# An Infinitary Generalization of the Brauer-Schur Theorem

Shahram Mohsenipour  School of Mathematics, Institute for Research in
Fundamental Sciences (IPM) P. O. Box 19395-5746, Tehran, Iran<EMAIL_ADDRESS>
###### Abstract.

We prove an infinitary version of the Brauer-Schur theorem.

###### 2010 Mathematics Subject Classification:

05D10

##  1\. Introduction

In [1], the following infinitary generalization of van der Waerden’s theorem
is considered and proved.

Let $2\leq k_{1}\leq k_{2}\leq k_{3}\leq\cdots$ be a sequence of positive
integers. Then, for any finite coloring of $\mathbb{N}$, there exist positive
integers $d_{1},d_{2},d_{3},\dots$ and a color $\gamma$ such that for every
$n\in\mathbb{N}$, there exists a positive integer $a_{n}$ such that the set

$\big{\\{}a_{n}+j_{1}d_{1}+\cdots+j_{n}d_{n}|0\leq j_{1}\leq
k_{1}-1,\dots,0\leq j_{n}\leq k_{n}-1\big{\\}}$

is monochromatic with the color $\gamma$.

In [2], we gave an elementary combinatorial proof. One may naturally ask
whether in the above theorem, we can include the set
$\\{d_{1},d_{2},d_{3},\dots\\}$ in the monochromatic set. For $n=1$, this is
Schur’s extension of van der Waerden’s theorem, which we call it here, the
Brauer-Schur theorem (See Section 2.4 of Prömel’s book [3] for some historical
remarks). For finite case, namely general fixed $n$, this is actually an
special case of Deuber’s $(m,n,p)$ theorem. In [1], we gave a simpler proof by
direct generalization of the proof for the case $n=1$. In this paper, we show
that the infinite case also holds true. More precisely, we prove

Main Theorem. Let $2\leq k_{1}\leq k_{2}\leq k_{3}\leq\cdots$ be a sequence of
positive integers. Then, for any finite coloring of $\mathbb{N}$, there exist
positive integers $d_{1},d_{2},d_{3},\dots$ and a color $\gamma$ such that for
every $n\in\mathbb{N}$, there exists a positive integer $a_{n}$ such that the
sets

$\big{\\{}a_{n}+j_{1}d_{1}+\cdots+j_{n}d_{n}|0\leq j_{1}\leq
k_{1}-1,\dots,0\leq j_{n}\leq k_{n}-1\big{\\}}$

as well as

$\\{d_{1},d_{2},d_{3},\dots\\}$

are all monochromatic with the color $\gamma$.

Our proof is elementary and combinatorial and consists of two steps. In
Section 2, we first generalize our previous theorem from [2] (Theorem 3.3) and
then in Section 3, we prove Theorem 3.1, from which our Main Theorem can
easily be derived by decreasing induction on the number of colors. Now, we
explain some notation and conventions. $\mathbb{N}$ denotes the set of
positive integers. Let $c$ be a positive integer, by a $c$-coloring, we mean a
coloring with $c$ many colors and furthermore, the set of the colors is
$[c]=\\{1,\dots,c\\}$. If $b$ is a non-negative integer and $A$ is a set of
integers, then $\mathbb{T}^{b}A$ will denote the set $\\{a+b|a\in A\\}$. For
positive integers $k\geq 2$, and $c\geq 1$, $\textbf{W}(k,c)$ will denote the
van der Waerden number which is the least integer $n$ such that for any
$c$-coloring of $[n]$, there exists a monochromatic arithmetic progression of
length $k$. For two finite subsets $A,B$ of $\mathbb{N}$, by $A<B$, we mean
that every element of $A$ is less than every element of $B$. If
$A=\\{a_{1},a_{2},\dots,a_{n}\\}$ with $a_{1}<a_{2}<\cdots<a_{n}$, then we may
denote $a_{t}$ by $A[t]$. Also we use the abbreviation AP for the term
“arithmetic progression”.

##  2\. Generalized Infinitary van der Waerden’s Theorem

The following proposition is in fact a generalization of Proposition 3.1 in
[2].

######  Proposition 2.1.

Let $a\in\mathbb{N}$, $c\geq 1$, $n\geq 1$, $2\leq k_{1}\leq
k_{2}\leq\cdots\leq k_{n}$ and consider the following sequence $\langle
W_{i}|i=0,\dots,n\rangle$ of the van der Waerden numbers with $W_{0}=1$ and
$W_{i}={\bf W}(k_{i},c^{W_{1}\cdots W_{i-1}})$ for $i=1,\dots,n$. Let
$l_{1},\dots,l_{n}$ be positive integers with $l_{1}>0$ and for $i=2,\dots,n$,
we have $l_{i}>(W_{1}-1)l_{1}+\cdots+(W_{i-1}-1)l_{i-1}$. Let

$A_{n}=\big{\\{}a+x_{1}l_{1}+\cdots+x_{n}l_{n}|0\leq x_{1}\leq
W_{1}-1,\dots,0\leq x_{n}\leq W_{n}-1\big{\\}}.$

Then, for any $c$-coloring of $A_{n}$, there exist positive integers $a_{n}$
and $d_{1},\dots,d_{n}$ with $d_{i}=\alpha_{i}l_{i}$ for some positive integer
$\alpha_{i}$ with $1\leq\alpha_{i}\leq\lfloor\frac{W_{i}-1}{k_{i}-1}\rfloor$,
for $i=1,\dots,n$, such that the following set

$P_{n}:=\big{\\{}a_{n}+x_{1}d_{1}+\cdots+x_{n}d_{n}|0\leq x_{1}\leq
k_{1}-1,\dots,0\leq x_{n}\leq k_{n}-1\big{\\}}$

is a subset of $A_{n}$ and is monochromatic.

###### Proof.

Before going into the proof, we make some simple observations. Define the
finite subsets $A_{0},A_{1},\dots,A_{n-1}$ of $A_{n}$ with $A_{0}=\\{a\\}$ and
for $i=1,\dots,n-1$, we have

$A_{i}=\big{\\{}a+x_{1}l_{1}+\cdots+x_{i}l_{i}|0\leq x_{1}\leq
W_{1}-1,\dots,0\leq x_{i}\leq W_{i}-1\big{\\}}.$

First observe that for $i=1,\dots,n$, we can represent $A_{i}$ as

$A_{i}=A_{i-1}\cup\mathbb{T}^{l_{i}}A_{i-1}\cup\mathbb{T}^{2l_{i}}A_{i-1}\cup\cdots\cup\mathbb{T}^{(W_{i}-1)l_{i}}A_{i-1}.$

Also note that the condition $l_{1}>0$ implies that for
$A_{1}=\\{a,a+l_{1},\dots,a+(W_{1}-1)l_{1}\\}$, we have
$a<a+l_{1}<\cdots<a+(W_{1}-1)l_{1}$, so $|A_{1}|=W_{1}$. Also it is easily
seen that for $i=1\dots,n$, we have $\min A_{i}=a$ and $\max
A_{i}=a+(W_{1}-1)l_{1}+\cdots+(W_{i}-1)l_{i}$, so
$\min\mathbb{T}^{l_{i+1}}A_{i}=a+l_{i+1}$ and the condition
$l_{i+1}>(W_{1}-1)l_{1}+\cdots+(W_{i}-1)l_{i}$, implies
$A_{i}<\mathbb{T}^{l_{i+1}}A_{i}$ which is preserved under any translation by
$\mathbb{T}^{l_{i+1}}$, thus, for $i=1,\dots,n$, we get

$A_{i-1}<\mathbb{T}^{l_{i}}A_{i-1}<\mathbb{T}^{2l_{i}}A_{i-1}<\cdots<\mathbb{T}^{(W_{i}-1)l_{i}}A_{i-1},$

from which it is easily deduced that $|A_{i}|=W_{1}\cdots W_{i}$. Now, we
start the proof which is by induction. First, let $n=1$. We have
$A_{1}=\\{a,a+l_{1},\dots,a+(W_{1}-1)l_{1}\\}$ with $l_{1}>0$. Let $\chi\colon
A_{1}\rightarrow[c]$ be a $c$-coloring. We define a corresponding $c$-coloring
$\chi^{{}^{\prime}}\colon\\{0,1,\dots,W_{1}-1\\}\rightarrow[c]$ as follows.
For $0\leq i<j\leq W_{1}-1$, we set
$\chi^{{}^{\prime}}(i)=\chi^{{}^{\prime}}(j)$ iff
$\chi(a+il_{1})=\chi(a+jl_{1})$. Recall the definition of $W_{1}={\bf
W}(k_{1},c^{W_{0}})={\bf W}(k_{1},c)$, so by van der Waerden’s theorem, there
exist $\chi^{{}^{\prime}}$-monochromatic AP of length $k_{1}$ in
$\\{0,1,\dots,W_{1}-1\\}$, say, $b_{1}<\cdots<b_{k_{1}}$. This means that
$a+b_{1}l_{1},a+b_{2}l_{1}\dots,a+b_{k_{1}}l_{1}$ have the same $\chi$-colors.
Let $d^{*}$ be the common difference
$d^{*}:=b_{2}-b_{1}=\cdots=b_{k_{1}}-b_{k_{1}-1}$. Obviously $1\leq
d^{*}\leq\lfloor\frac{W_{1}-1}{k_{1}-1}\rfloor$ and we can rewrite the above
$\chi$-monochromatic sequence as
$a+b_{1}l_{1},a+b_{1}l_{1}+d^{*}l_{1},\ldots,a+b_{1}l_{1}+(k_{1}-1)d^{*}l_{1}$.
Now, put $a_{1}:=a+b_{1}l_{1}$, $d_{1}:=d^{*}l_{1}$ and $\alpha_{1}=d^{*}$.
So, the case $n=1$ is done.

Now, suppose that $n\geq 1$ and the assertion is true for $n$, we prove it for
$n+1$. Let $\chi\colon A_{n+1}\rightarrow[c]$ be a $c$-coloring. Recall that

$A_{n+1}=A_{n}\cup\mathbb{T}^{l_{n+1}}A_{n}\cup\mathbb{T}^{2l_{n+1}}A_{n}\cup\cdots\cup\mathbb{T}^{(W_{n+1}-1)l_{n+1}}A_{n}$

as well as

$A_{n}<\mathbb{T}^{l_{n+1}}A_{n}<\mathbb{T}^{2l_{n+1}}A_{n}<\cdots<\mathbb{T}^{(W_{n+1}-1)l_{n+1}}A_{n},$

as well as

$|A_{n}|=|\mathbb{T}^{l_{n+1}}A_{n}|=|\mathbb{T}^{2l_{n+1}}A_{n}|=\cdots=|\mathbb{T}^{(W_{n+1}-1)l_{n+1}}A_{n}|=W_{1}\cdots
W_{n}.$

We define the corresponding coloring
$\chi^{{}^{\prime}}\colon\\{0,1,\dots,W_{n+1}-1\\}\rightarrow c^{W_{1}\cdots
W_{n}}$ as follows. For $0\leq i<j\leq W_{n+1}-1$, we set
$\chi^{{}^{\prime}}(i)=\chi^{{}^{\prime}}(j)$ iff for every $1\leq t\leq
W_{1}\cdots W_{n}$, we have

$\chi(\mathbb{T}^{il_{n+1}}A_{n}[t])=\chi(\mathbb{T}^{jl_{n+1}}A_{n}[t]).$

Having $W_{n+1}={\bf W}(k_{n+1},c^{W_{1}\cdots W_{n}})$, we can use van der
Waerden’s theorem to get a $\chi^{{}^{\prime}}$-monochromatic AP of length
$k_{n+1}$ in $\\{0,1,\dots,W_{n+1}-1\\}$, which we denote it by
$b_{1}<b_{2}<\cdots<b_{k_{n+1}}$. Let $d^{*}$ be the common difference
$d^{*}:=b_{2}-b_{1}=\cdots=b_{k_{n+1}}-b_{k_{n+1}-1}$. This means that for
$1\leq i<j\leq k_{n+1}$ and $1\leq t\leq W_{1}\cdots W_{n}$, we have
$\chi(\mathbb{T}^{b_{i}l_{n+1}}A_{n}[t])=\chi(\mathbb{T}^{b_{j}l_{n+1}}A_{n}[t])$.
By a simple computation, for $i=1,\dots,k_{n+1}$, we have

$\mathbb{T}^{b_{i}l_{n+1}}A_{n}[t]=\mathbb{T}^{b_{1}l_{n+1}}A_{n}[t]+(b_{i}-b_{1})l_{n+1}=\mathbb{T}^{b_{1}l_{n+1}}A_{n}[t]+(i-1)d^{*}l_{n+1}.$

Thus, we obtain that for every $1\leq t\leq W_{1}\cdots W_{n}$ and every
$0\leq x_{n+1}\leq k_{n+1}-1$, we have

$\chi(\mathbb{T}^{b_{i}l_{n+1}}A_{n}[t])=\chi(\mathbb{T}^{b_{1}l_{n+1}}A_{n}[t]+x_{n+1}d^{*}l_{n+1}).$

Now, we turn our attention to $\mathbb{T}^{b_{1}l_{n+1}}A_{n}$. In fact,
$\mathbb{T}^{b_{1}l_{n+1}}A_{n}$ is represented as

$\big{\\{}(b_{1}l_{n+1}+a)+x_{1}d_{1}+\cdots+x_{n}d_{n}|0\leq x_{1}\leq
W_{1}-1,\dots,0\leq x_{n}\leq W_{n}-1\big{\\}},$

therefore, we can use the induction hypothesis for it, namely, there exist
$a_{n},d_{1},\dots,d_{n}$ with $d_{i}=\alpha_{i}l_{i}$ and
$1\leq\alpha_{i}\leq\lfloor\frac{W_{i}-1}{k_{i}-1}\rfloor$ for $i=1,\dots,n$,
such that the set

$\big{\\{}a_{n}+x_{1}d_{1}+\cdots+x_{n}d_{n}|0\leq x_{1}\leq
k_{1}-1,\dots,0\leq x_{n}\leq k_{n}-1\big{\\}}$

is $\chi$-monochromatic and is included in $\mathbb{T}^{b_{1}l_{n+1}}A_{n}$.
Now, we set $d_{n+1}=d^{*}l_{n+1}$ and define $P_{n+1}$ as

$\big{\\{}a_{n}+x_{1}d_{1}+\cdots+x_{n+1}d_{n+1}|0\leq x_{1}\leq
k_{1}-1,\dots,0\leq x_{n+1}\leq k_{n+1}-1\big{\\}},$

which is $\chi$-monochromatic and is included in

$\mathbb{T}^{b_{1}l_{n+1}}A_{n}\cup\mathbb{T}^{b_{2}l_{n+1}}A_{n}\cup\cdots\cup\mathbb{T}^{b_{k_{n+1}}l_{n+1}}A_{n}$

which is included in $A_{n+1}$. Also it is clear that $1\leq
d^{*}\leq\lfloor\frac{W_{n+1}-1}{k_{n+1}-1}\rfloor$, thus, we set
$d^{*}=\alpha_{n+1}$. This completes the proof. ∎

Now, we are ready to state and prove our generalized infinitary van der
Waerden’s theorem.

