]=\mathcal{L}[\psi_{Z_{2}}]$.

The Meyer-Zheng topology is a weaker than the Skohorod topology.

######  Lemma B.1.

Let $(Z_{n}(t))_{t\geq 0}$ $n\in\mathbb{N}$ and $(Z(t))_{t\geq 0}$ be
stochastic processes with Polish state-space $E$. If

(B.13) $\lim_{n\to\infty}\mathcal{L}\left[(Z_{n}(t))_{t\geq
0}\right]=\mathcal{L}\left[(Z(t))_{t\geq 0}\right]\text{ in the Skohorod
topology},$

then

(B.14) $\lim_{n\to\infty}\mathcal{L}\left[(Z_{n}(t))_{t\geq
0}\right]=\mathcal{L}\left[(Z(t))_{t\geq 0}\right]\text{ in the Meyer-Zheng
topology}.$

###### Proof.

Since we do not know whether $\Psi$ is compact, the set $\mathcal{C}_{m}$ does
not have to be convergence determining. Therefore, via Skorohod’s theorem we
construct the process $\tilde{Z}^{n}$ and $\tilde{Z}$ on one probability
space, such that $\mathcal{L}[\tilde{Z}^{n}]=\mathcal{L}[{Z}^{n}]$ and
$\mathcal{L}[\tilde{Z}]=\mathcal{L}[{Z}]$, and

(B.15) $\lim_{n\to\infty}\tilde{Z}_{n}=\tilde{Z}\qquad a.s.$

This implies

(B.16) $\lim_{n\to\infty}\psi_{\tilde{Z}^{n}}=\psi_{\tilde{Z}}\qquad a.s.$

Consequently, for all $f\in\mathcal{C}_{b}(\Psi)$,

(B.17)
$\lim_{n\to\infty}\mathbb{E}[f(\psi_{\tilde{Z}^{n}})]=\mathbb{E}[f(\psi_{\tilde{Z}})].$

Note that, since $\mathcal{L}[\tilde{Z}^{n}]=\mathcal{L}[{Z}^{n}]$ and
$\mathcal{L}[\tilde{Z}]=\mathcal{L}[{Z}]$, we can use (B.9) and (B.12) to see
that the latter implies
$\mathcal{L}[\psi_{Z^{n}}]=\mathcal{L}[\psi_{\tilde{Z}^{n}}]$ and
$\mathcal{L}[\psi_{Z}]=\mathcal{L}[\psi_{\tilde{Z}}]$. Hence (B.17) indeed
implies that

(B.18) $\lim_{n\to\infty}\mathcal{L}[\psi_{Z^{n}}]=\mathcal{L}[\psi_{Z}].$

∎

###### Convergence in probability in the Meyer-Zheng topology.

Let $(S,d)$ be a metric space, $\mathcal{B}(S)$ denote the Borel-$\sigma$
algebra on $S$, and $\mathcal{P}(S)$ the set of probability measures on
$\mathcal{B}(S)$. Recall (see e.g. [EK86, Chapter 3]) that the Prohorov metric
$d_{P}$ on the space $\mathcal{P}(S)$ is given by

(B.19)
$d_{P}(\mathbb{P},\mathbb{Q})=\inf\left\\{\epsilon>0\colon\,\mathbb{P}(A)\leq\mathbb{Q}(A^{\epsilon})+\epsilon\,\,\forall\,A\in\mathcal{C}\right\\},$

where $\mathcal{C}\subset\mathcal{B}(S)$ is the set of all closed sets in $S$
and $A^{\epsilon}=\\{x\in S\colon\,\inf_{y\in A}d(x,y)<\epsilon\\}$.

######  Theorem B.2.

[EK86, Theorem 3.1.2] Let $(S,d)$ be separable and let
$\mathbb{P},\mathbb{Q}\in\mathcal{P}(S)$. Define
$\mathcal{M}(\mathbb{P},\mathbb{Q})$ to be the set of all
$\mu\in\mathcal{P}(S\times S)$ with marginals $\mathbb{P}$ and $\mathbb{Q}$,
i.e., $\mu(A\times S)=\mathbb{P}(A)$ and $\mu(S\times A)=\mathbb{Q}(A)$ for
all $A\in\mathcal{B}(S)$. Then

(B.20)
$d_{P}(\mathbb{P},\mathbb{Q})=\inf_{\mu\in\mathcal{M}(\mathbb{P},\mathbb{Q})}\inf\\{\epsilon>0\colon\,\mu(\\{(x,y)\colon\,d(x,y)\geq\epsilon\\})\leq\epsilon\\}.$

Moreover, [EK86, Theorem 3.3.1] states that convergence of measures in the
Prohorov distance, $\lim_{n\to\infty}d_{P}(\mathbb{P}_{n},\mathbb{P})=0$, is
the same as weak convergence $\mathbb{P}_{n}\Rightarrow\mathbb{P}$. Hence,
since convergence of pseudopaths is weak convergence, we can endow the space
of pseudopaths $\Psi$ with the metric $d_{P}$.

Let $(\Psi,d_{P})$ be the pseudopath space metrized by the Prohorov distance.
Let $(Z^{n}(t))_{t>0}$, $(Z(t))_{t>0}$ be stochastic processes on the state
space $E$, where $E$ is endowed with the metric $d(\cdot,\cdot)$. Note that
convergence in probability in the Meyer-Zheng topology means that

(B.21)
$\forall\,\delta>0\colon\quad\lim_{n\to\infty}\mathbb{P}\left[d_{P}\left(\psi_{Z^{n}},\psi_{Z}\right)>\delta\right]=0.$

###### Tightness.

Define the _conditional variation_ for an $\mathbb{R}$-valued process
$(U(t))_{t\geq 0}$ with natural filtration $(\mathcal{F}(t))_{t\geq 0}$ as
follows. For a subdivision $\tau\colon\,0=t_{0}<t_{1}<\cdots<t_{n}=\infty$,
set

(B.22) $V_{\tau}(U)=\sum\limits_{0\leq
i<n}\mathbb{E}\Big{[}\big{|}\mathbb{E}[U(t_{i+1})-U(t_{i})\mid
F(t_{i})]\big{|}\Big{]}$

(with $U(\infty)=0)$ and

(B.23) $V(U)=\sup\limits_{\tau}V_{\tau}(U).$

If $V(U)<\infty$, then $U$ is called a _quasi-martingale_. Note that we can
always stop the process at some finite time and work with compact time
intervals.

######  Lemma B.3.

[Tightness in the Meyer-Zheng topology]
If $(P_{n})_{n\in\mathbb{N}}$ is a sequence of probability laws on
$D([0,T],\mathbb{R})$ such that under $P_{n}$ the coordinate process
$(U(t))_{t\geq 0}$ is a quasi-martingale with a conditional variation
$V_{n}(U)$ that is bounded uniformly in $n$, then there exists a subsequence
$(P_{n_{k}})_{k\in\mathbb{N}}$ that converges weakly in the Meyer-Zheng
topology on $D([0,T],\mathbb{R})$ to a probability law $P$, and $(U(t))_{t\geq
0}$ is a quasi-martingale under $P$.

(See [MZ84, Theorem 7] for the identification of the limiting semi-
martingale.)

####  B.3 Proof of key lemmas

######  $\bullet$ Proof of Lemma 8.19.

###### Proof.

Fix $\delta>0$. Then

(B.24)
$\displaystyle\lim_{n\to\infty}\mathbb{P}\left[d_{P}(\psi_{Z_{n}},\psi_{Z})>\delta\right]$
$\displaystyle=\lim_{n\to\infty}\mathbb{P}\left[\inf_{\mu\in\mathcal{M}(\psi_{Z_{n}},\psi_{Z})}\inf\\{\epsilon>0\colon\,\mu(\\{(x,y)\colon\,d(x,y)\geq\epsilon\\})\leq\epsilon\\}>\delta\right]$
$\displaystyle=\lim_{n\to\infty}\mathbb{P}\left[\,\forall\mu\in\mathcal{M}(\psi_{Z_{n}},\psi_{Z}),\,\inf\\{\epsilon>0\colon\,\mu(\\{(x,y)\colon\,d(x,y)\geq\epsilon\\})\leq\epsilon\\}>\delta\right]$
$\displaystyle=\lim_{n\to\infty}\mathbb{P}\left[\,\forall\mu\in\mathcal{M}(\psi_{Z_{n}},\psi_{Z}),\,\mu(\\{(x,y)\colon\,d(x,y)\geq\delta\\})>\delta\right].$

Let $\mu_{n}\in\mathcal{P}(([0,\infty]\times E)^{2})$ be the measure defined
by

(B.25)
$\mu_{n}(A)=\int_{0}^{\infty}1_{A}\left((t,Z_{n}(t)),(t,Z(t))\right)\,{\rm
e}^{-t}{\rm d}t,\qquad A\in\mathcal{B}(([0,\infty]\times E)^{2}),$

such that, for $B\in\mathcal{B}([0,\infty]\times E)$,

(B.26) $\mu_{n}(B\times
S)=\int_{0}^{\infty}1_{B}(t,Z_{n}(t))1_{S}((t,Z(t))\,{\rm e}^{-t}{\rm
d}t=\psi_{Z^{n}}(B),$

and similarly $\mu_{n}(S\times B)=\psi_{Z}(B)$. Hence
$\mu_{n}\in\mathcal{M}(\psi_{Z_{n}},\psi_{Z})$ for all $n\in\mathbb{N}$, and
we obtain from (B.24) that

(B.27)
$\displaystyle\lim_{n\to\infty}\mathbb{P}\left[d_{P}(\psi_{Z_{n}},\psi_{Z})>\delta\right]$
$\displaystyle\leq\lim_{n\to\infty}\mathbb{P}\left[\mu_{n}(\\{(x,y):d(x,y)\geq\delta\\})>\delta\right]$
$\displaystyle\leq\lim_{n\to\infty}\mathbb{P}\left[\int_{0}^{\infty}1_{\\{(x,y):d(x,y)\geq\delta\\}}\left((t,Z_{n}(t)),(t,Z(t))\right){\rm
e}^{-t}{\rm d}t>\delta\right]$
$\displaystyle\leq\lim_{n\to\infty}\mathbb{P}\left[\int_{0}^{\infty}1_{\\{d(Z_{n}(t),Z(t))\geq\delta\\}}\,{\rm
e}^{-t}{\rm d}t>\delta\right]$
$\displaystyle\leq\lim_{n\to\infty}\frac{1}{\delta}\mathbb{E}\left[\int_{0}^{\infty}d(Z_{n}(t),Z(t))\,{\rm
e}^{-t}{\rm d}t\right]$
$\displaystyle=\lim_{n\to\infty}\frac{1}{\delta}\int_{0}^{\infty}\mathbb{E}\left[d(Z_{n}(t),Z(t))\right]\,{\rm
e}^{-t}{\rm d}t=0.$

∎

######  $\bullet$ Proof of Lemma 8.20.

###### Proof.

We have to show that

(B.28)
$\lim_{n\to\infty}\mathcal{L}\left[\psi_{(X_{n},Y_{n})}\right]=\mathcal{L}\left[\psi_{(X,c)}\right].$

Hence we must show that, for all $f\in\mathcal{C}_{b}(\Psi)$,

(B.29)
$\lim_{n\to\infty}\mathbb{E}[f(\psi_{(X_{n},Y_{n})})]=\mathbb{E}[f(\psi_{(X,c)})].$