######  Theorem 2.2.

Let $a\in\mathbb{N}$, $c\geq 1$, and let $\langle k_{i},i\in\mathbb{N}\rangle$
be a sequence of positive integers with $2\leq k_{1}\leq k_{2}\leq\cdots$ and
consider the following sequence $\langle
W_{i}|i\in\mathbb{N}\cup\\{0\\}\rangle$ of the van der Waerden numbers with
$W_{0}=1$ and $W_{i}={\bf W}(k_{i},c^{W_{1}\cdots W_{i-1}})$ for
$i\in\mathbb{N}$. Let $l_{1},l_{2},\dots$ be positive integers with $l_{1}>0$
and for $i\geq 2$, we have $l_{i}>(W_{1}-1)l_{1}+\cdots+(W_{i-1}-1)l_{i-1}$.
Define the finite subsets $A_{0},A_{1},A_{2}\dots$, in such a way that
$A_{0}=\\{a\\}$ and for $i\in\mathbb{N}$, we have

$A_{i}=\big{\\{}a+x_{1}l_{1}+\cdots+x_{i}l_{i}|0\leq x_{1}\leq
W_{1}-1,\dots,0\leq x_{i}\leq W_{i}-1\big{\\}}.$

Then, for any $c$-coloring of $A=\bigcup_{i\in\mathbb{N}}A_{i}$, there exist
positive integers $a_{i},d_{i},i\in\mathbb{N}$ with $d_{i}=\alpha_{i}l_{i}$
for some positive integer $\alpha_{i}$ with
$1\leq\alpha_{i}\leq\lfloor\frac{W_{i}-1}{k_{i}-1}\rfloor$, such that the
following subsets

$\big{\\{}a_{i}+x_{1}d_{1}+\cdots+x_{i}d_{i}|0\leq x_{1}\leq
k_{1}-1,\dots,0\leq x_{i}\leq k_{i}-1\big{\\}}$

are all monochromatic with the same color.

###### Proof.

Let $\chi\colon A\rightarrow[c]$ be a $c$-coloring. By Proposition 2.1, for
each positive integer $m\in\mathbb{N}$, there exist $b_{m}$ and
$d_{1}^{m},d_{2}^{m},\dots,d_{m}^{m}$ such that

$P_{m}=\big{\\{}b_{m}+x_{1}d_{1}^{m}+\cdots+x_{m}d_{m}^{m}|0\leq x_{1}\leq
k_{1}-1,\dots,0\leq x_{m}\leq k_{m}-1\big{\\}}$

is $\chi$-monochromatic and for $i=1,\dots,m$, we have
$d_{i}^{m}=\alpha_{i}^{m}l_{m}$ for some positive integer $\alpha_{i}^{m}$
with $1\leq\alpha_{i}^{m}\leq\lfloor\frac{W_{m}-1}{k_{m}-1}\rfloor$. This
means that for any positive integer $i$, all
$d_{i}^{i},d_{i}^{i+1},d_{i}^{i+2},d_{i}^{i+3},\dots$ are bounded by
$(W_{i}-1)l_{i}$. Now by a simple iterated use of pigeonhole argument, we
obtain a color $\gamma\in[c]$ and a strictly increasing sequence of positive
integers $n_{1}<n_{2}<n_{3}<n_{4}<\cdots$ such that
$P_{n_{1}},P_{n_{2}},P_{n_{3}},P_{n_{4}},\dots$ are all monochromatic with
color $\gamma$ and also we have

$\displaystyle d_{1}^{n_{1}}=d_{1}^{n_{2}}=d_{1}^{n_{3}}=d_{1}^{n_{4}}=$
$\displaystyle\cdots$ $\displaystyle
d_{2}^{n_{2}}=d_{2}^{n_{3}}=d_{2}^{n_{4}}=$ $\displaystyle\cdots$
$\displaystyle d_{3}^{n_{3}}=d_{3}^{n_{4}}=$ $\displaystyle\cdots$
$\displaystyle d_{4}^{n_{4}}=$ $\displaystyle\cdots$ $\displaystyle\cdots$

Now, for $i\in\mathbb{N}$, we put $a_{i}:=b_{n_{i}}$ and
$d_{i}:=d_{i}^{n_{i}}$, which finishes the proof. ∎

We now introduce a very useful notation. Let for $i\in\mathbb{N}$,
$a_{i},d_{i}$ be positive integers. Suppose also that for positive integers
$k_{i},i\in\mathbb{N}$, we have $2\leq k_{1}\leq k_{2}\leq\dots$. For
$i\in\mathbb{N}$, set

$A_{i}=\big{\\{}a_{i}+x_{1}d_{1}+\cdots+x_{i}d_{i}|0\leq x_{1}\leq
k_{1}-1,\dots,0\leq x_{i}\leq k_{i}-1\big{\\}},$

and set $A=\bigcup_{i\in\mathbb{N}}A_{i}$. We denote $A$ as a triple of
sequences, from which $A$ and $A_{i}$ can be reproduced in the obvious way:

$\begin{cases}A^{1}=\langle a_{1},a_{2},a_{3},\dots\rangle,\\\ A^{2}=\langle
d_{1},d_{2},d_{3},\dots\rangle,\\\ A^{3}=\langle
k_{1},k_{2},k_{3},\dots\rangle.\end{cases}$

Then, we can reformulate Theorem 2.2 as follows.

Theorem 2.2. Let $a,c,k_{i},W_{i},l_{i},i\in\mathbb{N}$, satisfying the
conditions of Theorem 2.2. Suppose that $\chi\colon A\rightarrow[c]$ is a
$c$-coloring, where $A$ is given by

$\begin{cases}A^{1}=\langle a,a,a,\dots\rangle,\\\ A^{2}=\langle
l_{1},l_{2},l_{3},\dots\rangle,\\\ A^{3}=\langle
W_{1},W_{2},W_{3},\dots\rangle.\end{cases}$

Then, for $i\in\mathbb{N}$, there exist positive integers $a_{i},d_{i}$, with
$d_{i}=\alpha_{i}l_{i}$ for some positive integer $\alpha_{i}$ with
$1\leq\alpha_{i}\leq\lfloor\frac{W_{i}-1}{k_{i}-1}\rfloor$ such that
$B\subseteq A$ and $B$ is $\chi$-monochromatic, where $B$ is given by the
triplet

$\begin{cases}B^{1}=\langle a_{1},a_{2},a_{3},\dots\rangle,\\\ B^{2}=\langle
d_{1},d_{2},d_{3},\dots\rangle,\\\ B^{3}=\langle
k_{1},k_{2},k_{3},\dots\rangle.\end{cases}$

##  3\. Infinitary Brauer-Schur Theorem

The proof of Main Theorem, as in the case of the classical Brauer-Schur
theorem and also our previous finite generalization of it, proceeds by
induction on the number of colors. In fact, the following theorem allows us to
do a decreasing induction on the number of colors. We leave it to the reader
to observe that Main Theorem is easily implied.

######  Theorem 3.1.

Let $c\geq 1$, and $a\in\mathbb{N}$ and let $\langle
k_{i},i\in\mathbb{N}\rangle$ be an unbounded sequence of positive integers
with $2\leq k_{1}\leq k_{2}\leq\cdots$ and consider the following sequence
$\langle W_{i}|i\in\mathbb{N}\cup\\{0\\}\rangle$ of the van der Waerden
numbers according to the square sequence $k_{1}^{2}\leq k_{2}^{2}\leq\cdots$,
where $W_{0}=1$ and $W_{i}={\bf W}(k_{i}^{2},c^{W_{1}\cdots W_{i-1}})$ for
$i\in\mathbb{N}$. Suppose that $l_{1},\dots,l_{n},\dots$ are positive integers
with $l_{1}>0$ and for $i\geq 2$, we have
$l_{i}>(W_{1}-1)l_{1}+\cdots+(W_{i-1}-1)l_{i-1}$. Also Suppose that
$\chi\colon A\cup B\rightarrow[c]$ is a $c$-coloring, where $A,B$ are given by

$\begin{cases}A^{1}=\langle a,a,a,\dots\rangle,\\\ A^{2}=\langle
l_{1},l_{2},l_{3},\dots\rangle,\\\ A^{3}=\langle
W_{1},W_{2},W_{3},\dots\rangle,\end{cases}\begin{cases}B^{1}=\langle
0,0,0,\dots\rangle,\\\ B^{2}=\langle l_{1},l_{2},l_{3},\dots\rangle,\\\
B^{3}=\langle W_{1},W_{2},W_{3},\dots\rangle.\end{cases}$

Then, $\underline{either}$ there exist positive integers
$a_{i},d_{i},i\in\mathbb{N}$ such that $C\subseteq A$,
$\\{d_{i}|i\in\mathbb{N}\\}\subseteq B$ and $C\cup\\{d_{i}|i\in\mathbb{N}\\}$
is $\chi$-monochromatic with a color, say $c$, where $C$ is given by

$\begin{cases}C^{1}=\langle a_{1},a_{2},a_{3},\dots\rangle,\\\ C^{2}=\langle
d_{1},d_{2},d_{3},\dots\rangle,\\\ C^{3}=\langle
k_{1},k_{2},k_{3},\dots\rangle,\end{cases}$

$\underline{or}$ there exist positive integers $b,l_{i}^{*},i\in\mathbb{N}$
with $l_{1}^{*}>0$ and for $i\geq 2$, we have

$l_{i}^{*}>(W_{1}-1)l_{1}^{*}+\cdots+(W_{i-1}-1)l_{i-1}^{*},$

such that for $D,E$ defined below

$\begin{cases}D^{1}=\langle b,b,b,\dots\rangle,\\\ D^{2}=\langle
l_{1}^{*},l_{2}^{*},l_{3}^{*},\dots\rangle,\\\ D^{3}=\langle
W_{1},W_{2},W_{3},\dots\rangle,\end{cases}\begin{cases}E^{1}=\langle
0,0,0,\dots\rangle,\\\ E^{2}=\langle
l_{1}^{*},l_{2}^{*},l_{3}^{*},\dots\rangle,\\\ E^{3}=\langle
W_{1},W_{2},W_{3},\dots\rangle,\end{cases}$

we have $D\cup E\subseteq B$ and the range of the restriction of $\chi$ to
$D\cup E$ is a subset of $[c-1]$.

###### Proof.

For $c=1$, the theorem is trivial. We assume $c\geq 2$. By Theorem 2.2 (the
reformulated version), there exist positive integers $b_{1},b_{2},b_{3},\dots$
and $d_{1}^{*},d_{2}^{*},d_{3}^{*},\dots$ with $C_{1}\subseteq A$ and $C_{1}$
is $\chi$-monochromatic with a color, say $c$, where $C_{1}$ is given by the
triplet

$\begin{cases}C_{1}^{1}=\langle b_{1},b_{2},b_{3},\dots\rangle,\\\
C_{1}^{2}=\langle d_{1}^{*},d_{2}^{*},d_{3}^{*},\dots\rangle,\\\
C_{1}^{3}=\langle k_{1}^{2},k_{2}^{2},k_{3}^{2},\dots\rangle.\end{cases}$

Note that for $i\in\mathbb{N}$, we have $d_{i}^{*}=\alpha_{i}l_{i}$ with
$1\leq\alpha_{i}\leq\lfloor\frac{W_{i}-1}{k_{i}-1}\rfloor$. Hence, for $C_{2}$
given by

$\begin{cases}C_{2}^{1}=\langle 0,0,0,\dots\rangle,\\\ C_{2}^{2}=\langle
d_{1}^{*},d_{2}^{*},d_{3}^{*},\dots\rangle,\\\ C_{2}^{3}=\langle
k_{1}^{2},k_{2}^{2},k_{3}^{2},\dots\rangle,\end{cases}$

we also have $C_{2}\subseteq B$ and therefore, all of its elements receive a
$\chi$-color.

There are two cases corresponding to the two cases of the theorem:

Case 1: There exist infinitely many non-empty finite subsets
$I_{1}<I_{2}<I_{3}<\cdots$ of $\mathbb{N}$, with the following property. For
each $i\in\bigcup_{j\in\mathbb{N}}I_{j}$, there exists $x_{i}^{*}$ with
$0<x_{i}^{*}\leq k_{i}-1$ such that

$\chi\big{(}\sum_{i\in
I_{1}}x_{i}^{*}d_{i}^{*}\big{)}=c,\,\,\,\chi\big{(}\sum_{i\in
I_{2}}x_{i}^{*}d_{i}^{*}\big{)}=c,\,\,\,\chi\big{(}\sum_{i\in
I_{3}}x_{i}^{*}d_{i}^{*}\big{)}=c,\,\,\,\ldots.$

Notice that this is well-defined, since all $\sum_{i\in
I_{1}}x_{i}^{*}d_{i}^{*}$, $\sum_{i\in I_{2}}x_{i}^{*}d_{i}^{*}$, $\sum_{i\in
I_{3}}x_{i}^{*}d_{i}^{*}$, etc, are elements of $C_{2}$ and therefore receive
a $\chi$-color. Now, we define the desired $d_{i},i\in\mathbb{N}$ as follows.

$d_{1}=\sum_{i\in I_{1}}x_{i}^{*}d_{i}^{*},\,\,\,d_{2}=\sum_{i\in
I_{2}}x_{i}^{*}d_{i}^{*},\,\,\,d_{3}=\sum_{i\in
I_{3}}x_{i}^{*}d_{i}^{*},\,\,\,\ldots.$

For $i\in\mathbb{N}$, we put $\min I_{i}=n_{i}$ and $\max I_{i}=m_{i}$.
Obviously for each $t\in\mathbb{N}$, we have $n_{t}\geq t$ and therefore, for
each $i\in I_{t}$, we have $t\leq n_{t}\leq i$. Now, we claim that for any
integers $x_{1},\dots,x_{t}$ with $0\leq x_{1}\leq k_{1}-1$,$\dots$, $0\leq
x_{t}\leq k_{t}-1$, there are integers $y_{i}$ for $i\in I_{1}\cup\cdots\cup
I_{t}$, with $0\leq y_{i}\leq k_{i}^{2}-1$ such that

$\sum_{i=1}^{t}x_{i}d_{i}\,\,\,\,=\sum_{i\in I_{1}\cup\cdots\cup
I_{t}}y_{i}d_{i}^{*}.$

To see this, we have that

$\displaystyle\sum_{i=1}^{t}x_{i}d_{i}$ $\displaystyle=$
$\displaystyle\sum_{i=1}^{t}x_{i}(\sum_{j\in
I_{i}}x_{j}^{*}d_{j}^{*})=\sum_{i=1}^{t}\sum_{j\in
I_{i}}(x_{i}x_{j}^{*})d_{j}^{*}$ $\displaystyle=$ $\displaystyle\sum_{j\in
I_{1}}(x_{1}x_{j}^{*})d_{j}^{*}+\sum_{j\in
I_{2}}(x_{2}x_{j}^{*})d_{j}^{*}+\cdots+\sum_{j\in
I_{t}}(x_{t}x_{j}^{*})d_{j}^{*}.$

Note that for $1\leq i\leq t$ and $j\in I_{i}$, we have $j\geq i$, thus the
following inequalities hold.

$\forall j\in I_{1}:0\leq
x_{1}x_{j}^{*}\leq(k_{1}-1)(k_{j}-1)\leq(k_{j}^{2}-1),$ $\forall j\in
I_{2}:0\leq x_{2}x_{j}^{*}\leq(k_{2}-1)(k_{j}-1)\leq(k_{j}^{2}-1),$
$\dots\dots\dots$ $\forall j\in I_{t}:0\leq
x_{t}x_{j}^{*}\leq(k_{t}-1)(k_{j}-1)\leq(k_{j}^{2}-1).$

Now, for $j\in I_{1}$, we put $y_{j}=x_{1}x_{j}^{*}$ and for $j\in I_{2}$, we
put $y_{j}=x_{2}x_{j}^{*}$, $\dots$, and for $j\in I_{t}$, we put
$y_{j}=x_{t}x_{j}^{*}$. This proves the claim. Now, from $I_{1}<I_{2}<\cdots
I_{t}$, we infer that $m_{t}=\max I_{t}$ is the maximum element of
$I_{1}\cup\cdots\cup I_{t}$, so according to the claim, we have the following
inclusions (recall $b_{1},b_{2},\dots$):

$\big{\\{}b_{m_{t}}+x_{1}d_{1}+\cdots+x_{t}d_{t}|0\leq x_{1}\leq
k_{1}-1,\dots,0\leq x_{t}\leq k_{t}-1\big{\\}}$
$\subseteq\big{\\{}b_{m_{t}}+\sum_{i\in I_{1}\cup\cdots\cup
I_{t}}y_{i}d_{i}^{*}|0\leq y_{i}\leq k_{i}^{2}-1\big{\\}}$
$\subseteq\big{\\{}b_{m_{t}}+z_{1}d_{1}^{*}+\cdots+z_{m_{t}}d_{m_{t}}^{*}|0\leq
z_{1}\leq k_{1}^{2}-1,\dots,0\leq z_{m_{t}}\leq k_{m_{t}}^{2}-1\big{\\}}$
$\subseteq(C_{1})_{m_{t}}.$

Now, put $a_{1}=b_{m_{1}}$,$a_{2}=b_{m_{2}}$,$a_{3}=b_{m_{3}}$ and therefore,
for $C$ given by

$\begin{cases}C^{1}=\langle a_{1},a_{2},a_{3},\dots\rangle=\langle
b_{m_{1}},b_{m_{2}},b_{m_{3}},\dots\rangle,\\\ C^{2}=\langle
d_{1},d_{2},d_{3},\dots\rangle,\\\ C^{3}=\langle
k_{1},k_{2},k_{3},\dots\rangle,\end{cases}$

we get that $C\subseteq C_{1}$ as well as $C\cup\\{d_{1},d_{2},\dots\\}$ is
$\chi$-monochromatic with the color $c$ and consequently, we have proved the
theorem in Case 1.