We can write

(B.30)
$\displaystyle|\mathbb{E}[f(\psi_{(X_{n},Y_{n})})-f(\psi_{(X,c)})]|\leq|\mathbb{E}[f(\psi_{(X_{n},Y_{n})})-f(\psi_{(X_{n},c)})]|+|\mathbb{E}[f(\psi_{(X_{n},c)})-f(\psi_{(X,c)})]|.$

Since $\lim_{n\to\infty}\mathbb{E}[d(Y_{n}(t),c)]=0$ implies
$\lim_{n\to\infty}\mathbb{E}[d((X_{n}(t),Y_{n}(t)),(X_{n}(t),c))]=0$, it
follows from Lemma 8.19 that, for all $\delta>0$,

(B.31)
$\lim_{n\to\infty}\mathbb{P}\left[d_{P}\left(\psi_{(X_{n},Y_{n})},\psi_{(X_{n},c)}\right)\right]=0.$

Hence, for all $f\in\mathcal{C}_{b}(\Psi)$,

(B.32)
$\lim_{n\to\infty}|\mathbb{E}[f(\psi_{(X_{n},Y_{n})})-f(\psi_{(X_{n},c)})]|=0.$

To see that the second term in the right-hand side of (B.30) tends to zero,
note that we can define

(B.33) $\tilde{f}(\psi_{x})=f(\psi_{x,c}).$

We show that $\tilde{f}$ is continuous.

Recall that convergence in the Meyer-Zheng topology is simply convergence in
Lebesgue measure. Hence, for two paths $(t,x_{n}(t))$ and $(t,x(t))\in
M_{E}[0\infty)$ we have $\psi_{x_{n}}\to\psi_{x}$ if and only if, for all
$\delta>0$,

(B.34)
$\lim_{n\to\infty}\int_{0}^{\infty}1_{\\{d(x_{n}(t),x(t))>\delta\\}}\,{\rm
e}^{-t}{\rm d}t=0.$

Therefore $\psi_{x_{n}}\to\psi_{x}$ implies that, for all $\delta>0$,

(B.35)
$\lim_{n\to\infty}\int_{0}^{\infty}1_{\\{d((x_{n}(t),c),(x(t),c))>\delta\\}}\,{\rm
e}^{-t}{\rm d}t=0,$

and hence $\psi_{x_{n},c}\to\psi_{x,c}$. Therefore

(B.36)
$\lim_{n\to\infty}\tilde{f}(\psi_{x_{n}})=\lim_{n\to\infty}f(\psi_{(x_{n},c)})=f(\psi_{(x,c)})=\tilde{f}(\psi_{x})$

and we conclude that $f\in\mathcal{C}_{b}(\Psi)$. Since
$\mathcal{L}[X_{n}]=\mathcal{L}[X]$ in the Meyer-Zheng topology, we have, for
all $f\in\mathcal{C}_{b}(\Psi)$,

(B.37)
$\displaystyle\lim_{n\to\infty}|\mathbb{E}[f(\psi_{(X_{n},c)})-f(\psi_{(X,c)})]|=\lim_{n\to\infty}|\mathbb{E}[\tilde{f}(\psi_{(X_{n})})-\tilde{f}(\psi_{(X)})]|=0.$

Therefore also the second term on the right-hand side of (B.30) tends to $0$.
∎

######  $\bullet$ Proof of Lemma 8.21.

###### Proof.

For part (a), suppose that $\lim_{n\to\infty}\psi_{x_{n}}=\psi_{x}$. Then,
since convergence in pseudopath space is convergence in measure, we have, for
all $\delta>0$,

(B.38)
$\lim_{n\to\infty}\int_{0}^{\infty}1_{\\{d(x_{n}(t),x(t))>\delta\\}}\,{\rm
e}^{-t}{\rm d}t=0.$

Since $f$ is a continuous function, this implies that, for all $\epsilon>0$,

(B.39)
$\lim_{n\to\infty}\int_{0}^{\infty}1_{\\{d(f(x_{n}(t)),f(x(t)))>\epsilon\\}}\,{\rm
e}^{-t}{\rm d}t=0.$

and we conclude that $\lim_{n\to\infty}\psi_{f(x_{n})}=\psi_{f(x)}$. Hence $h$
is indeed continuous.

For part (b), recall that

(B.40) $\lim_{n\to\infty}\mathcal{L}[X_{n}]=\mathcal{L}[X]\text{ in the Meyer-
Zheng topology}$

implies that, for all $g\in\mathcal{C}_{b}(\Psi)$,

(B.41) $\lim_{n\to\infty}\mathbb{E}[g(\psi_{X_{n}})]=\mathbb{E}[g(\psi_{X})].$

Since $h\colon\,\Psi\to\Psi$ is continuous, we have for all
$g\in\mathcal{C}_{b}(\Psi)$ that $g\circ h\in\mathcal{C}_{b}(\Psi)$. Hence

(B.42)
$\lim_{n\to\infty}\mathbb{E}[g(\psi_{f(X_{n})}]=\lim_{n\to\infty}\mathbb{E}[g\circ
h(\psi_{X_{n}})]=\mathbb{E}[g\circ h(\psi_{X})]=\mathbb{E}[g(\psi_{f(X)})].$

We conclude that

(B.43) $\lim_{n\to\infty}\mathcal{L}[f(X_{n})]=\mathcal{L}[f(X)]\text{ in the
Meyer-Zheng topology.}$

∎

######  $\bullet$ Proof of Lemma 10.17.

Supposethat $\lim_{n\to\infty}\psi_{(x_{n},y_{n})}=\psi_{(x,y)}$. Since
convergence of pseudopaths is convergence in Lebesgue measure, we have