Case 2: Suppose that Case 1, doesn’t occur. This means that there is
$p\in\mathbb{N}$, such that all elements of the following subset $G$ of
$C_{2}$, get their $\chi$-colors from $\\{1,\dots,c-1\\}$.

$G=\big{\\{}x_{p}d_{p}^{*}+\cdots+x_{p+t}d_{p+t}^{*}|t\in\mathbb{N},0\leq
x_{p}\leq k_{p}-1,\dots,0\leq x_{p+t}\leq k_{p+t}-1\big{\\}}.$

Now, we pass to the following subsets $D_{1},E_{1}$ of $G$ given by

$\begin{cases}D_{1}^{1}=\langle d_{p}^{*},d_{p}^{*},d_{p}^{*},\dots\rangle\\\
D_{1}^{2}=\langle d_{p+1}^{*},d_{p+2}^{*},d_{p+3}^{*},\dots\rangle,\\\
D_{1}^{3}=\langle
k_{p+1},k_{p+2},k_{p+3},\dots\rangle,\end{cases}\begin{cases}E_{1}^{1}=\langle
0,0,0,\dots\rangle\\\ E_{1}^{2}=\langle
d_{p+1}^{*},d_{p+2}^{*},d_{p+3}^{*},\dots\rangle,\\\ E_{1}^{3}=\langle
k_{p+1},k_{p+2},k_{p+3},\dots\rangle.\end{cases}$

For getting $D,E,$ satisfying the conclusion of the theorem, we first define a
sequence of triples of integers $\\{\langle
t_{i},m_{i},l_{i}^{*}\rangle|i\in\mathbb{N}\\}$ inductively, as follows. Set
$t_{1}=0$ and let $m_{1}$ be the least integer with $m_{1}>p$ and
$k_{m_{1}}\geq W_{1}$. This is possible as the sequence
$k_{1},k_{2},k_{3},\dots$ is unbounded. Now put $l_{1}^{*}=d_{m_{1}}^{*}$ and
let $m_{2}$ be the least integer with $m_{2}>m_{1}+t_{1}=m_{1}$, such that
$k_{m_{2}}\geq W_{2}$ and let $t_{2}$ be the least integer with

$d_{m_{2}}^{*}+d_{m_{2}+1}^{*}+\cdots+d_{m_{2}+t_{2}}^{*}>(W_{1}-1)l_{1}^{*}.$

Then, put
$l_{2}^{*}=d_{m_{2}}^{*}+d_{m_{2}+1}^{*}+\cdots+d_{m_{2}+t_{2}}^{*}$. To
continue, let $m_{3}$ be the least integer with $m_{3}>m_{2}+t_{2}$ such that
$k_{m_{3}}\geq W_{3}$ and let $t_{3}$ be the least integer with

$d_{m_{3}}^{*}+d_{m_{3}+1}^{*}+\cdots+d_{m_{3}+t_{3}}^{*}>(W_{1}-1)l_{1}^{*}+(W_{2}-1)l_{2}^{*}.$

Put $l_{3}^{*}=d_{m_{3}}^{*}+d_{m_{3}+1}^{*}+\cdots+d_{m_{3}+t_{3}}^{*}$ and
so on. Clearly this can be continued and we obtain the sequence $\langle
l_{i}^{*}|i\in\mathbb{N}\rangle$ with $l_{1}^{*}>0$ and for $i\geq 2$, we have

$l_{i}^{*}>(W_{1}-1)l_{1}^{*}+\cdots+(W_{i-1}-1)l_{i-1}^{*}.$

Also for $i\in\mathbb{N}$, we have $k_{m_{i}}\geq W_{i}$. Now, put
$b=d_{p}^{*}$ and observe that the construction of $\\{\langle
t_{i},m_{i},l_{i}^{*}\rangle|i\in\mathbb{N}\\}$, implies (by an easy
computation) that $D_{2}\subseteq D_{1}$ and $E_{2}\subseteq E_{1}$, where
$D_{2},E_{2}$ are given by

$\begin{cases}D_{2}^{1}=\langle b,b,b,\dots\rangle\\\ D_{2}^{2}=\langle
l_{1}^{*},l_{2}^{*},l_{3}^{*},\dots\rangle,\\\ D_{2}^{3}=\langle
k_{m_{1}},k_{m_{2}},k_{m_{3}},\dots\rangle,\end{cases}\begin{cases}E_{2}^{1}=\langle
0,0,0,\dots\rangle\\\ E_{2}^{2}=\langle
l_{1}^{*},l_{2}^{*},l_{3}^{*},,\dots\rangle,\\\ E_{2}^{3}=\langle
k_{m_{1}},k_{m_{2}},k_{m_{3}},\dots\rangle.\end{cases}$

Now, for the final step of the proof, note that in the third sequences
$D_{2}^{3},E_{2}^{3}$, above, we can actually put smaller elements and pass to
subsets. Hence, we get the desired $E,D$ as follows

$\begin{cases}D^{1}=\langle b,b,b,\dots\rangle\\\ D^{2}=\langle
l_{1}^{*},l_{2}^{*},l_{3}^{*},\dots\rangle,\\\ D^{3}=\langle
W_{1},W_{2},W_{3},\dots\rangle,\end{cases}\begin{cases}E^{1}=\langle
0,0,0,\dots\rangle\\\ E^{2}=\langle
l_{1}^{*},l_{2}^{*},l_{3}^{*},,\dots\rangle,\\\ E^{3}=\langle
W_{1},W_{2},W_{3},\dots\rangle,\end{cases}$

which completes the proof of the theorem in Case 2. ∎

### Acknowledgment

The research of the author was in part supported by a grant from IPM (No.
02030403).

## References

  * [1] T. Brown and S. Mohsenipour, _Two extensions of Hilbert’s cube lemma_ , Integers 21A (2021), 9 pp.
  * [2] S. Mohsenipour, _On the infinitary van der Waerden’s theorem_ , arXiv:2106.14383.
  * [3] H. J. Prömel, _Ramsey Theory for Discrete Structures_ , Springer, Cham, 2013.


# The process of fluctuations of the giant component of an Erdős-Rényi graph

Nathanaël Enriquez, Gabriel Faraud, Sophie Lemaire Université Paris-Saclay,
CNRS, Laboratoire de mathématiques d’Orsay, 91405, Orsay, France; DMA, École
normale supérieure, Université PSL, CNRS, 75005 Paris, France; email:
<EMAIL_ADDRESS>Paris Nanterre, CNRS,
Laboratoire MODAL’X, 92000, Nanterre, France; email:
<EMAIL_ADDRESS>Paris-Saclay, CNRS, Laboratoire de
mathématiques d’Orsay, 91405, Orsay, France; email: sophie.lemaire@universite-
paris-saclay.fr.

###### Abstract

We present a detailed study of the evolution of the giant component of the
Erdős-Rényi graph process as the mean degree increases from 1 to infinity. It
leads to the identification of the limiting process of the rescaled
fluctuations of its order around its deterministic asymptotic. This process is
Gaussian with an explicit covariance.

Keywords. random graph process, Erdős-Rényi random graph, invariance
principle.
AMS MSC 2020. Primary 05C80. Secondary 60F17.

##  1 Introduction

The Erdős-Rényi graph $\mathrm{ER}(n,p_{n})$ is, without any contest, the most
popular random graph. The model goes back to Gilbert in [6] and was first
studied by Erdős and Rényi, who presented the main features of a slight
variation of this model in a seminal paper [5]. For an integer $n\geq 1$ and
$p_{n}\in(0,1)$, it is defined as a graph with $n$ vertices, where any pair of
vertices are connected by an edge independently with probability $p_{n}$. The
most striking phenomenon occurs at the scale $p_{n}=\frac{c}{n}$ where this
model exhibits a phase transition at $c=1$. Namely, with high probability, for
$c<1$, all connected components of the graph are of order $O(\ln(n))$, whereas
for $c>1$, a single component is of order of magnitude $n$ and all the others
are of order $O(\ln(n))$. More precisely, the largest component, commonly
called the ‘giant component’ is asymptotically of order $\rho(c)n$ where
$\rho(c)$ denotes the probability that a Galton-Watson tree with Poisson
reproduction distribution of parameter $c$ is infinite. The constant $\rho(c)$
is the only solution in $(0,1)$ of the equation

$1-x=e^{-cx}.$ (1)

Few years later, the question of the fluctuations of this giant component in
the case $c>1$ was raised. Different approaches have been proposed to prove
that the fluctuation of the order of the giant component around $\rho(c)$ is
asymptotically Gaussian with variance
$\dfrac{\rho(c)(1-\rho(c))}{(1-c(1-\rho(c)))^{2}}n$. The first proof was
written by Stepanov [15], who adopted a combinatorial approach and sketched
the proof of a local limit theorem. Later, Pittel [12] obtained the asymptotic
normality of the giant component from the asymptotic normality of the number
of components of any given tree-like shape. Barraez, Boucheron and de la Vega
[1] further established an upper bound for the rate of convergence and a
bivariate result for both the order of the giant component and the number of
edges by analysing an exploration process of the graph. Independently,
Puhalskii in [14] also used an exploration process to study the joint
distribution of three variables: the giant component, the number of excess
edges and the number of connected components. The same year, Pittel and
Wormald in [13] extended the study of the fluctuations of the giant component
for $c=1+\epsilon_{n}$ with $\epsilon_{n}\rightarrow 0$ and
$\epsilon_{n}n^{1/3}\rightarrow+\infty$. Using combinatorial methods, they
showed a trivariate result on the fluctuation of the 2-core that implies that
the asymptotic fluctuation of the giant component is Gaussian with variance
$\frac{2n}{\epsilon_{n}}$. Later, this last result was directly proven by
Bollobas and Riordan in [4] with the help of an exploration process.

All the previously mentioned articles focus on the study of the Erdős-Rényi
graph for a fixed parameter $c$. There is however a very natural construction
that makes it possible to couple the graphs $\mathrm{ER}(n,\frac{c}{n})$ for
different values of $c$ and obtain a graph-valued process. Namely, we can
attach to every pair $(i,j)$ of vertices a weight given by independent uniform
random variables $w_{i,j}$. We consider the graph $\mathscr{G}_{n}(t)$ whose
edges are the pairs of vertices with weight smaller than $\frac{t}{n}$. With
this construction, for any $t$, $\mathscr{G}_{n}(t)$ follows the distribution
of an $\mathrm{ER}(n,\frac{t}{n})$ graph. There are remarkably few results
concerning the dynamics of the features of the graph $\mathscr{G}_{n}(t)$,
seen as a stochastic process. An important reference on these questions are
the works by Janson [7, 8, 9]. In [7, 8], Janson develops a sophisticated
general method based on chaos decomposition of martingales to get functional
limit theorems for random graph statistics. In [9], he uses this method to get
the fluctuations of the infinite family of the number of isolated trees of
order $k$. He conjectures ([9], Remark 1.3) that this approach could be used
to show that the fluctuations of the size of the giant component converge to a
Gaussian process. We are going to prove this result and identify the limiting
process. Our approach however does not rely on the techniques proposed by
Janson; it relies on a careful analysis of the infinitesimal increment of the
giant component and leads to a stochastic differential equation that we can
explicitly solve.

In order to state the precise result, let us introduce some notation.
Throughout the article we let $\mathscr{C}_{n,x}(t)$ define the connected
component of the vertex $x$ in $\mathscr{G}_{n}(t)$ and
$|\mathscr{C}_{n,x}(t)|$ its order, that is its number of vertices. We set
$L_{n}(t)=\max(|\mathscr{C}_{n,x}(t)|,\,x\in\llbracket{n}\rrbracket)$. Among
components of order $L_{n}(t)$, we choose one with a fixed rule (for instance
the component of the first vertex having $L_{n}(t)$ vertices) and denote it
$\mathscr{C}^{max}_{n}(t)$.

We deal with the fluctuation process of $L_{n}(t)$, namely
$X_{n}(t):=\frac{1}{\sqrt{n}}(L_{n}(t)-n\rho(t))$ where the function $\rho$ is
defined by (1). As proven in the above references, it converges in
distribution for a fixed $t>1$, to a Gaussian distribution with variance

$\sigma^{2}(t)=\frac{\rho(t)(1-\rho(t))}{(1-t(1-\rho(t)))^{2}}.$ (2)

Our main result identifies the limit of $X_{n}$ as a process.

######  Theorem 1.1.

Let $\\{B(t),\,t\geq 0\\}$ be a standard Brownian motion on $\mathbb{R}$. Set
for all $t>1$,

$u(t)=\frac{1}{1-\rho(t)}-t\text{ and }v(t)=\frac{1}{1-\rho(t)}-1.$

Let $t_{0},t_{1}$ be two real numbers such that $1<t_{0}<t_{1}$. The process
$\\{X_{n}(t),\;t\in[t_{0},t_{1}]\\}$ converges weakly, with respect to the
Skorohod topology, to the Gaussian process $\\{X(t),\,t\in[t_{0},t_{1}]\\}$
defined by:

$\forall t\in[t_{0},t_{1}],\quad X(t)=\frac{1}{u(t)}B(v(t)).$ (3)

Note that the limiting process is Gaussian and does not have independent
increments. Its variance at time $t$ is $\sigma^{2}(t)$; it explodes at
$1^{+}$ and is bounded away from 0 and $+\infty$ on the interval
$[t_{0},t_{1}]$.

A heuristic of how the giant component evolves during an infinitesimal time
interval is displayed in Section 2 in order to identify the limiting process
$X$. The first tool to prove Theorem 1.1 is a representation of the variation
of the order of the giant component during a small time interval detailed in
Section 3. We show in Section 4 the finite-dimensional convergence of the
sequence $(uX_{n})\circ v^{(-1)}$ towards a standard Brownian motion, based on
an extension of the three-moments method of Norman [10]. Finally, we prove its
tightness in Section 5, which finishes the proof of Theorem 1.1.

##  2 Heuristic

We propose a heuristic to identify the limiting process of $X_{n}$ as a
solution of a stochastic differential equation (SDE). Our main idea is to
understand precisely what happens to the largest component during an
infinitesimal step $h$. Between time $t$ and $t+h$ a random number $N_{n}(t)$
of edges is added to the graph. This number is distributed as a
$\mathrm{Bin}(\frac{1}{2}n(n-1),\frac{h}{n})$ random variable.