(B.44)
$\lim_{n\to\infty}\int_{0}^{\infty}1_{\\{d[(x_{n},y_{n}),(x,y)]>\delta\\}}{\rm
e}^{-t}{\rm d}t=0$

and, consequently,

(B.45) $\lim_{n\to\infty}\int_{0}^{\infty}1_{\\{d[x_{n},x]>\delta\\}}{\rm
e}^{-t}{\rm d}t=0.$

Therefore $\lim_{n\to\infty}\psi_{x_{n}}=\psi_{x}$. Suppose that
$f\in\mathcal{C}_{b}(\Psi(E))$, so $f$ is bounded continuous function on the
space of pseudopaths on $[0,\infty]\times E$. Define the function $\tilde{f}$
on the space of pseudopaths on $[0,\infty]\times E^{2}$, i.e., $\tilde{f}$ is
a function on $\Psi(E^{2})$, by

(B.46) $\tilde{f}(\psi_{(x,y)})=f(\psi_{x}).$

Then $\tilde{f}\in\mathcal{C}_{b}(\Psi(E^{2}))$ and

(B.47)
$\lim_{n\to\infty}\tilde{f}(\psi_{(x_{n},y_{n})})=\lim_{n\to\infty}f(\psi_{x_{n}})=f(\psi_{x})=\tilde{f}(\psi_{(x_{n},y_{n})}).$

Hence $\tilde{f}$ is indeed a continuous function on $\Psi(E^{2})$. Moreover,
since $f$ is bounded, it follows that $\tilde{f}$ is bounded and we conclude
that $\tilde{f}\in\mathcal{C}_{b}(\Psi(E^{2}))$.

Therefore, if $X_{n},Y_{n}$ are continuous-time stochastic processes on $E$
and

(B.48)
$\lim_{n\to\infty}\mathcal{L}\left[(X_{n}(s),Y_{n}(s))_{s>0}\right]=\mathcal{L}\left[(X(s),Y(s))_{s>0}\right]\,\text{
in Meyer Zheng topology},$

then for all $f\in\mathcal{C}_{b}(\Psi(E^{2}))$ we have

(B.49)
$\lim_{n\to\infty}\mathbb{E}[f(\psi_{(X_{n},Y_{n})})]=\mathbb{E}[f(\psi_{(X,Y)})].$

Since for each $f\in\mathcal{C}_{b}(\Psi(E))$ we can construct a function
$\tilde{f}\in\mathcal{C}_{b}(\Psi(E^{2}))$ as in (B.46), we obtain for all
$f\in\mathcal{C}_{b}(\Psi(E))$ that

(B.50)
$\displaystyle\lim_{n\to\infty}\mathbb{E}[f(\psi_{(X_{n})})]=\lim_{n\to\infty}\mathbb{E}[\tilde{f}(\psi_{(X_{n},Y_{n})})]=\mathbb{E}[\tilde{f}(\psi_{(X,Y)})]=\mathbb{E}[\tilde{f}(\psi_{X})].$

We conclude that

(B.51)
$\lim_{n\to\infty}\mathcal{L}[(X_{n}(s))]=\mathcal{L}[(X(s))_{s>0}]\,\text{ in
Meyer-Zheng topology}$

and, similarly,

(B.52)
$\lim_{n\to\infty}\mathcal{L}[(Y_{n}(s))]=\mathcal{L}[(Y(s))_{s>0}]\,\text{ in
Meyer-Zheng topology.}$

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# Carbon dioxide emissions trading and hierarchical structure in worldwide
finance and commodities markets

Zeyu Zheng  Department of Environmental Sciences, Tokyo University of
Information Sciences, Chiba 265-8501, Japan  Kazuko Yamasaki  Department of
Environmental Sciences, Tokyo University of Information Sciences, Chiba
265-8501, Japan  Center for Polymer Studies and Department of Physics, Boston
University, Boston, MA 02215, USA  Joel Tenenbaum  Center for Polymer Studies
and Department of Physics, Boston University, Boston, MA 02215, USA  H. Eugene
Stanley  Center for Polymer Studies and Department of Physics, Boston
University, Boston, MA 02215, USA

###### Abstract

In a highly interdependent economic world, the nature of relationships between
financial entities is becoming an increasingly important area of study.
Recently, many studies have shown the usefulness of minimal spanning trees
(MST) in extracting interactions between financial entities. Here, we propose
a modified MST network whose metric distance is defined in terms of cross-
correlation coefficient absolute values, enabling the connections between
anticorrelated entities to manifest properly. We investigate 69 daily time
series, comprising 3 types of financial assets: 28 stock market indicators, 21
currency futures, and 20 commodity futures. We show that though the resulting
MST network evolves over time, the financial assets of similar type tend to
have connections which are stable over time. In addition, we find a
characteristic time lag between the volatility time series of the stock market
indicators and those of the EU CO2 emission allowance (EUA) and crude oil
futures (WTI). This time lag is given by the peak of the cross-correlation
function of the volatility time series EUA (or WTI) with that of the stock
market indicators, and is markedly different ($>20$ days) from 0, showing that
the volatility of stock market indicators today can predict the volatility of
EU emissions allowances and of crude oil in the near future.

###### pacs:

PACS numbers:89.65.Gh, 89.20.-a, 02.50.Ey

##  I Introduction and Method

In the study of complex systems, there has been much work demonstrating the
usefulness of extracting underlying structure from the correlations found in
statistical data Mantegna ; Bonanno2 ; Bonanno1 ; Onnela1 ; Onnela2 ; Song ;
Micciche ; Tumminello ; ManandStan ; Wang ; laloux ; Plerou1 ; Plerou2 ;
Utsugi . Any means of selection of statistically reliable information from
correlation matrices has been dubbed a “filtering procedure” Tumminello .
Useful examples of filtering procedures that use a correlation matrix from
return time series are: hierarchical clustering Mantegna ; Bonanno2 ; Bonanno1
; Onnela1 ; Onnela2 ; Song ; Micciche , procedures based on the random matrix
theory Wang ; laloux ; Plerou1 ; Plerou2 ; Utsugi , and networks from minimum
spanning trees Mantegna ; Bonanno2 ; Bonanno1 ; Onnela1 ; Onnela2 .