Each edge contributes to increasing the order of the giant component between
$t$ and $t+h$ if it connects one vertex of the largest component to a vertex
outside of it. This event has probability

$\zeta_{n}(t):=\frac{2L_{n}(t)(n-L_{n}(t))}{n(n-1)}=2\rho(t)(1-\rho(t))+2(1-2\rho(t))\frac{X_{n}(t)}{\sqrt{n}}+O(\frac{X_{n}(t)^{2}}{n}).$

Whenever this event is fulfilled, the order of the giant component is
increased by the order of the connected component of the ‘new’ vertex. It is
known that the order of such a component is approximately distributed as the
total progeny of a Bienaymé-Galton-Watson (BGW for short) tree with
$\mathrm{Bin}(n-L_{n}(t),\frac{t}{n})$ reproduction distribution. We denote by
$\mu_{n}(t)$ this distribution. The limiting expectation of $\mu_{n}(t)$ is

$\lambda(t):=t(1-\rho(t)).$ (4)

Let us note that $\lambda(t)$ verifies $-\lambda(t)e^{-\lambda(t)}=-te^{-t}$
and thus $-\lambda(t)=W_{0}(-te^{-t})$ where $W_{0}$ is the principal branch
of the Lambert W-function. In particular, $\lambda(1)=1$, the function
$\lambda$ is smaller than 1 and bounded away from 1 on the interval
$[t_{0},t_{1}]$.

Summing over all the added edges between $t$ and $t+h$ we have the following
representation:

$L_{n}(t+h)-L_{n}(t)=\sum_{i=1}^{N_{n}(t)}\epsilon_{i}T_{i},$ (5)

where $\epsilon_{i}$ are Bernoulli$(\zeta_{n}(t))$ random variables that are
equal to 1 when the associated edge connects the giant component to a vertex
outside of it, and the random variables $T_{i}$ are $\mu_{n}(t)$ distributed
and represent the order of components outside of $\mathscr{C}^{max}_{n}(t)$.

The expectation of (5) yields

$\mathbb{E}(N_{n})\mathbb{E}(\epsilon)\mathbb{E}(T)=\frac{nh}{2}\left(\frac{2\rho(t)(1-\rho(t))}{1-\lambda(t)}+2\left(\frac{1-2\rho(t)}{1-\lambda(t)}-\frac{\rho(t)(1-\rho(t))t}{(1-\lambda(t))^{2}}\right)\frac{X_{n}(t)}{\sqrt{n}}\right)\\\
+O\left(ht^{2}X_{n}(t)^{2}\right).$

Note that the first order term $nh\frac{\rho(t)(1-\rho(t))}{1-\lambda(t)}$
coincides with $nh\rho^{\prime}(t)$ and therefore will compensate the
increment of $n\rho(t)$ in the variation of
$L_{n}(t)-n\rho(t)=\sqrt{n}X_{n}(t)$. The second term
$\sqrt{n}h\left(\frac{1-2\rho(t)}{1-\lambda(t)}-\frac{\rho(t)(1-\rho(t))t}{(1-\lambda(t))^{2}}\right){X_{n}(t)}$
will give the drift term in the limiting stochastic differential equation.

We are now interested in the fluctuations in (5). They come from the three
sources of randomness $N_{n}(t),\epsilon_{i},T_{i}$ which will turn out to be
asymptotically independent. Informally we can describe these fluctuations with
the help of 3 standard Brownian motions $C^{(N)}$, $C^{(T)}$ and
$C^{(\epsilon)}$:

  * •

fluctuations of $N_{n}(t)$: $\sqrt{n/2}\mathbb{E}(\epsilon
T)dC^{(N)}(t)=\sqrt{2n}\frac{\rho(t)(1-\rho(t))}{1-\lambda(t)}dC^{(N)}(t)$,

  * •

fluctuations due to $\epsilon_{i}$:

$\sqrt{\mathrm{Var}(\epsilon)\mathbb{E}(N_{n}(t))\mathbb{E}(T)^{2}}dC^{(\epsilon)}(t)=\sqrt{n\rho(t)(1-\rho(t))\frac{1-2\rho(t)(1-\rho(t))}{(1-\lambda(t))^{2}}}dC^{(\epsilon)}(t),$

  * •

fluctuations due to $T_{i}$:

$\sqrt{\mathbb{E}(N_{n}(t)\epsilon)\mathrm{Var}(T)}dC^{(T)}(t)=\sqrt{n\rho(t)(1-\rho(t))\frac{\lambda(t)}{(1-\lambda(t))^{3}}}dC^{(T)}(t).$

Therefore, the limiting equation for $(X_{n}(t))_{t}$ is:

$\displaystyle dX(t)=$
$\displaystyle\left(\frac{1-2\rho(t)}{1-\lambda(t)}-\frac{\rho(t)(1-\rho(t))t}{(1-\lambda(t))^{2}}\right)X(t)dt+\sqrt{2}\frac{\rho(t)(1-\rho(t))}{1-\lambda(t)}dC^{(N)}(t)$
$\displaystyle+\sqrt{\rho(t)(1-\rho(t))\frac{1-2\rho(t)(1-\rho(t))}{(1-\lambda(t))^{2}}}dC^{(\epsilon)}(t)+\sqrt{\rho(t)(1-\rho(t))\frac{\lambda(t)}{(1-\lambda(t))^{3}}}dC^{(T)}(t).$

Assuming the independence of the three above Brownian motions, we can regroup
them under a single Brownian motion $C$ and get

$dX(t)=\left(\frac{1-2\rho(t)}{1-\lambda(t)}-\frac{\rho(t)(1-\rho(t))t}{(1-\lambda(t))^{2}}\right)X(t)dt+\sqrt{\frac{\rho(t)(1-\rho(t))}{(1-\lambda(t))^{3}}}dC(t).$
(6)

Solving this equation gives

$X(t)=\frac{1-\rho(t)}{1-\lambda(t)}B(\frac{\rho(t)}{1-\rho(t)}),$ (7)

where $B$ is a standard Brownian motion.

##  3 A representation of the infinitesimal evolution of the giant component

Let $h>0$. The goal of this section is to give a representation of the
variation of $L_{n}(t)$ between $t$ and $t+h$, in order to treat this
variation as a sum of i.i.d. random variables, up to a tractable error.

First note that, conditional on the situation at time $t$, $L_{n}(t+h)$ only
depends on the edges added between $t$ and $t+h$, and not on the order in
which we add them. Therefore we can add them in a strategic order. Call
$\mathscr{G}_{n}^{e}(t)$ the subgraph induced in $\mathscr{G}_{n}(t)$ by
vertices outside of $\mathscr{C}^{max}_{n}(t)$. We partition the edges added
between $t$ and $t+h$ in three categories:

  * •

the edges which connect vertices of $\mathscr{C}^{max}_{n}(t)$, they do not
affect $L_{n}(t)$ and can be ignored;

  * •

the edges that connect vertices of $\mathscr{G}_{n}^{e}(t)$;

  * •

the edges that connect $\mathscr{C}^{max}_{n}(t)$ to $\mathscr{G}_{n}^{e}(t)$.

Note that there is not a complete independence between the connections in
$\mathscr{C}^{max}_{n}(t)$ and $\mathscr{G}_{n}^{e}(t)$, since the orders of
the connected components of $\mathscr{G}_{n}^{e}(t)$ have to be smaller than
$|\mathscr{C}^{max}_{n}(t)|$. To overcome this difficulty, we introduce two
graphs, denoted $\tilde{\mathscr{G}}_{n}(t)$ and
$\tilde{\mathscr{G}}_{n}(t,h)$, for which the pair of vertices outside the
giant component are connected independently with probability respectively
$\frac{t}{n}$ and $\frac{t+h}{n}$.

First, we construct $\tilde{\mathscr{G}}_{n}(t)$ on $\llbracket{n}\rrbracket$,
having $\mathscr{C}^{max}_{n}(t)$ as a connected component in the following
way:

  * •

every edge between vertices inside $\mathscr{C}^{max}_{n}(t)$ is present in
$\tilde{\mathscr{G}}_{n}(t)$ if and only if it is present in
$\mathscr{G}_{n}(t)$;

  * •

$\tilde{\mathscr{G}}_{n}(t)$ has no edge between a vertex in
$\mathscr{C}^{max}_{n}(t)$ and another outside;

  * •

an edge between two vertices outside $\mathscr{C}^{max}_{n}(t)$, is added with
probability $\frac{t}{n}$ independently from every other edge. We call them
external edges.

Secondly, the graph $\tilde{\mathscr{G}}_{n}(t,h)$ is defined from
$\tilde{\mathscr{G}}_{n}(t)$ by adding independently with probability
$\frac{\frac{h}{n}}{1-\frac{t}{n}}=\frac{h}{n-t}$ all the edges that are not
in $\tilde{\mathscr{G}}_{n}(t)$. The steps of the construction of
$\tilde{\mathscr{G}}_{n}(t,h)$ from $\mathscr{G}_{n}(t)$ are illustrated in
Fig. 5, 5 and 5. Let us call a bridge, an edge in
$\tilde{\mathscr{G}}_{n}(t,h)$ between a vertex in $\mathscr{C}^{max}_{n}(t)$
and another one outside of $\mathscr{C}^{max}_{n}(t)$. Let $Y_{n}$ denote
their number. Let $\tilde{\mathscr{G}}_{n}^{e}(t,h)$ be the subgraph induced
in $\tilde{\mathscr{G}}_{n}(t,h)$ by vertices which are not in
$\mathscr{C}^{max}_{n}(t)$. It is obvious from the construction that
$\tilde{\mathscr{G}}_{n}^{e}(t,h)$ is distributed as an
$\mathrm{ER}(n-L_{n}(t),\frac{t+h}{n})$-random graph and that $Y_{n}$ is a
$\mathrm{Bin}(L_{n}(t)(n-L_{n}(t)),\frac{h}{n-t})$ random variable.

Let $L_{n}(t,h)$ be the order of the component of
$\tilde{\mathscr{G}}_{n}(t,h)$ containing $\mathscr{C}^{max}_{n}(t)$. We will
see that $L_{n}(t,h)$ behaves similarly to $L_{n}(t+h)$.

An analog construction was made in [2], in the setting of random uniform
hypergraphs in order to prove that the order of the giant component follows a
local limit theorem at some fixed time. We are going to use a modified version
of Lemma 10 of [2], namely dealing with estimations that are valid uniformly
on a time interval. This will make it possible to reduce the study of $\Delta
L_{n}(t):=L_{n}(t+h)-L_{n}(t)$ to the study of
$\tilde{\Delta}L_{n}(t):=L_{n}(t,h)-L_{n}(t)$, on the event

$\mathcal{E}_{n}(t):=\\{|L_{n}(t)-n\rho(t)|\leq n^{0.7}\\}=\\{|X_{n}(t)|\leq
n^{0.2}\\}.$

We state it in our framework:

######  Lemma 3.1 (Lemma 10 of [2]).

There is a number $n_{0}>0$ and a constant $C_{1}$ such that, for
$t\in[t_{0},t_{1}],$ for all integers $n>n_{0}$, all integers $n_{1}$ such
that $|n_{1}-\rho(t)n|<n^{0.7},$ and all integers $\nu$ we have

$|\mathbb{P}(\Delta
L_{n}(t)=\nu|L_{n}(t)=n_{1})-\mathbb{P}(\tilde{\Delta}L_{n}(t)=\nu|L_{n}(t)=n_{1})|\leq
C_{1}n^{-10}.$

As a consequence, $|\mathbb{E}[(\Delta L_{n}(t))^{k}\mid
L_{n}(t)=n_{1}]-\mathbb{E}[(\tilde{\Delta}L_{n}(t))^{k}|L_{n}(t)=n_{1}]|\leq
C_{1}n^{k-9}$ for $k\in\\{1,2,3\\}$.

The only difference between this statement and the one presented in [2] is the
fact that the bound is uniform in $t\in[t_{0},t_{1}]$. To get this statement,
the proof in [2] is still valid; the only missing ingredient is a uniform (in
$t$) upper bound for the probability of existence of components of order
larger than $(\ln n)^{2}$ outside of the giant one. We show this bound in the
Appendix (Lemma A.1).

For a fixed time $t>1$, it is known that the complement of
$\mathcal{E}_{n}(t)$ denoted by $\mathcal{E}^{c}_{n}(t)$ appears rarely:
$\mathbb{P}(\mathcal{E}^{c}_{n}(t))=O(\exp(-Cn^{0.4}))$ (see for example A.
Puhalskii [14], Thm 2.3). However we need a bound of
$\mathbb{P}(\mathcal{E}^{c}_{n}(t))$ that is uniform in $t\in[t_{0},t_{1}]$.
To the best of our knowledge such a uniform bound has never been shown so we
shall prove in Lemma A.1 of the Appendix the following uniform estimate:

$\text{Uniformly on
}t\in[t_{0},t_{1}],\quad\mathbb{P}(\mathcal{E}^{c}_{n}(t))=O(\exp(-Cn^{0.4})).$
(8)

The proof relies on some results by V.E. Stepanov [15].

To study $\tilde{\Delta}L_{n}(t)$, we give a description of the edges that
contribute to it. We call $x_{1},x_{2},\ldots,x_{Y_{n}}$ the sequence (in an
arbitrary order) of the endpoints of the bridges in
$\tilde{\mathscr{G}}_{n}(t,h)$ which are outside of
$\mathscr{C}^{max}_{n}(t)$. We denote by $C_{x_{1}},\ldots,C_{x_{Y_{n}}}$
their connected components in $\tilde{\mathscr{G}}_{n}^{e}(t,h)$.

We have to take into account the unlikely but possible event that two bridges
point to the same component of $\tilde{\mathscr{G}}_{n}^{e}(t,h)$. We
introduce the set

$\mathcal{R}:=\\{2\leq j\leq Y_{n}\text{ s.t. }C_{x_{j}}=C_{x_{i}}\text{ for
some }i<j\\}$

and define

$C^{*}_{x_{i}}:=\begin{cases}\emptyset&\text{ if }i\in\mathcal{R}\\\
C_{x_{i}}&\text{ otherwise}.\end{cases}$ (9)

(for the graph depicted in Fig. 5, $\mathcal{R}=\\{3\\}$)

The variables $|C_{x_{i}}|$ are classically compared to the total progeny of a
Bienyaimé-Galton-Watson process with a binomial reproduction distribution. The
purpose of the following lemma is to use this comparison in order to give
properties of $\tilde{\Delta}L_{n}(t)$ that will lead to upper and lower
bounds of the moments of $\tilde{\Delta}L_{n}(t)$ in the next section.

######  Lemma 3.2.

Let $\mathrm{BGW}(n,p)$ denote a Bienaymé-Galton-Watson process with
$\mathrm{Bin}(n,p)$ reproduction distribution. With the previously introduced
notation,

$\tilde{\Delta}L_{n}(t)=\sum_{\begin{subarray}{c}i=1,\\\
i\not\in\mathcal{R}\end{subarray}}^{Y_{n}}|C_{x_{i}}|=\sum_{i=1}^{Y_{n}}|C^{*}_{x_{i}}|.$

  1. 1.

There exists a sequence $(M_{i})_{i\in\mathbb{N}^{*}}$ of i.i.d. random
variables independent of $Y_{n}$ such that

    1. (a)

$\forall 1\leq j\leq
Y_{n},\quad\sum_{i=1}^{j}|C_{x_{i}}^{*}|\leq\sum_{i=1}^{j}M_{i}$,

    2. (b)

the distribution of $M_{i}$ conditional on $L_{n}(t)$ is the same as the total
progeny of a $\mathrm{BGW}(n-L_{n}(t),\frac{t+h}{n})$;

we write accordingly

$\tilde{\Delta}L_{n}(t)\leq\min\left(\sum_{i=1}^{Y_{n}}M_{i},n\right).$

  2. 2.

Moreover,

    1. (a)

the conditional distribution of $Y_{n}$ given $L_{n}(t)$ is
$\mathrm{Bin}(L_{n}(t)(n-L_{n}(t)),\frac{h}{n-t})$;

    2. (b)

the conditional distribution of $|C_{x_{i}}|$ given $(Y_{n},L_{n}(t))$
dominates the conditional distribution of the minimum of $\ln^{2}(n)$ and of
the total progeny of ${\mathrm{BGW}(n-L_{n}(t)-\ln^{2}(n),\frac{t+h}{n})}$;

    3. (c)

Let $\mathcal{A}_{n}(t):=\\{\text{no component of
}\tilde{\mathscr{G}}_{n}^{e}(t,h)\text{ has size greater than }\ln^{2}(n)\\}.$

The conditional distribution of $|\mathcal{R}|\mathds{1}_{\mathcal{A}_{n}(t)}$
given $(Y_{n},L_{n}(t))$ is dominated by the binomial distribution
$\mathrm{Bin}\left(Y_{n},\dfrac{Y_{n}(\ln(n))^{2}}{n-L_{n}(t)}\right)$.