Correlation structure studies are not limited only to stock return time series
Mantegna ; Bonanno2 ; Bonanno1 , but also extend to quasi-synchronously
recorded time series of worldwide stock exchange market indices Bonanno1  and
stock return volatility increments Micciche .

Financial time series can include not only stock price, but also many other
types of data, such as commodities price, treasury yield, market index, etc….
Investigation of multi-type quasi-synchronous financial data may yield
insights into the interdependent relationships of markets and commodities.
Moreover, a relationship map of financial assets can highlight the movement of
speculative capital.

We investigate 69 daily financial series from the time period spanning January
2007 to September 2011. The data set includes 21 currency futures, 20
commodity futures which are taken from http://data.theice.com, and 28 stock
market indicators, which are taken from http://finance.yahoo.com (see
Appendix).

Recently, papers have shown the usefulness of the correlation structure
described by an ultrametric space and a corresponding hierarchical
organization for financial return time series Mantegna ; Bonanno2 ; Bonanno1 .
The approach requires the definition of a metric distance. Because correlation
does not fulfill the three metric  axioms that define a metric, the
aforementioned papers use the Mantegna-Sornette distance defined by

$d_{ij}=\sqrt{2(1-\rho_{ij})}$ (1)

for each pair of elements $i$ and $j$, where $\rho_{ij}$ is the correlation
coefficient of the two time series given by feller

$\rho_{ij}\equiv\frac{\langle Y_{i}Y_{j}\rangle-\langle Y_{i}\rangle\langle
Y_{j}\rangle}{\sqrt{(\langle{Y_{i}}^{2}\rangle-\langle
Y_{i}\rangle^{2})(\langle{Y_{j}}^{2}\rangle-\langle
Y_{j}\rangle^{2})}}=\frac{\Big{\langle}\bigg{(}Y_{i}-\langle
Y_{i}\rangle\bigg{)}\bigg{(}Y_{j}-\langle
Y_{j}\rangle\bigg{)}\Big{\rangle}}{\sigma_{Y_{i}}\sigma_{Y_{j}}},$ (2)

where $\langle...\rangle$ denotes the mean.

This distance $d_{ij}$ fulfills the three axioms metric  of a metric: i)
$d_{ij}=0$ if and only if $i=j$, ii) $d_{ij}=d_{ji}$, and iii) $d_{ij}\leq
d_{ik}+d_{kj}$ Mantegna ; Bonanno1 ; Bonanno2 .

In this work, we make a modification to the metric above, based on the
reasoning that if two time series have a very large negative correlation, they
should still be considered close to each other in ultrametric “correlation”
space, since this would indicate a strong connection, regardless of the sign.
Likewise, weak correlations of either sign indicate a weaker connection. On
this basis, strong correlations of either sign should be considered closer
than weak ones.

The value of this modification can be readily seen in situations where two
entities have strong anti-correlations, as has been observed in the
relationship between bond and stock markets (e.g. between UK gilts and the
FTSEMIB), between stocks and currency futures, and between industries and
their inputs (e.g. the price of oil and the value of airline stock) Norden ;
Kwan ; Hammoudeh ; businessweek . Here, use of the conventional Mantenga-
Sornette metric would likely result in any of these two entities manifesting
on opposite sides of a tree, even though we know them to be very closely
linked. Use of our generalized metric ensures that such entities will be
placed nearby when such appropriately strong relations exist.

For this reason, we replace the Pearson correlation coefficient in Eq. 1 with
the absolute value of the Pearson correlation coefficient, defining a new
distance as

$d_{ij}=\sqrt{2(1-|\rho_{ij}|)}$ (3)

with the $\rho_{ij}$ defined as before in Eq. 2 as the correlation coefficient
of assets $i$ and $j$.

This equation also fulfills three axioms of a metric distance. Because the set
of assets considered has no cases such that $\rho_{ij}=-1$, the first axiom is
satisfied on this set in that $d_{ij}=0$ if and only if the correlation is
total ($\rho=1$, meaning that the stocks perform the same stochastic process).
The second axiom (that of symmetry) is trivially satisfied because
$\rho_{ij}=\rho_{ji}$ by definition of the Pearson correlation coefficient.
For the validity of axiom (iii), consider three time series ${Y_{i}}$,
${Y_{j}}$ and ${Y_{k}}$, which have means equal to 0 and standard deviations
equal to 1. All times series have the same length. In order to prove
$d_{ij}\leq d_{ik}+d_{kj}$, firstly we define two new time series as

$Y^{\prime}_{i}\equiv\begin{cases}Y_{i},&\mbox{if }\rho_{Y_{i},Y_{k}}\geq 0\\\
-Y_{i},&\mbox{if }\rho_{Y_{i},Y_{k}}<0)\end{cases}$ (4)
$Y^{\prime}_{j}\equiv\begin{cases}Y_{j},&\mbox{if }\rho_{Y_{j},Y_{k}}\geq 0\\\
-Y_{j},&\mbox{if }\rho_{Y_{j},Y_{k}}<0.\end{cases}$ (5)

So we can rewrite $d_{ik}+d_{kj}$ of distance Eq. 3 by using $Y^{\prime}_{i}$
and $Y^{\prime}_{j}$ as