######  Remark 3.3.

  * •

The general idea behind this lemma is that, up to negligible effects,
$\tilde{\Delta}L_{n}(t)$ is close to a sum of $Y_{n}$ i.i.d. random variables
$M_{i}$. The difference between $\tilde{\Delta}L_{n}(t)$ and
$\sum_{i=1}^{Y_{n}}M_{i}$ comes from two sources: the indices in $\mathcal{R}$
and the difference between $M_{i}$ and $|{C}_{x_{i}}|$. Using the last two
points of the lemma, we will see in the next section that this difference is
very small.

  * •

We will show that, uniformly on $t\in[t_{0},t_{1}]$, on the event
$\mathcal{E}_{n}(t)$,

$\mathbb{P}(\mathcal{A}^{c}_{n}(t)\mid\sigma(X_{n}(s),s\leq t))=O(n^{-10}).$

This is a consequence of usual bounds on the order of the components of a sub-
critical E.R. graph. A detailed proof is given in Section A.1.

Before starting the proof of Lemma 3.2, let us recall some classical
inequalities between the order of a component and the total progeny of a
Binomial BGW process, derived from the exploration of a component in a
breadth-first order.

######  Theorem (Theorems 4.2 and 4.3 in [16]).

For a positive integer $n$ and a real $p\in]0,1[$, let $L_{n,p}$ denote the
size of the component of a vertex in $\mathrm{ER}(n,p)$ and let $T_{n,p}$
denote the total progeny of a $\mathrm{BGW}(n,p)$. Then for every
$k\in\\{0,1,\ldots,n\\}$,

$\displaystyle\mathbb{P}(L_{n,p}\geq k)\leq\mathbb{P}(T_{n,p}\geq k),$ (10)
$\displaystyle\mathbb{P}(L_{n,p}\geq k)\geq\mathbb{P}(T_{n-k,p}\geq k).$ (11)

###### Proof of Lemma 3.2.

To prove Assertions 1.(a) and 1.(b), we define recursively a sequence of
graphs $(\tilde{\mathscr{G}}_{n}^{e,i}(t,h))_{i}$ and a sequence
$(\tilde{C}_{x_{i}})_{i}$ such that $\tilde{C}_{x_{i}}$ is a component of
$x_{i}$ in $\tilde{\mathscr{G}}_{n}^{e,i-1}(t,h)$ and

$\forall j\geq
1,\quad\bigcup_{i=1}^{j}C^{\star}_{x_{i}}\subset\bigcup_{i=1}^{j}\tilde{C}_{x_{i}}.$

We do it as follows111See Fig. 5 and Fig. 5 for a realization of the sequence
$(\tilde{\mathscr{G}}_{n}^{e,i}(t,h))_{i}$ for the graph
$\tilde{\mathscr{G}}_{n}(t,h)$ depicted in Fig. 5.

First set $\tilde{C}_{x_{1}}=C^{*}_{x_{1}}$. Then call
$\tilde{\mathscr{G}}_{n}^{e,1}(t,h)$ the graph obtained from
$\tilde{\mathscr{G}}_{n}^{e}(t,h)$ by redrawing every edges containing a
vertex of $C^{*}_{x_{1}}$. More precisely, we remove all those edges and then
add each pair of vertices containing an element of $C^{*}_{x_{1}}$ with
probability $\frac{t+h}{n}$ independently of anything else. Note that
$\tilde{\mathscr{G}}_{n}^{e,1}(t,h)$ is independent of $C^{*}_{x_{1}}$.

We now call $\tilde{C}_{x_{2}}$ the component of $x_{2}$ in
$\tilde{\mathscr{G}}_{n}^{e,1}(t,h)$. It follows from the construction that
$\tilde{C}_{x_{1}}$ and $\tilde{C}_{x_{2}}$ are independent and have the same
distribution.

We also note that

$C^{\star}_{x_{2}}\subset\tilde{C}_{x_{2}}.$

Indeed either $C^{*}_{x_{2}}=\emptyset$ and the statement is trivial or
$x_{2}$ and $x_{1}$ are not in the same component in
$\tilde{\mathscr{G}}_{n}^{e}(t,h)$, which means that when redrawing the edges
we did not change edges between vertices in $C_{x_{2}}$. It is however
possible that by redrawing the edges we add edges connecting for instance
$C_{x_{2}}$ to $C_{x_{1}}$ in $\tilde{\mathscr{G}}_{n}^{e,1}(t,h)$. Observe
that it may occur that this redrawing creates also paths between $C_{x_{2}}$
and $C_{x_{i}}$ for some $i\geq 3$, via some intermediate vertex belonging to
$C_{x_{1}}$.

For $i\geq 2$,

  * •

if $x_{i+1}\in\tilde{C}_{x_{l}}$ for some $l\leq i$, then
$C^{*}_{x_{i+1}}\subset\tilde{C}_{x_{l}}$. We set
$\tilde{C}_{x_{i+1}}:=\emptyset$ and define
$\tilde{\mathscr{G}}_{n}^{e,i}(t,h)$ as
$\tilde{\mathscr{G}}_{n}^{e,i-1}(t,h)$.

  * •

if $x_{i+1}\notin\cup_{l=1}^{i}\tilde{C}_{x_{l}}$, then $C^{*}_{x_{i+1}}$ is
disjoint of $\cup_{l=1}^{i}\tilde{C}_{x_{l}}$. We redraw the edges having one
extreme point in $\cup_{l=1}^{i}\tilde{C}_{x_{l}}$. This defines a new graph
denoted by $\tilde{\mathscr{G}}_{n}^{e,i}(t,h)$. This graph is again
distributed as an $\mathrm{ER}(n-L_{n}(t),\frac{t+h}{n})$ and is independent
of $\cup_{l=1}^{i}\tilde{C}_{x_{l}}$. We now call $\tilde{C}_{x_{i+1}}$ the
component of $x_{i+1}$ in $\tilde{\mathscr{G}}_{n}^{e,i}(t,h)$.

$\mathscr{C}^{max}_{n}(t)$

Figure 1:  A realization of graph $\mathscr{G}_{n}(t)$ for which
$L_{n}(t)=20$.

$\mathscr{C}^{max}_{n}(t)$

Figure 2:  A realization of graph $\tilde{\mathscr{G}}_{n}(t)$ obtained from
$\mathscr{G}_{n}(t)$ by resampling independently with probability
$\frac{t}{n}$, all edges ‘external’ to $\mathscr{C}^{max}_{n}(t)$.

$a_{5}$$a_{1}$$a_{3}$$a_{4}$$a_{2}$$x_{1}$$x_{2}$$x_{3}$$x_{4}$$x_{5}$$C_{x_{1}}$$C_{x_{2}}=C_{x_{3}}$$C_{x_{4}}$$C_{x_{5}}$

Figure 3: Graph $\tilde{\mathscr{G}}_{n}(t,h)$ obtained from
$\tilde{\mathscr{G}}_{n}(t)$ by adding edges (drawn as dashed lines)
independently with probability $\frac{h}{n-t}$. In this realization, there are
${Y_{n}=5}$ bridges drawn as green dashed lines and nine external edges drawn
in blue. Its largest component is the union of $\mathscr{C}^{max}_{n}(t)$ with
$\cup_{i=1}^{5}C_{x_{i}}$.

$x_{1}$$x_{2}$$x_{3}$$x_{4}$$x_{5}$$\tilde{C}_{x_{1}}$$C_{x_{2}}=C_{x_{3}}$$C_{x_{4}}$$C_{x_{5}}$$\tilde{C}_{x_{2}}$

Figure 4:  Subgraph $\tilde{\mathscr{G}}_{n}^{e,1}(t,h)$. New edges with
respect to $\tilde{\mathscr{G}}_{n}^{e}(t,h)$ are drawn in red, and obtained
from an independent resampling of edges having an endpoint in
$\tilde{C}_{x_{1}}$, with probability $\frac{t+h}{n}$. In this example,
vertices $x_{3}$ and $x_{4}$ are in $\tilde{C}_{x_{2}}$, therefore
$\tilde{C}_{x_{3}}=\tilde{C}_{x_{4}}=\emptyset$ and
$\tilde{\mathscr{G}}_{n}^{e,3}(t,h)=\tilde{\mathscr{G}}_{n}^{e,2}(t,h)=\tilde{\mathscr{G}}_{n}^{e,1}(t,h)$.

$x_{1}$$x_{2}$$x_{3}$$x_{4}$$x_{5}$$\tilde{C}_{x_{1}}$$C_{x_{5}}$$\tilde{C}_{x_{2}}$$\tilde{C}_{x_{5}}$

Figure 5:  Subgraph $\tilde{\mathscr{G}}_{n}^{e,4}(t,h)$. New edges with
respect to $\tilde{\mathscr{G}}_{n}^{e,3}(t,h)$ are drawn in orange, and
obtained from an independent resampling of edges having an endpoint in
$\cup_{i=1}^{4}\tilde{C}_{x_{i}}$, with probability $\frac{t+h}{n}$.

Inequality (10) implies that the conditional distribution of
$|\tilde{C}_{x_{i}}|$ is dominated by the total progeny of a
$\mathrm{BGW}(n-L_{n}(t),\frac{t+h}{n})$. Thus there exists a family of i. i.
d. random variables $M_{i}$ independent of $Y_{n}$, having the same
distribution as the total progeny of a
$\mathrm{BGW}(n-L_{n}(t),\frac{t+h}{n})$ such that $|\tilde{C}_{x_{i}}|\leq
M_{i}$ for every $i$. This ends the proof of Assertions 1.(a) and 1.(b).
Assertion 2.(a) is a direct consequence of the construction as already
mentioned.

To prove Assertion 2.(b), we first note that inequality (11) implies that the
variable $\min(L_{n,p},\ln^{2}(n))$ stochastically dominates
$\min(T_{n-\ln^{2}(n),p},\ln^{2}(n))$ since the total progeny of a
$\mathrm{BGW}(j,p)$ stochastically dominates the total progeny of a
$\mathrm{BGW}(j-1,p)$ for every $j\in\mathbb{N}^{*}$. Therefore, the
conditional distribution of $|C_{x_{i}}|$ dominates the conditional
distribution of the minimum of $\ln(n)^{2}$ and of the total progeny of a
$\mathrm{BGW}(n-L_{n}(t)-\ln(n)^{2},\frac{t+h}{n})$.

It remains to prove the domination of $|\mathcal{R}|$ (Assertion 2.(c)).
Recall that $\mathcal{R}$ is the set of indices $1\leq i\leq Y_{n}$ such that
$C_{x_{i}}$ has already been visited by a bridge of index $j<i$, i.e.
$x_{i}\in\cup_{j=1}^{i-1}C_{x_{j}}$.

Since the targets are picked uniformly outside $\mathscr{C}^{max}_{n}(t)$, the
probability of this event is at most

$\frac{\sum_{j<i}|C_{x_{j}}|}{n-L_{n}(t)}.$

On the event $\mathcal{A}_{n}(t)$, this probability is at most

$\frac{Y_{n}(\ln{n})^{2}}{n-L_{n}(t)},$

which implies Assertion 2.(c). ∎

##  4 Convergence of the finite dimensional distributions

The aim of this section is to prove that
$X_{n}(t):=\frac{1}{\sqrt{n}}(L_{n}(t)-n\rho(t))$ converges in the sense of
finite dimensional (fi-di) distributions towards

$X(t)=\frac{1-\rho(t)}{1-t(1-\rho(t))}B\left(\frac{\rho(t)}{1-\rho(t)}\right).$

As already mentioned (see (6) in section 2), this process is actually a
diffusion with drift
$a(t,x):=\left(\frac{1-2\rho(t)}{1-\lambda(t)}-\frac{\rho(t)(1-\rho(t))t}{(1-\lambda(t))^{2}}\right)x$
and diffusion coefficient
$b(t,x):=\frac{{\rho(t)(1-\rho(t))}}{(1-t(1-\rho(t)))^{3}}$.

To show this convergence, we are going to adapt a theorem by F. Norman [10] to
a non-compact state space. In the case of a sequence of compact-valued
processes, its statement makes it possible to reduce the proof of the fi-di
convergence of a sequence of random processes, to the study of the first three
moments of their infinitesimal increments, in the case where the limiting
process is a diffusion. The extension of this statement we shall apply is:

######  Proposition 4.1.

For each $n\geq 1$, let $\\{Z_{n}(t),t\geq 0\\}$ be a stochastic process in
$\mathbb{R}$ adapted to a filtration $\\{\mathcal{F}_{n}(t),\ t\geq 0\\}$.
Assume there exist two $C^{3}$ functions $\alpha$ and $\beta$, three sequences
of functions $(e_{1,n})_{n}$, $(e_{2,n})_{n}$, $(e_{3,n})_{n}$ and a sequence
$(N_{n})_{n}$ of positive integers converging to $\infty$, satisfying, for
every $t\geq 0$ and every $h>0$:

$\displaystyle\mathbb{E}(Z_{n}(t+h)-Z_{n}(t)\mid\mathcal{F}_{n}(t))=h\alpha(Z_{n}(t))+e_{1,n}(t,h,Z_{n}(t))$
$\displaystyle\mathbb{E}((Z_{n}(t+h)-Z_{n}(t))^{2}\mid\mathcal{F}_{n}(t))=h\beta(Z_{n}(t))+e_{2,n}(t,h,Z_{n}(t))$
(Assumption A1)
$\displaystyle\mathbb{E}(|Z_{n}(t+h)-Z_{n}(t)|^{3}\mid\mathcal{F}_{n}(t))=e_{3,n}(t,h,Z_{n}(t))$
$\displaystyle\forall 0<s<t,$
$\displaystyle\sum_{k=0}^{N_{n}-1}\sum_{i=1}^{3}\mathbb{E}\left(\left|e_{i,n}\left(s+k\frac{t-s}{N_{n}},\frac{t-s}{N_{n}},Z_{n}(s+k\frac{t-s}{N_{n}})\right)\right|\right)\underset{n\rightarrow+\infty}{\rightarrow}0.$

Let $(Z(t))_{t\geq 0}$ be the diffusion process with drift function $\alpha$
and diffusion coefficient $\beta$; denote its semigroup by $\\{T_{t},t\geq
0\\}$. Define the three functions $\epsilon_{1},\epsilon_{2},\epsilon_{3}$ by,
for every $t\geq 0$ and $h>0$,

$\displaystyle\mathbb{E}(Z(t+h)-Z(t)\mid
Z(t))=h\alpha(Z(t))+\epsilon_{1}(h,Z(t))$
$\displaystyle\mathbb{E}((Z(t+h)-Z(t))^{2}\mid
Z(t))=h\beta(Z(t))+\epsilon_{2}(h,Z(t))$
$\displaystyle\mathbb{E}(|Z(t+h)-Z(t)|^{3}\mid Z(t))=\epsilon_{3}(h,Z(t)).$

Let $\mathcal{H}$ denote the set of bounded $C^{3}$ functions defined on
$\mathbb{R}$ whose derivatives are also bounded. Assume the following
properties:

(Assumption A2) $\displaystyle\forall
f\in\mathcal{H},\,T_{t}f\in\mathcal{H}\text{ and }\forall\tau>0\sup_{0\leq
t\leq\tau}\sum_{k=0}^{3}||(T_{t}f)^{(k)}||_{\infty}<+\infty$ (Assumption A3)
$\displaystyle\forall
0<s<t,\,\sum_{k=0}^{N_{n}-1}\sum_{i=1}^{3}\mathbb{E}\left(\left|\epsilon_{i}\left(\frac{t-s}{N_{n}},Z_{n}(s+k\frac{t-s}{N_{n}})\right)\right|\right)\underset{n\rightarrow+\infty}{\rightarrow}0$
(Assumption A4) $\displaystyle(Z_{n}(0))_{n}\text{ converges in distribution
to }Z(0).$

Then for every $k\geq 1$ and for every $0<s_{1}<\ldots<s_{k}$,
$(Z_{n}(s_{1}),\ldots,Z_{n}(s_{k}))_{n}$ converges in distribution to
$(Z(s_{1}),\ldots,Z(s_{k}))$.

The proof of this Proposition is postponed to Section A.2 of the Appendix.

Note that this Proposition is restricted to the case where the limiting
process is homogeneous in time, which $X$ is not. To overcome this difficulty,
we consider the auxiliary process $Z_{n}:=(uX_{n})\circ v^{(-1)}$ and use
Proposition 4.1 to show its convergence to a standard Brownian motion. In this
case, Proposition 4.1 will apply with diffusion coefficients $\alpha\equiv 0$,
$\beta\equiv 1$. This argument is detailed in Section 4.2. However, before
applying this time change we need to evaluate the first three moments of the
increments of the original process
$X_{n}(t):=\frac{1}{\sqrt{n}}(L_{n}(t)-n\rho(t))$. This is the goal of Section
4.1.

###  4.1 The first three moments of $X_{n}(t+h)-X_{n}(t)$

Set $\Delta X_{n}(t)=X_{n}(t+h)-X_{n}(t)$. Let $\\{\mathcal{F}_{n}(t),\,t\geq
t_{0}\\}$ denote the filtration generated by the process $\\{X_{n}(t),\,t\geq
t_{0}\\}$ for every $n\in\mathbb{N}$. In the sequel, we will use the notation

$\mathbb{P}_{t}(\cdot)=\mathbb{P}(\cdot\mid\mathcal{F}_{n}(t))\text{ and
}\mathbb{E}_{t}(\cdot)=\mathbb{E}(\cdot\mid\mathcal{F}_{n}(t)).$

Unless specified, the notation $O(.)$ is always uniform in
$t\in[t_{0},t_{1}]$.

##### First moment.

We begin by showing that the expectation of the conditional increment of
$X_{n}(t)$ between $t$ and $t+h$ verifies:

######  Lemma 4.2.

There exists a constant $C>0$, such that for all $t\in[t_{0},t_{1}],$

$\mathbb{E}_{t}(\Delta X_{n}(t))=ha(t,X_{n}(t))+r_{1,n}(t,h,X_{n}(t))$

with

$\displaystyle a(t,x)$ $\displaystyle=$
$\displaystyle\left(\frac{1-2\rho(t)}{1-\lambda(t)}-\frac{\rho(t)(1-\rho(t))t}{(1-\lambda(t))^{2}}\right)x$
$\displaystyle|r_{1,n}(t,h,x)|$ $\displaystyle\leq$ $\displaystyle
C\left(\sqrt{n}(\ln
n)^{2}h^{2}+\frac{|x|^{2}h}{\sqrt{n}}+n^{-8}\right)\mathds{1}_{\\{|x|\leq
n^{0.2}\\}}+\sqrt{n}\mathds{1}_{\\{|x|>n^{0.2}\\}}.$

In particular if $h\leq n^{-1},$ there exists some $\eta>0$ and a constant $C$
only depending on $t_{0}$ and $t_{1}$ such that

$|r_{1,n}(t,h,x)|\leq
C(hn^{-\eta}+n^{-8}+\sqrt{n}\mathds{1}_{\\{|x|>n^{0.2}\\}}).$

###### Proof.

Recall that

$\Delta X_{n}(t)=\frac{1}{\sqrt{n}}\left(\Delta
L_{n}(t)-n(\rho(t+h)-\rho(t))\right).$

On the event $\mathcal{E}_{n}(t)$ which belongs to $\mathcal{F}_{n}(t),$ Lemma
3.1 gives

$\mathbb{E}_{t}[\Delta
L_{n}(t)]=\mathbb{E}_{t}[\tilde{\Delta}L_{n}(t)]+O(n^{-8}).$

We now use Lemma 3.2 to get an upper bound of
$\mathbb{E}_{t}[\tilde{\Delta}L_{n}(t)]$. On the event $\mathcal{E}_{n}(t)$,
we denote $m^{+}_{n,1}$ the conditional first moment of the total progeny of
the subcritical $\mathrm{BGW}(n-L_{n}(t),\frac{t+h}{n})$ given
$\mathcal{F}_{t}$. On the event $\mathcal{E}_{n}(t)$,

$\displaystyle\mathbb{E}_{t}[\tilde{\Delta}L_{n}(t)]\leq\mathbb{E}_{t}(\sum_{i=1}^{Y_{n}}M_{i})$
$\displaystyle=\mathbb{E}_{t}(Y_{n})m^{+}_{n,1}=L_{n}(t)(n-L_{n}(t))\frac{h}{n-t}\frac{1}{1-(n-L_{n}(t))\frac{t+h}{n}}$
$\displaystyle=nhf_{1}\left(t+h,\rho(t)+\frac{X_{n}(t)}{\sqrt{n}}\right)\left(1+O(\frac{1}{n})\right)$
(12)

with $f_{1}(t,x)=\frac{x(1-x)}{1-t(1-x)}$.

This induces the following upper bound for $\mathbb{E}_{t}[\Delta X_{n}(t)]$
on the event $\mathcal{E}_{n}(t)$,

$\mathbb{E}_{t}[\Delta
X_{n}(t)]\leq\sqrt{n}\left(hf_{1}\left(t+h,\rho(t)+\frac{X_{n}(t)}{\sqrt{n}}\right)(1+O(\frac{1}{n}))-(\rho(t+h)-\rho(t))+O(n^{-8.5})\right).$

Noticing that $\rho^{\prime}(t)=f_{1}(t,\rho(t))$, on the event
$\mathcal{E}_{n}(t)$, this upper bound is equal to

$\displaystyle\mathbb{E}_{t}[\Delta X_{n}(t)]$ $\displaystyle\leq$
$\displaystyle\sqrt{n}h\left(f_{1}\left(t+h,\rho(t)+\frac{X_{n}(t)}{\sqrt{n}}\right)-f_{1}(t,\rho(t))\right)$
(13)
$\displaystyle-\sqrt{n}(\rho(t+h)-\rho(t)-h\rho^{\prime}(t))+O\left(\frac{h|X_{n}|}{n}+n^{-8.5}\right)$
$\displaystyle=$ $\displaystyle ha(t,X_{n}(t))+r^{+}_{1,n}(t,h,X_{n}(t))$

with

$\displaystyle a(t,X_{n}(t))$ $\displaystyle=$ $\displaystyle
X_{n}(t)\frac{\partial f_{1}}{\partial
x}(t,\rho(t))=\left(\frac{1-2\rho(t)}{1-\lambda(t)}-\frac{\rho(t)(1-\rho(t))t}{(1-\lambda(t))^{2}}\right)X_{n}(t)$
$\displaystyle r^{+}_{1,n}(t,h,X_{n}(t))$ $\displaystyle=$
$\displaystyle\sqrt{n}h\left(f_{1}\left(t+h,\rho(t)+\frac{X_{n}(t)}{\sqrt{n}}\right)-f_{1}\left(t,\rho(t)+\frac{X_{n}(t)}{\sqrt{n}}\right)\right)$
$\displaystyle+\sqrt{n}h\left(f_{1}\left(t,\rho(t)+\frac{X_{n}(t)}{\sqrt{n}}\right)-f_{1}(t,\rho(t))-\frac{X_{n}(t)}{\sqrt{n}}\frac{\partial
f_{1}}{\partial x}(t,\rho(t))\right)$
$\displaystyle-\sqrt{n}\left(\rho(t+h)-\rho(t)-h\rho^{\prime}(t)\right)+O\left(\frac{h|X_{n}|}{n}+n^{-8.5}\right)$
$\displaystyle=$ $\displaystyle
O\left(\sqrt{n}h^{2}+\frac{|X_{n}(t)|^{2}h}{\sqrt{n}}+n^{-8.5}\right).$

We now want to show that this upper bound is close to the true expectation of
$\Delta X_{n}(t)$ conditional on $\mathcal{F}_{n}(t)$. The only place where we
used an inequality is in (4.1). Accordingly, we want to bound
$\mathbb{E}_{t}\left[\sum_{i=1}^{Y_{n}}M_{i}-\tilde{\Delta}L_{n}(t)\right]$.

Note that a priori $\sum_{i=1}^{Y_{n}}M_{i}$ is not bounded. Therefore, we
write

$\mathbb{E}_{t}\left(\sum_{i=1}^{Y_{n}}M_{i}-\tilde{\Delta}L_{n}(t)\right)=\mathbb{E}_{t}\left(\left(\sum_{i=1}^{Y_{n}}M_{i}-n^{3}\right)\mathds{1}_{\\{\sum_{i=1}^{Y_{n}}M_{i}>n^{3}\\}}\right)\\\
+\mathbb{E}_{t}\left(\min\left(\sum_{i=1}^{Y_{n}}M_{i},n^{3}\right)-\tilde{\Delta}L_{n}(t),\right).$
(14)

We first show that the first term is negligible. Using Cauchy-Schwarz
inequality we have

$\displaystyle\mathbb{E}_{t}\left(\left(\sum_{i=1}^{Y_{n}}M_{i}-n^{3}\right)\mathds{1}_{\\{\sum_{i=1}^{Y_{n}}M_{i}>n^{3}\\}}\right)$
$\displaystyle\leq\mathbb{E}_{t}\left(\left(\sum_{i=1}^{Y_{n}}M_{i}\right)^{2}\right)^{1/2}\mathbb{P}_{t}\left(\sum_{i=1}^{Y_{n}}M_{i}>n^{3}\right)^{1/2}$
$\displaystyle=O\left(n^{2}\mathbb{P}_{t}(\tilde{{\mathcal{A}}}^{c}_{n}(t))^{1/2}\right),$

with $\tilde{{\mathcal{A}}}_{n}(t)=\\{M_{i}\leq(\ln(n))^{2},\forall 1\leq
i\leq n^{2}\\}$. We will show in Section A.1 that on the event
${\mathcal{E}}_{n}(t),$ $\mathbb{P}_{t}(\tilde{{\mathcal{A}}}^{c}_{n}(t))\leq
n^{-20}$.

To bound the second term, using (4.1), we write on the event
$\mathcal{E}_{n}(t)$,

$0\leq\mathbb{E}_{t}\left(\min\left(\sum_{i=1}^{Y_{n}}M_{i},n^{3}\right)-\tilde{\Delta}L_{n}(t)\right)\leq\mathbb{E}_{t}\left(\left(\sum_{i=1}^{Y_{n}}M_{i}-\tilde{\Delta}L_{n}(t)\right)\mathds{1}_{\mathcal{A}_{n}(t)}\right)+n^{3}\mathbb{P}_{t}\left(\mathcal{A}_{n}(t)^{c}\right).$

Let us introduce $f_{2}(t,x)=\frac{1}{1-t(1-x)}$. On the event
$\mathcal{E}_{n}(t)$, let denote by $m^{-}_{n,1}$ the conditional expectation
of the minimum of $\ln(n)^{2}$ and of the total progeny of a
$\mathrm{BGW}(n-L_{n}(t)-\ln(n)^{2},\frac{t+h}{n})$. Using Lemma 3.2, on the
event $\mathcal{E}_{n}(t)$,

$\mathbb{E}_{t}\left[\left(\sum_{i=1}^{Y_{n}}M_{i}-\tilde{\Delta}L_{n}(t)\right)\mathds{1}_{\mathcal{A}_{n}(t)}\mid
Y_{n}\right]\leq Y_{n}(m^{+}_{n,1}-m^{-}_{n,1})+(\ln
n)^{2}\mathbb{E}_{t}[|\mathcal{R}|\mathds{1}_{\mathcal{A}_{n}(t)}\mid Y_{n}].$

Let us recall that if $T$ is the total progeny of a $\mathrm{BGW}(n,p)$ for
$p<1/n$, its first moment is $\frac{1}{1-np}$ and there exists a constant $C$
such that $\mathbb{E}(T\mathds{1}_{\\{T\geq\ln(n)^{2}\\}})\leq
C\mathbb{P}(T\geq\ln(n)^{2})^{1/2}=O(n^{-9})$ since $T$ has finite exponential
moments. In particular,

$\mathbb{E}(\min(T,\ln^{2}(n))=\mathbb{E}(T)-\mathbb{E}(T\mathds{1}_{\\{T>\ln^{2}(n)\\}}+\ln^{2}(n)\mathbb{P}(T>\ln^{2}(n))\geq\mathbb{E}(T)+O(n^{-9}).$

One yields on the event $\mathcal{E}_{n}(t)$:

$m^{+}_{n,1}-m^{-}_{n,1}\leq
f_{2}\left(t+h,\rho(t)+\frac{X_{n}(t)}{\sqrt{n}}\right)-f_{2}\left(t,\rho(t)+\frac{X_{n}(t)}{\sqrt{n}}+\frac{\ln(n)^{2}}{n}\right)+O(n^{-9}).$

By property 2.c of Lemma 3.2

$\mathbb{E}_{t}[|\mathcal{R}|\mathds{1}_{\mathcal{A}_{n}(t)}\mid
Y_{n}]\leq\frac{Y_{n}^{2}(\ln(n))^{2}}{n\left(1-(\rho(t)+\frac{X_{n}(t)}{\sqrt{n}})\right)}.$

Therefore, on $\mathcal{E}_{n}(t)$,

$\displaystyle\mathbb{E}_{t}\Big{(}\Big{(}\sum_{i=1}^{Y_{n}}M_{i}$
$\displaystyle-\tilde{\Delta}L_{n}(t)\Big{)}\mathds{1}_{\mathcal{A}_{n}(t)}\mid
Y_{n}\Big{)}$ $\displaystyle\leq$ $\displaystyle
Y_{n}\left(f_{2}\left(t+h,\rho(t)+\frac{X_{n}(t)}{\sqrt{n}}\right)-f_{2}\left(t,\rho(t)+\frac{X_{n}(t)}{\sqrt{n}}+\frac{\ln(n)^{2}}{n}\right)\right)$
$\displaystyle+Y_{n}O(n^{-9})+\frac{Y_{n}^{2}(\ln(n))^{4}}{n\left(1-(\rho(t)+\frac{X_{n}(t)}{\sqrt{n}})\right)}$
$\displaystyle\leq$ $\displaystyle
C\left(hY_{n}+\frac{Y_{n}^{2}(\ln(n))^{4}}{n}\right).$ (15)

The last inequality comes from the fact that for $t>t_{0}$, the denominator of
$f_{2}(t,\rho(t))$ i.e. ${1-t(1-\rho(t))}$ is away from 0, as well as the
quantity $1-t(1-\frac{L_{n}(t)}{n})$ on the event $\mathcal{E}_{n}(t)$, and a
fortiori $1-t(1-\frac{L_{n}(t)}{n}-a)$ for all $a>0$. Taking the expectation
in the last upper bound, we obtain on $\mathcal{E}_{n}(t)$,

$\displaystyle\mathbb{E}_{t}\left[\min\left(\sum_{i=1}^{Y_{n}}M_{i},n^{3}\right)-\tilde{\Delta}L_{n}(t)\right]\leq
Cn(\ln n)^{4}h^{2}+n^{3}\mathbb{P}_{t}(\mathcal{A}_{n}(t)^{c}),$ (16)

and therefore

$\displaystyle\mathbb{E}_{t}\left[\sum_{i=1}^{Y_{n}}M_{i}-\tilde{\Delta}L_{n}(t)\right]\leq
Cn(\ln n)^{4}h^{2}+n^{3}\mathbb{P}_{t}(\mathcal{A}_{n}(t)^{c})+Cn^{-8}.$ (17)

Recalling (13), we obtain, using Lemmas 3.1 and A.1

$\mathbb{E}_{t}(\Delta X_{n}(t))=ha(t,X_{n}(t))+r_{1,n}(t,h,X_{n}(t))$ (18)

with $|r_{1,n}(t,h,X_{n}(t))|\leq C\left(\sqrt{n}(\ln
n)^{4}h^{2}+\frac{|X_{n}|^{2}h}{\sqrt{n}}+n^{-8}\right)\mathds{1}_{\mathcal{E}_{n}(t)}+\sqrt{n}\mathds{1}_{\mathcal{E}^{c}_{n}(t)}$.
∎

##### Second moment.

We now turn to the estimation of $\mathbb{E}_{t}(\Delta X_{n}(t)^{2}).$ We are
going to prove

######  Lemma 4.3.

There exists a constant $C>0$, such that for all $t\in[t_{0},t_{1}],$

$\mathbb{E}_{t}((\Delta X_{n}(t))^{2})=hb(t)+r_{2,n}(t,h,X_{n}(t))$ (19)

with

$\displaystyle b(t)$ $\displaystyle=$
$\displaystyle\frac{{\rho(t)(1-\rho(t))}}{(1-t(1-\rho(t)))^{3}}$
$\displaystyle|r_{2,n}(t,h,x)|$ $\displaystyle\leq$ $\displaystyle
C\left(\frac{|x|}{\sqrt{n}}h+h^{2}+\frac{h}{n}+nh^{3}+\sqrt{n}h^{2}|x|+(\ln
n)^{4}n^{3}h^{2}+n^{-7}\right)\mathds{1}_{\\{|x|\leq n^{0.2}\\}}$
$\displaystyle+n\mathds{1}_{\\{|x|>n^{0.2}\\}}.$

In particular if $h\leq n^{-4}$, there exists some $\eta>0$ and a constant
$C>0$ only depending on $t_{0}$ and $t_{1}$ such that

$|r_{2,n}(t,h,x)|\leq C(hn^{-\eta}+n^{-7}+n\mathds{1}_{\\{|x|>n^{0.2}\\}}).$

###### Proof.

By definition,

$\displaystyle\mathbb{E}_{t}[\Delta X_{n}(t)^{2}]$ $\displaystyle=$
$\displaystyle\mathbb{E}_{t}[\frac{(\Delta
L_{n}(t))^{2}}{n}+n(\rho(t+h)-\rho(t))^{2}-2\Delta
L_{n}(t)(\rho(t+h)-\rho(t))]$

As before we make use of Lemma 3.1. From now on, the computations will be
performed only on the event $\mathcal{E}_{n}(t)$.