$\displaystyle d_{ik}+d_{kj}$
$\displaystyle=\sqrt{2(1-|\rho_{Y_{i},Y_{k}}|)}+\sqrt{2(1-|\rho_{Y_{j},Y_{k}}|)}$
(6)
$\displaystyle=\sqrt{2(1-\rho_{Y^{\prime}_{i},Y_{k}})}+\sqrt{2(1-\rho_{Y^{\prime}_{j},Y_{k}})}$
(7) $\displaystyle\geq\sqrt{2(1-\rho_{Y^{\prime}_{i},Y^{\prime}_{j}})}$ (8)
$\displaystyle\begin{cases}=\sqrt{2(1-|\rho_{Y^{\prime}_{i},Y^{\prime}_{j}}|)},&\mbox{if
}\rho_{Y^{\prime}_{i},Y^{\prime}_{j}}\geq 0\\\
\geq\sqrt{2(1-|\rho_{Y^{\prime}_{i},Y^{\prime}_{j}}|)},&\mbox{if
}\rho_{Y^{\prime}_{i},Y^{\prime}_{j}}<0\\\ \end{cases}$ (9)
$\displaystyle=\sqrt{2(1-|\rho_{Y_{i},Y_{j}}|)}\equiv d_{ij}.$ (10)

Thus $d_{ik}+d_{kj}\geq d_{ij}$ , and our metric satisfies the three axioms of
a metric.

For each of the 22 financial time series, we calculate the return time series,
defined as the change of logarithmic price of time series $i$

$R_{i}(t)\equiv\ln(Y_{i}(t+1))-\ln(Y_{i}(t)).$ (11)

Here $Y_{i}(t)$ is the daily price time series of financial asset $i$. For
each of the 22 time series, we also calculate the volatility time series which
is defined simply as the absolute value of the return $|R_{i}|$

$V_{i}(t)\equiv|R_{i}(t)|=|\ln(Y_{i}(t+1))-\ln(Y_{i}(t))|.$ (12)

Additionally, we define the cross-correlation for our analysis. Consider two
time series $\\{y_{t}\\}$ and $\\{y^{\prime}_{t}\\}$. The cross-correlation
between $\\{y_{t}\\}$ and $\\{y^{\prime}_{t}\\}$ is given by

$C_{y,y^{\prime}}(n)\equiv\overline{(y_{t}-\mu)(y^{\prime}_{t+n}-\mu^{\prime}))}/(\sigma\sigma^{\prime}),$
(13)

where $\mu$ and $\mu^{\prime}$ are the respective means and $\sigma$ and
$\sigma^{\prime}$ are the respective standard deviations of the series
$\\{y_{i}\\}$ and $\\{y_{i}^{\prime}\\}$.

The efficient market hypothesis, a basic tenet of modern economics, states
that markets are approximately efficient, meaning that one cannot consistently
achieve returns better than the market because all information about an asset
is already incorporated into that asset’s price Samuelson ; ManandStan . As a
result, it is believed that the long-range memory cannot exist in any return
time series. Suppose that long-range auto-correlations exist in a return time
series: investors may then obtain benefits by using information, which stands
in contradiction to the principle of an efficient market. Consider the cross-
correlation function (Eq. (13)) between return time series of asset $i$ and
asset $j$. Any significant cross-correlations $C_{R_{i},R_{j}}(n)$ in $n\neq
0$ of two return time series would also contradict the existence of an
efficient market. Therefore we can assume that significant cross-correlations
$C_{R_{i},R_{j}}(n)$ will only exist for the case $n=0$.

However, because trading occurs at different times in different cities, some
markets are open when others are closed. The effect of non-synchronous trading
in time series analysis has been well stated Lo ; Lin . In fact, the highest
degree of correlation between different markets may be detected at a one-day
time lag because of the time difference.

Therefore, significant cross-correlations $C_{R_{i},R_{j}}(n)$ may also exist
for $n=-1$ or $n=1$. Additionally, we only care about the magnitude and not
the sign of cross-correlation. Thus, we define the absolute correlation
coefficient as

$\rho_{ij}=\max\left(\left|C_{R_{i},R_{j}}(n)\right|\right)$ (14)

for $n=-1,0,1$.

In volatility time series, long-range correlations $C_{V_{i},V_{j}}(n)$ have
been shown to exist [10-13]. It follows that significant cross-correlations
$C_{V_{i},V_{j}}(n)$ may exist for $n\gg 0$ or $n\ll 0$. Additionally, the
existence of long-range negative correlations between past returns and future
volatility Bouchaud ; Perello , known as the leverage effect, has also been
reported. This correlation is moderate and decays exponentially over the long
term. However, while both of these types of correlations may help predict the
financial risks on a long-range time interval, we point out that neither the
negative correlation between returns and volatilities nor long-range
autocorrelation of volatility can be used to obtain benefits. This is because
the price volatility does not include the direction of price changes, and so
neither contradicts the efficient market hypothesis.

As a sample, the correlation functions of volatility $C_{V_{i},V_{j}}(n)$ and
return $C_{R_{i},R_{j}}(n)$ between the FTSE100 and DJIA are shown in Fig. 1.
The peaks (highest correlations) of both correlation functions are near n=0;
however, the correlation function for returns is fast decaying, quickly
approaching 0 for $n\neq 0$, while the volatility correlation function is slow
decaying with $C_{V_{i},V_{j}}(n)>0$ for $n>-50$ and $n<50$.

We now turn our attention to the stability and structure of MSTs, which are
made using the distance based defined in Eq. 3. Following our discussion on
MSTs, we will show the correlation function graphs of volatility that relate
the correlation $C(n)$ to the time-lag $n$, specifically focusing on the value
of $n$ that gives the LOWESS (the smoother which uses locally weighted
polynomial regression) equal to its maximum value. Here, the LOWESS is defined
by a complex algorithm, proposed by W. S Cleveland Cleveland  in 1981. For
each value, we define 10th nearest neighbor values as the local region, which
is used to calculate the LOWESS value.