$\displaystyle\mathbb{E}_{t}[\Delta X_{n}(t)^{2}]$ $\displaystyle=$
$\displaystyle\mathbb{E}_{t}[\frac{(\tilde{\Delta}L_{n}(t))^{2}}{n}]+n(\rho(t+h)-\rho(t))^{2}-2\left(\rho(t+h)-\rho(t)\right)\mathbb{E}_{t}[\tilde{\Delta}L_{n}(t)]$
$\displaystyle+O(n^{-7}).$

Using Lemma 3.2, the term $\mathbb{E}_{t}[(\tilde{\Delta}L_{n}(t))^{2}]$ is
bounded from above by the second moment of the sum of $Y_{n}$ i.i.d. copies of
the total progeny of a $\mathrm{BGW}(n-L_{n}(t),\frac{t+h}{n})$. Hence,
denoting by $v_{n}^{+}$ the variance of the total progeny of a
$\mathrm{BGW}(n-L_{n}(t),\frac{t+h}{n})$, classical variance computations
give,

$\mathbb{E}_{t}[(\tilde{\Delta}L_{n}(t))^{2}]\leq\mathbb{E}_{t}\left[\left(\sum_{i=1}^{Y_{n}}M_{i}\right)^{2}\right]=(\mathbb{E}_{t}[{Y_{n}}]m_{n,1}^{+})^{2}+\mathrm{Var}(Y_{n}\mid\mathcal{F}_{t})(m_{n,1}^{+})^{2}+\mathbb{E}_{t}[Y_{n}]v_{n}^{+},$
(20)

In the right hand side of (20), the second and third terms will give the main
contribution and yield the diffusion term $nhb(t,X_{t})$. Since the
expectation and variance of the total progeny of Galton-Watson process with
offspring distribution $\mathrm{Bin}(n,p)$ are equal to $\frac{1}{1-np}$ and
$\frac{np(1-p)}{(1-np)^{3}}$ respectively, we can write using the function
$f_{3}(t,x)=\frac{x(1-x)}{(1-t(1-x))^{3}}$,

$\displaystyle\mathrm{Var}(Y_{n}\mid$
$\displaystyle\mathcal{F}_{t})(m_{n,1}^{+})^{2}+\mathbb{E}_{t}[Y_{n}]v_{n}^{+}$
$\displaystyle=$ $\displaystyle
L_{n}(t)(n-L_{n}(t))\frac{h}{n-t}\left(1-\frac{h}{n-t}\right)\frac{1}{\left(1-(n-L_{n}(t))\frac{t+h}{n}\right)^{2}}$
$\displaystyle+L_{n}(t)(n-L_{n}(t))\frac{h}{n-t}\frac{(n-L_{n}(t))\frac{t+h}{n}(1-\frac{t+h}{n})}{\left(1-(n-L_{n}(t))\frac{t+h}{n}\right)^{3}}$
$\displaystyle=$ $\displaystyle
nhf_{3}\left(t+h,\frac{L_{n}(t)}{n}\right)\left(1-\frac{h}{n-t}-\frac{t}{n}(1-\frac{h}{n-t})(1-\frac{L_{n}(t)}{n})(t+h)\right)$
$\displaystyle=$ $\displaystyle
nh\left(f_{3}(t,\rho(t))+O\left(\frac{|X_{n}|}{\sqrt{n}}\left|\frac{\partial
f_{3}}{\partial x}(t,\rho(t))\right|+h\left|\frac{\partial f_{3}}{\partial
t}(t,\rho(t))\right|\right)\right)\left(1+O\left(\frac{1}{n}\right)\right)$
$\displaystyle=$ $\displaystyle
nh\left(f_{3}(t,\rho(t))+O\left(\big{|}\frac{X_{n}}{\sqrt{n}}\big{|}+h+\frac{1}{n}\right)\right).$
(21)

The other contributions in the estimation of the upper bound of
$\mathbb{E}_{t}[\Delta X_{n}(t)^{2}]$ are the three following terms, where the
first one will compensate the two last ones through a Taylor expansion.
Indeed,

$(\mathbb{E}_{t}[{Y_{n}}]m_{n,1}^{+})^{2}+n^{2}(\rho(t+h)-\rho(t))^{2}-2n(\rho(t+h)-\rho(t))\mathbb{E}_{t}[\tilde{\Delta}L_{n}(t)]\\\
=n^{2}h^{2}\left(f_{1}(t+h,\frac{L_{n}(t)}{n})\right)^{2}-n^{2}(\rho(t+h)-\rho(t))^{2}-2n^{3/2}(\rho(t+h)-\rho(t))\mathbb{E}_{t}[\Delta
X_{n}(t)]+O(n^{-7}).$

Since $\rho^{\prime}(t)=f_{1}(t,\rho(t))$, the zero orders of the first two
terms cancel. The Taylor expansion of the function $f$ at order 1 together
with Lemma 4.2, giving the expression of $\mathbb{E}_{t}[\Delta X_{n}(t)]$,
and (4.1) yield:

$\displaystyle\mathbb{E}_{t}[\Delta X_{n}(t)^{2}]$ $\displaystyle\leq$
$\displaystyle hf_{3}(t,\rho(t))$
$\displaystyle+O\left(\frac{h|X_{n}(t)|}{\sqrt{n}}+h^{2}+\frac{h}{n}+nh^{3}+h\sqrt{n}(h|X_{n}(t)|+r_{1,n}(t,h,X_{n}(t)))\right)$
$\displaystyle+O(n^{-7})+n\mathds{1}_{\mathcal{E}^{c}_{n}(t)}.$

Since we based the upper bound of $\mathbb{E}_{t}[\Delta X_{n}(t)^{2}]$ on the
inequality

$\mathbb{E}_{t}[\tilde{\Delta}L_{n}(t)^{2}]\leq\mathbb{E}_{t}\left[\left(\sum_{i=1}^{Y_{n}}M_{i}\right)^{2}\right],$

we just have to upper bound
$\mathbb{E}_{t}\left[\left(\sum_{i=1}^{Y_{n}}M_{i}\right)^{2}\right]-\mathbb{E}_{t}[(\tilde{\Delta}L_{n}(t))^{2}]$
in order to get a lower bound of $\mathbb{E}_{t}[\Delta X_{n}(t)^{2}]$. We
proceed as in (14).

$\displaystyle\mathbb{E}_{t}\left[\left(\sum_{i=1}^{Y_{n}}M_{i}\right)^{2}-\tilde{\Delta}L_{n}(t)^{2}\right]$
$\displaystyle=$
$\displaystyle\mathbb{E}_{t}\left[\left(\left(\sum_{i=1}^{Y_{n}}M_{i}\right)^{2}-n^{6}\right)\mathds{1}_{\sum_{i=1}^{Y_{n}}M_{i}>n^{3}}\right]$
$\displaystyle+\mathbb{E}_{t}\left[\min\left(\sum_{i=1}^{Y_{n}}M_{i},n^{3}\right)^{2}-\tilde{\Delta}L_{n}(t)^{2}\right].$

Using Cauchy-Schwarz inequality and the fact that $Y_{n}$ is bounded by
$n^{2}$, the first term is smaller than
$Cn^{4}\mathbb{P}_{t}(\sum_{i=1}^{Y_{n}}M_{i}>n^{3})^{\frac{1}{2}}$ which in
turn is bounded by
$Cn^{4}\mathbb{P}_{t}({\mathcal{A}}^{c}_{n}(t))^{\frac{1}{2}}$. To upper bound
the second term we note that

$\displaystyle\mathbb{E}_{t}\left[\left(\min\left(\sum_{i=1}^{Y_{n}}M_{i},n^{3}\right)\right)^{2}\right]$
$\displaystyle-$ $\displaystyle\mathbb{E}_{t}[(\tilde{\Delta}L_{n}(t))^{2}]$
$\displaystyle\leq$ $\displaystyle
2\mathbb{E}_{t}\left[\min\left(\sum_{i=1}^{Y_{n}}M_{i},n^{3}\right)\left(\min\left(\sum_{i=1}^{Y_{n}}M_{i},n^{3}\right)-\tilde{\Delta}L_{n}(t)\right)\right]$
$\displaystyle\leq$ $\displaystyle
2n^{3}\mathbb{E}_{t}\left[\left(\min\left(\sum_{i=1}^{Y_{n}}M_{i},n^{3}\right)-\tilde{\Delta}L_{n}(t)\right)\right]$

By equation (16), this second term is bounded by $Cn^{4}(\ln
n)^{4}h^{2}+n^{6}\mathbb{P}_{t}(\mathcal{A}_{n}(t)^{c})$.

Gathering all the above estimations we obtain

$\mathbb{E}_{t}((\Delta X_{n}(t))^{2})=hb(t)+r_{2,n}(t,h,X_{n}(t))$

with

$\displaystyle b(t)$ $\displaystyle=$ $\displaystyle
f_{3}(t,\rho(t))=\dfrac{\rho(t)(1-\rho(t))}{(1-t(1-\rho(t)))^{3}}$
$\displaystyle r_{2,n}(t,h,X_{n})$ $\displaystyle=$ $\displaystyle
O\left(\big{|}\frac{X_{n}(t)}{\sqrt{n}}\big{|}h+h^{2}+\frac{h}{n}+nh^{3}+\sqrt{n}h^{2}|X_{n}(t)|+(\ln
n)^{4}n^{3}h^{2}\right)$
$\displaystyle+O(n^{-7}+n\mathds{1}_{\mathcal{E}_{n}^{c}(t)}).$

∎

##### Third moment.

For the third moment, we only need to obtain an upper bound
for$\mathbb{E}_{t}[|\Delta X_{n}(t)|^{3}]$.

######  Lemma 4.4.

There exists a constant $C>0$, such that for all $t\in[t_{0},t_{1}],$

$\mathbb{E}_{t}(|\Delta X_{n}(t)|^{3})=r_{3,n}(t,h,X_{n})$ (22)

with

$\displaystyle|r_{3,n}(t,h,x)|$ $\displaystyle\leq$ $\displaystyle C\left((\ln
n)^{6}n^{\frac{3}{2}}h^{3}+n^{-6}\right)\mathds{1}_{\\{|x|\leq
n^{0.2}\\}}+n^{\frac{3}{2}}\mathds{1}_{\\{|x|>n^{0.2}\\}}.$

In particular, if $h\leq n^{-1},$ there exists some $\eta>0$ and a constant
$C$ only depending on $t_{0}$ and $t_{1}$ such that

$|r_{3,n}(t,h,x)|\leq
C(hn^{-\eta}+n^{-6})+n^{\frac{3}{2}}\mathds{1}_{\\{|x|>n^{0.2}\\}}.$

###### Proof.

As before, we first restrict the study to the event $\mathcal{E}_{n}(t)$. On
this event,

$\displaystyle\mathbb{E}_{t}[|\Delta X_{n}(t)^{3}|]=$
$\displaystyle\mathbb{E}_{t}[|n^{-\frac{3}{2}}(\Delta
L_{n}(t))^{3}-3n^{-\frac{1}{2}}\Delta L_{n}(t))^{2}(\rho(t+h)-\rho(t))$
$\displaystyle+3n^{\frac{1}{2}}\Delta
L_{n}(t)(\rho(t+h)-\rho(t))^{2}-n^{\frac{3}{2}}(\rho(t+h)-\rho(t))^{3}]$

Using again lemma 3.1, and using the fact that on $\mathcal{A}_{n}(t)$,
$|\tilde{\Delta}L_{n}(t)|\leq(\ln n)^{2}Y_{n},$ we have on
$\mathcal{E}_{n}(t)$:

$\mathbb{E}_{t}[|\Delta X_{n}(t)^{3}|]\leq
n^{-\frac{3}{2}}\ln(n)^{6}\mathbb{E}_{t}[Y^{3}_{n}]+3n^{-\frac{1}{2}}\ln(n)^{4}\mathbb{E}_{t}[Y^{2}_{n}]|\rho(t+h)-\rho(t)|\\\
+3n^{\frac{1}{2}}\ln(n)^{2}\mathbb{E}_{t}[Y_{n}](\rho(t+h)-\rho(t))^{2}+n^{\frac{3}{2}}|\rho(t+h)-\rho(t)|^{3}+n^{\frac{3}{2}}\mathbb{P}_{t}(\mathcal{A}_{n}^{c}(t))+O(n^{-6}).$

As the conditional distribution of $Y_{n}$ is stochastically dominated by
$\mathcal{B}(\frac{n^{2}}{4},\frac{h}{n-t})$, there exists a constant $C$ such
that for all $t_{0}\leq t\leq t_{1}$, $\mathbb{E}_{t}[Y^{k}_{n}]\leq
Cn^{k}h^{k}$ for $k\in\\{1,2,3\\}$. Therefore, there exists a constant $C$
such that

$\mathbb{E}_{t}[|\Delta X_{n}(t)^{3}|]\leq
Cn^{\frac{3}{2}}(\ln(n))^{6}h^{3}+n^{\frac{3}{2}}\mathds{1}_{\mathcal{E}_{n}(t)^{c}}+O(n^{-6}).$

∎

###  4.2 Assumptions of Proposition 4.1 for the sequence $(uX_{n})\circ
v^{(-1)}$

Let $Z_{n}:=(uX_{n})\circ v^{(-1)}$. This section is devoted to checking that
the sequence of processes $Z_{n}$, satisfies the assumptions of Proposition
4.1 where the process $Z$ is nothing but the standard Brownian motion. For
this purpose we want to check these assumptions with $\alpha\equiv 0$,
$\beta\equiv 1$, and take the origin of time at $v(t_{0})$. In this case, the
remainder functions are $\epsilon_{1}\equiv 0$, $\epsilon_{2}\equiv 0$ and
$\epsilon_{3}\equiv\frac{4}{\sqrt{2}\pi}h^{3/2}$.