An MST is defined as the set of $n-1$ links that connect a set of $n$ elements
in the smallest possible total distance Graham . MSTs have been used in prior
papers Mantegna ; Bonanno2 ; Bonanno1 ; Onnela1 ; Onnela2  to connect
financial data, illustrating the MST’s usefulness in highlighting the
interactions between a number of financial time series.

In Fig. 2, we find that, although MSTs show significantly different structures
in different calendar years, the same type of financial assets tend to group
together consistently over time.

We also find that the stock market indicators (blue in Fig. 2) and currency
futures (green) groups show stronger interconnections than the commodities
(red) group. For stock market indicators and currency futures, financial
factors are the predominant reason for price changes. On the other hand,
commodity futures may be just as much affected by investment as they are by
actual supply and demand. Speculation in commodity futures may alter pricing
in a way that contradicts the law of supply and demand. The existence of such
contradictions depends on commodity type and the calendar year and therefore
serve to decrease the stability of the MST. If this reasoning is correct,
increasing the time span of the time series for cross-correlation should make
the observed connections more stable. In Fig. 3, we show the MST using the
time series from January 2007 to September 2011. We note that only two coal
futures are not connected to the commodity group and the stability is greater
than for single-year time series.

In Fig.4, we describe the cross-correlation functions of the volatility time
series. We show the cross-correlation function of main stock market indicators
with EUA in (a), and with WTI in (b). A systemic time shift between EUA or
WTI, and stock market indicators can clearly be seen. Since the maximum cross-
correlation coefficient in most functions is not much greater than 0.2, the
connections are not so strong, but certainly they are significant. Further
tests of Granger causality also show that the volatility of stock indices is
useful in forecasting the volatility of EUA and WTI approximately 20 to 120
days in advance.

As mentioned before, the correlation function of volatility is a slow-decaying
function. It is much more slowly decaying than the correlation function of
return time series (see Fig. 1), meaning that a long-range cross-correlation
relationship exists. If we consider significant cross-correlations between
different volatility time series to be an information transfer between
different assets, the time lag corresponding to the highest values of
correlation gives the time lag of that information transfer. It is worth
pointing out that the time lag between each pair of stock markets is
approximately 0, such as the time lag between DJIA and other 27 stocks
indicators are shown in Fig. 4 (d) . We show a simple summary of such time lag
in Fig. 4 (c) and (d) If there is information affecting stock markets at a
time of 0 days, the EUA and WTI crude oil futures will be affected by this
information roughly 30 and 90 days later respectively. Because both crude oil
and the ability to emit carbon are major inputs in the world economy, the
existence of this time lag can have strong implications in terms of potential
economic feedback loops.

##  II Discussion

The drawn MSTs are reflective of a number of easily reasoned underlying
economic relationships, both through the stability and specificity of the
links.

As expected, we find a stable tendency for like financial assets to cluster.
Even during the otherwise anomalous 2007 subprime lending crisis and 2008
global financial crisis, this clustering tendency is preserved. This indicates
the existence of strong stable connections, which come out of the strong
cross-correlations, reflective of basic economic features and interactions.
These connections are stable over time and not affected by market conditions.

On the other hand, certain portions of the MSTs are consistently unstable,
like those relating to coal. This also may be reflective of economic
relationships. Unlike other commodities like oil, speculation in coal is
limited, so the movement of the coal futures may be simple supply and demand,
as opposed to driven by speculation. Coal’s lack of strong connections to
other commodities may be a result of investor’s low speculation in stock when
building their commodities portfolios.

Certain connections in our MSTs reveal underlying relationships that are
created by regulation. For example, we find that EUA futures mostly connect
with the base electricity and natural gas futures, which show stable
correlations among them. EU allowance permits, as a part of the European Union
Emission Trading Scheme, are either allocated or auctioned and allow a firm to
emit a designated amount of carbon dioxide Stavins . Since power generation
accounts for about one-quarter of total emissions of carbon dioxide, and
natural gas is the most resource of electricity generation in UK, the stable
connections of EUA to UK base electricity futures and UK natural gas futures
in our MST graph is reasonable.

Also intuitive is the location specific clustering of stock market indices.
AORD, an Australian index, consistently appears closely linked to those of
nearby countries like New Zealand, Japan, and China. Similarly, the HSI not
only keeps connections with most of the Asian stock indices like the JK11,
000001.SS, BSESN, and TWII, but also keeps connections with or otherwise stays
closely connected to indices from America and Australia. Thus, the MST created
reflects the common knowledge that Hong Kong is the financial center of Asia.

One benefit of the novelty of our approach is that it connects indices of
dissimilar type, yielding new insights. The index that connects most to coal
is OSEAX, that of Norway. Norway is rich in oil and natural gas, which
explains why the Norway stock index appears as the most “coal-like” of the
national indices. Similarly, the most “currency-like” of the national stock
indices seems to be AEX, the Dutch securities index.

We also note the relationship between the centrality in the network and real
world geographical knowledge. SOK/SEK is a currency future that is among the
furthest from the center. This is intuitive, since currency trading between
Norway and Sweden has little to do with financial activity in the rest of the
world. The same principle applies to the trading of Euros with the British
pound, shown as EUR/GBP. The Australian dollar, on the other hand, plays a
central role in its exchanges with far away currencies like the US dollar,
Euro, Japanese yen, and Canadian dollar.

##  III Conclusion

In this paper, we have analyzed the correlation function of return and
volatility time series, constructed MSTs based on return time series, and
found consistent time lags in the correlation functions of the volatilities.
From these analysis, we have two main conclusions. (i) The stability of MST
structure clustering between like commodities reflects a basic rule of
economic activity, that the interaction between economic actors is not easily
affected by capital movement. The method of absolute cross-correlation
coefficient based MSTs has strong implications in the ongoing debate about the
relationships of different financial commodity time series. (ii) We find that
a time lag of correlation functions of volatility appears between stock
markets and EUA and WTI. From this finding, we hypothesize that there may be
systemic differences in the spread of financial risk, most often quantified as
volatility. In other words, as concerns risk, different types of markets may
have different sensitivities to economic information and other influencing
factors. It would be interesting, from theoretical point of view, to
generalize this time lag to predict the financial risks on a much longer time
interval. However, much more would need to be understood first, such as the
properties and mechanisms for this time lag. Hence, further empirical study is
needed first. We endeavor to address this question more in future work.

##  IV Appendix: Set of Data

The data under investigation includes 28 stock market indicators, 21 currency
futures, and 20 commodity futures.

The stock market indicators investigated are:

Symbol | Meaning | Notes | Symbol | Meaning | Notes
---|---|---|---|---|---
000001.SS | SSE Composite Index | Shanghai stocks | ISEQ | Irish Stock Exchange Quotient |
AEX | Amsterdam Exchange Index | Dutch securities | JKII | Jakarta Islamic index |
AORD | All Ordinaries | Australian stocks | KLSE | Kuala Lumpur Stock Exchange |
ATX | Austrian Traded Index |  | KS11 | Seoul Composite (South Korea) |
BSESN | SENSEX | Bombay stocks | MXX | Mexican Stock Exchange IPC |
BVSP | Bovespa Index | São Paulo stocks | N225 | NIKKEI 225 | Tokyo stocks
CAC | CAC 40 | French stocks | NZ50 | NZX 50 Index | New Zealand index
DJIA | Dow Jones Industrial Average | American index | OMX | OMX Stockholm 30 |
FTSE | FTSE 100 | London stocks | OMXC20 | OMX Copenhagen 20 |
FTSEMIB | FTSE MIB | Italian stocks | OSEAX | Oslo Børs All Share Index |
GDAXI | Deutscher Aktien Index | German blue chips | STI | Straits Times Index | Singapore stocks
GSPTSE | S&P/TSX Composite Index | Toronto stocks | SSMI | Swiss Market Index |
HSI | Hang Seng Index | Hong Kong stocks | TA100 | Tel Aviv 100 |
IBEX | IBEX 35 | Spanish stocks | TWII | TSEC weighted index | Taiwanese stocks

The commodity and currency futures investigated are all traded in the markets
of intercontinental exchange. The commodity futures are:

Symbol | Meaning | Notes | Symbol | Meaning | Notes
---|---|---|---|---|---
Barley | Western Barley Futures |  | Gasoline | RBOB Gasoline Futures |
BrCrude | Brent Crude Futures | North Sea crude oil | HeatOil | Heating Oil Futures |
Canola | Canola Futures |  | NatGas | UK Natural Gas Futures |
CCI | Consumer Confidence Index Futures |  | PeakElec | UK Peak Electricity Futures |
Cocoa | Cocoa Futures |  | RBCoal | Richards Bay Coal Futures |
Cotton | Cotton No. 2 Futures |  | RCoal | Rotterdam Coal Futures |
FCOJA | FCOJ-A Futures | Florida orange juice | RJ/CBR | Thomson Reuters/Jefferies CRB Index | assorted commodities
Electric | UK Base Electricity Futures |  | Sugar | Sugar No. 11 Futures |
Coffee | Coffee “C” Futures |  | WTI | West Texas Intermediate | Texas crude oil
GasOil | Gas Oil Futures |  | EUA | EU emission allowance |

Additionally, the currency futures in the form A/B refers to the value of
currency A in units of currency B. For example, USD/EUR would be the value of
US dollars in units of Euros. The currency futures investigated are:

Symbol | Currency | Traded in units of
---|---|---
AUD | Australian dollar | CAD, JPY, NZD, USD
CAD | Canadian dollar | JPY
EUR | Euro | AUD, CAD, GBP, JPY
GDP | British pound | AUD, CAD, JPY, NOK, NZD, ZAR, SEK
JPY | Japanese yen |
NOK | Norwegian krone | JPY, SEK
NZD | New Zealand dollar | JPY, USD
SEK | Swedish krona | JPY
ZAR | South African rand |

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Figure 1:  The cross-correlation function $C(n)$ of volatility (a) and return
(b) time series between the DJIA and FTSE100. Both show statistically
significant correlation coefficients at their maxima near time lag=0 (dotted
curve). Solid lines show the LOWESS (locally weighted scatter plot smoothing)
values of $C(n)$, smoothed over a span of 30 days.
Figure 2: The MST obtained from the absolute correlation coefficients
$|\rho_{ij}|$ of the set of 68 return time series during in individual
calendar years (a) 2007 (b) 2008 (c) 2009 (d) 2010. Red indicates commodity
futures, blue indicates currency futures, and green represents stock market
indicators. See Appendix for a listing of symbols and their meanings.
Figure 3: The minimal-spanning tree (MST), similar to Fig. 2, but for the
longer time span January 2007 to September 2011. See Appendix for symbol
definitions.
Figure 4: Cross-correlation function $C(n)$ of volatility daily time series
between 5 main stock market indices and (a) EUA (European carbon emissions
permits), or (b) WTI (light sweet crude oil). Red lines indicate the locally
weighted scatter plot smoothing values (LOWESS) of $C(n)$. Graphs show
systemic time shift for the highest cross-correlation value. This time shift
is observed in most stock markets. (c) indicates the average time-lag between
EUA, WTI, and 28 stock market indicators, with error bars showing the standard
deviations. (d) indicates the average time-lag between the DJIA and other 27
stock market indicators, with error bars showing the standard deviations. The
time lags (in days) are calculated from the time lag $n$ of highest LOWESS
values.