In this setting, checking Assumption A2 is straightforward, and the sum in
Assumption A3 converges to 0 since it is of order $N_{n}^{-1/2}$. The
convergence of $X_{n}(t_{0})$ to a centered Gaussian variable with variance
$\sigma^{2}(t_{0})=\frac{\rho(t_{0})(1-\rho(t_{0}))}{(1-t_{0}(1-\rho(t_{0})))^{2}}=\frac{v(t_{0})}{u(t_{0})^{2}}$
is a consequence of the classical central limit Theorem for the order of the
giant component (see [1, 12, 15]). As $v$ is an increasing smooth function
from $(1,+\infty)$ to $(0,+\infty)$ and $u$ is a smooth function on
$(1,+\infty)$ that is never equal to zero, this implies that $Z_{n}(v(t_{0}))$
converges to a centrered Gaussian random variable with variance $v(t_{0})$,
which implies Assumption A4.

Let us now prove Assumption A1, concerning the first three moments of the
increment of $Z_{n}$. The estimations of the increment of $Z$ are based on the
identity:

$\displaystyle\Delta Z_{n}(t):=Z_{n}(t+h)-Z_{n}(t)$ $\displaystyle=$
$\displaystyle u\circ v^{(-1)}(t+h)\left(X_{n}\circ v^{(-1)}(t+h)-X_{n}\circ
v^{(-1)}(t)\right)$ (23) $\displaystyle+X_{n}\circ v^{(-1)}(t)\left(u\circ
v^{(-1)}(t+h)-u\circ v^{(-1)}(t)\right)$ $\displaystyle=$
$\displaystyle(u\circ v^{(-1)}(t)+O(h))\left(X_{n}\circ
v^{(-1)}(t+h)-X_{n}\circ v^{(-1)}(t)\right)$ $\displaystyle+X_{n}\circ
v^{(-1)}(t)\left(h(u\circ v^{(-1)})^{\prime}(t)+O(h^{2})\right)$

##### The first moment of $\Delta Z_{n}$.

By Lemma 4.2,

$\mathbb{E}_{v^{(-1)}(t)}\left[X_{n}\circ v^{(-1)}(t+h)-X_{n}\circ
v^{(-1)}(t)\right]\\\ =\left(\frac{h}{v^{\prime}\circ
v^{(-1)}(t)}+O(h^{2})\right)a(v^{(-1)}(t),X_{n}\circ
v^{(-1)}(t))+\tilde{r}_{1,n}(t,h,X_{n}\circ v^{(-1)}(t))$

where
$\tilde{r}_{1,n}(t,h,X_{n})=r_{1,n}(v^{(-1)}(t),v^{(-1)}(t+h)-v^{(-1)}(t),X_{n}\circ
v^{(-1)}(t))$.

Using the relation $a(t,x)=-\frac{u^{\prime}(t)}{u(t)}x$,

$(u\circ v^{(-1)})(t)\frac{a(v^{(-1)}(t),X_{n}\circ
v^{(-1)}(t))}{v^{\prime}\circ v^{(-1)}(t)}+X_{n}\circ v^{(-1)}(t)(u\circ
v^{(-1)})^{\prime}(t)=0.$

Hence, the expectation of the zero order term of (23) cancels. Therefore,

$\displaystyle\mathbb{E}_{v^{(-1)}(t)}\left[\Delta Z_{n}(t)\right]$
$\displaystyle=$ $\displaystyle O(h^{2}|X_{n}\circ
v^{(-1)}(t)|+|\tilde{r}_{1,n}(t,h,X_{n})|).$

Hence, if $h\leq n^{-1},$ there exists $\eta>0$ such that

$\displaystyle\mathbb{E}_{v^{(-1)}(t)}\left[\Delta
Z_{n}(t)\right]=O(hn^{-\eta}+n^{-8}+\sqrt{n}\mathds{1}_{|X_{n}\circ
v^{(-1)}(t)|>n^{0.2}}).$ (24)

##### The second moment of $\Delta Z_{n}$.

By (23),

$\displaystyle\mathbb{E}_{v^{(-1)}(t)}((\Delta Z_{n}(t))^{2})$
$\displaystyle=$ $\displaystyle(u\circ
v^{(-1)}(t)+O(h))^{2}\mathbb{E}_{v^{(-1)}(t)}\left[(\Delta(X_{n}\circ
v^{(-1)})(t))^{2}\right]$ $\displaystyle+(X_{n}\circ
v^{(-1)}(t))^{2}\left(h(u\circ v^{(-1)})^{\prime}(t)+O(h^{2})\right)^{2}$
$\displaystyle+O(h)X_{n}\circ
v^{(-1)}(t)\mathbb{E}_{v^{(-1)}(t)}\left[\Delta(X_{n}\circ
v^{(-1)})(t)\right]$

By Lemma 4.2, the last two terms are $O\left(h^{2}(X_{n}\circ
v^{(-1)}(t))^{2}+h|X_{n}\circ v^{(-1)}(t)r_{1,n}(t,h,X_{n})|\right)$.
Moreover, by Lemma 4.3,

$\displaystyle\mathbb{E}_{v^{(-1)}(t)}\left[(\Delta(X_{n}\circ
v^{(-1)})(t))^{2}\right]$ $\displaystyle=$
$\displaystyle\left(\frac{h}{v^{\prime}(v^{(-1)}(t)}+O(h^{2})\right)b(v^{(-1)}(t))$
$\displaystyle+r_{2,n}(v^{(-1)}(t),v^{(-1)}(t+h)-v^{(-1)}(t),X_{n}\circ
v^{(-1)}(t))$

Noticing that $b=\frac{v^{\prime}}{u^{2}}$, we obtain that if $h\leq n^{-4},$
there exists $\eta>0$ such that

$\displaystyle\mathbb{E}_{v^{(-1)}(t)}\left((\Delta
Z_{n}(t))^{2}\right)=h+O(hn^{-\eta}+n^{-7}+n\mathds{1}_{|X_{n}\circ
v^{(-1)}(t)|>n^{0.2}}).$ (25)

##### The third moment of $\Delta Z_{n}$.

By equation (23),

$\displaystyle|\Delta Z_{n}(t))|=O(|\Delta X_{n}\circ v^{-1}(t)|+h|X_{n}\circ
v^{-1}(t)|).$

By bounding separately the four terms obtained in the expansion of the cube of
the right hand side, and using Lemmas 4.2, 4.3, 4.4, we obtain that if $h\leq
n^{-4},$ there exists $\eta>0$ such that

$\displaystyle\mathbb{E}_{v^{(-1)}(t)}\left[|\Delta
Z_{n}(t)|^{3}\right]=O(hn^{-\eta}+n^{-6}+n^{3/2}\mathds{1}_{|X_{n}\circ
v^{(-1)}(t)|>n^{0.2}}).$ (26)

##### Assumption A1.

We now use the previous estimates to check Assumption (A1) of Proposition 4.1.
We have to show that there exists a sequence $N_{n}\to\infty$ such that, for
any $0<s<t$ ,

$\sum_{k=0}^{N_{n}-1}\sum_{i=1}^{3}\mathbb{E}\left(\left|e_{i,n}\left(s+k\frac{t-s}{N_{n}},\frac{t-s}{N_{n}},Z_{n}(s+k\frac{t-s}{N_{n}})\right)\right|\right)\underset{n\rightarrow+\infty}{\rightarrow}0,$

where, whenever $n^{-5.5}\leq h\leq n^{-4},$ using (24)

$e_{1,n}\left(t,h,Z_{n}(t)\right)=O(hn^{-\eta}+\sqrt{n}\mathds{1}_{|X_{n}\circ
v^{(-1)}(t)|>n^{0.2}}),$

using (25)

$e_{2,n}\left(t,h,Z_{n}(t)\right)=O(hn^{-\eta}+n\mathds{1}_{|X_{n}\circ
v^{(-1)}(t)|>n^{0.2}}),$

and, using (26),

$e_{3,n}\left(t,h,Z_{n}(t)\right)=O(hn^{-\eta}+n^{3/2}\mathds{1}_{|X_{n}\circ
v^{(-1)}(t)|>n^{0.2}}).$

Fixing $N_{n}=n^{5},$ assumption A1 is true if we show that

$3n^{3/2}\sum_{k=0}^{N_{n}-1}\mathbb{P}(|X_{n}\circ
v^{(-1)}(s+k\frac{t-s}{N_{n}})|>n^{0.2})\underset{n\rightarrow+\infty}{\rightarrow}0.$

This is an easy consequence of Lemma A.1.

##  5 Tightness

Our approach to prove the tightness of $Z_{n}$ relies on the criterion by
Billingsley (Theorem 13.5 [3]). Namely we have to prove that, there exist
$\alpha>1$ and a constant $C>0$ such that for $t_{0}<r<s<t<t_{1}$

$\mathbb{E}((Z_{n}(s)-Z_{n}(r))^{2}(Z_{n}(t)-Z_{n}(s))^{2})\leq
C(t-r)^{\alpha}.$ (27)

We will treat separately the case where $(t-r)$ is very small comparatively to
$n$ and when it is not. More precisely we will distinguish between $(t-r)\leq
n^{-5}$ and $(t-r)>n^{-5}$.

### The case $(t-r)\leq n^{-5}$

As $Z_{n}(t)=(uX_{n})(v^{-1}(t)),$ and $v$ and $v^{-1}$ are Lipschitz on the
intervals we are considering, to prove (27) for $(t-r)\leq n^{-5}$ it suffices
to prove for some constant $D$ that, if $(t-r)\leq Dn^{-5}$, then

$\mathbb{E}(((uX_{n})(s)-(uX_{n})(r))^{2}((uX_{n})(t)-(uX_{n})(s))^{2})\leq
C(t-r)^{\alpha}.$ (28)

Since

$(uX_{n})(s)-(uX_{n})(r)=\frac{(u(s)-u(r))L_{n}(s)+u(r)(L_{n}(s)-L_{n}(r))-(u(s)\rho(s)-u(r)\rho(r))n}{\sqrt{n}},$

we have by straightforward calculation

$((uX_{n})(s)-(uX_{n})(r))^{2}\leq
O(n(s-r)^{2})+C\frac{(L_{n}(s)-L_{n}(r))^{2}}{n}.$ (29)

The same is evidently true for $((uX_{n})(t)-(uX_{n})(s))^{2}$. Observing that
for $(t-r)\leq n^{-5}$,

$n(s-r)^{2}\leq n(t-r)^{0.9}(t-r)^{1.1}\leq\frac{(t-r)^{1.1}}{n^{3.5}},$ (30)

and bounding $L_{n}$ by $n$, we are reduced to show that for some $\alpha>1$,

$\mathbb{E}\left[\frac{(L_{n}(s)-L_{n}(r))^{2}(L_{n}(t)-L_{n}(s))^{2}}{n^{2}}\right]\leq
C(t-r)^{\alpha}.$

As the difference $t-r$ is assumed very small, the term in the expectation
will be zero with a very high probability. Indeed for it to be non-zero it is
necessary that one edge is drawn during each of the intervals $[r,s]$ and
$[s,t]$. Brutally bounding $L_{n}(t)-L_{n}(r)$ by $n$ and noting that the
number of edges drawn in two distinct intervals are negatively correlated, we
have

$\mathbb{E}\left[\frac{(L_{n}(s)-L_{n}(r))^{2}(L_{n}(t)-L_{n}(s))^{2}}{n^{2}}\right]\leq
n^{2}\mathbb{P}\left(\mathrm{Bin}(n^{2},(t-r)/n)>0\right)^{2}\\\
=O(n^{4}(t-r)^{2})=O(1)(t-r)^{1.1}.$ (31)

Hence (27) holds.

### The case $(t-r)>n^{-5}$

In this case our strategy is to subdivide $t-r$ so that we can apply our
results from section 4. Thus, we take as increment $h=n^{-5}$. Set
$k=\lfloor\frac{t-s}{h}\rfloor$, $s_{i}=s+ih$ for $0\leq i\leq k$,
$s_{k+1}=t$. We construct similarly $r_{i}=r+ih$ for $0\leq
i\leq\lfloor\frac{r-s}{h}\rfloor$.

We have

$\mathbb{E}[(Z_{n}(s)-Z_{n}(r))^{2}(Z_{n}(t)-Z_{n}(s))^{2}]=\mathbb{E}\left[(Z_{n}(s)-Z_{n}(r))^{2}\mathbb{E}_{v^{(-1)}(s)}[(Z_{n}(t)-Z_{n}(s))^{2}]\right].$

We first determine an upper bound for
$\mathbb{E}_{v^{(-1)}(s)}[(Z_{n}(t)-Z_{n}(s))^{2}]$.

We write, for any function $g$, $\Delta_{i}g:=g(s_{i+1})-g(s_{i}).$ The
decomposition

$Z_{n}(t)-Z_{n}(s)=\sum_{i=0}^{k}\left(\Delta_{i}Z_{n}-\mathbb{E}_{v^{(-1)}(s_{i})}[\Delta_{i}Z_{n}]\right)+\sum_{i=0}^{k}\mathbb{E}_{v^{(-1)}(s_{i})}[\Delta_{i}Z_{n}]$

implies

$(Z_{n}(t)-Z_{n}(s))^{2}\leq
2\left(\sum_{i=0}^{k}\left(\Delta_{i}Z_{n}-\mathbb{E}_{v^{(-1)}(s_{i})}[\Delta_{i}Z_{n}]\right)\right)^{2}+2\left(\sum_{i=0}^{k}\mathbb{E}_{v^{(-1)}(s_{i})}[\Delta_{i}Z_{n}]\right)^{2}.$

Thus, by orthogonality of the centered terms:

$\mathbb{E}_{v^{(-1)}(s)}[(Z_{n}(t)-Z_{n}(s))^{2}]\leq 2(S_{1}+S_{2}).$

with

$S_{1}=\sum_{i=0}^{k}\mathbb{E}_{v^{(-1)}(s)}\left(\left(\Delta_{i}Z_{n}-\mathbb{E}_{v^{(-1)}(s_{i})}(\Delta_{i}Z_{n})\right)^{2}\right)$

and

$S_{2}=\mathbb{E}_{v^{(-1)}(s)}\left(\left(\sum_{i=0}^{k}\mathbb{E}_{v^{(-1)}(s_{i})}(\Delta_{i}Z_{n})\right)^{2}\right).$

To find an upper bound of $S_{2}$ we use (24) and get

$\mathbb{E}_{v^{(-1)}(s_{i})}[\Delta_{i}Z_{n}]=O(hn^{-\eta}+n^{-8}+\sqrt{n}\mathds{1}_{|X_{n}\circ
v^{(-1)}(s_{i})|>n^{0.2}}).$

Note that for $i=k,$ this estimate remains true when replacing $h$ by
$s_{k+1}-s_{k}<h$. Hence, recalling $h=n^{-5}$

$\left(\sum_{i=0}^{k}\mathbb{E}_{v^{(-1)}(s_{i})}(\Delta_{i}Z_{n})\right)^{2}=O\left(((k+1)h)^{2}+n\left(\sum_{i=0}^{k}\mathds{1}_{\mathcal{E}_{n}^{c}(v^{(-1)}(s_{i}))}\right)^{2}\right),$

whence

$S_{2}=O\left((t-r)^{2}+n(k+1)^{2}\max_{0<i<k}\mathbb{P}_{v^{(-1)}(s)}(\mathcal{E}_{n}^{c}(s_{i}))\right).$

It remains to deal with $S_{1}$. Using (25), the conditional second moment of
$\Delta Z_{n}(s_{i})$ is bounded by:

$E[(\Delta_{i}Z_{n})^{2}\mid{\mathcal{F}}_{n}(s_{i})]\leq
O(h+n^{-7}+n\mathds{1}_{|X_{n}\circ v^{(-1)}(s_{i})|>n^{0.2}}).$

Hence,

$\displaystyle S_{1}$ $\displaystyle=$
$\displaystyle\sum_{i=0}^{k}\mathbb{E}_{v^{(-1)}(s)}\left(\left(\Delta_{i}Z_{n}-\mathbb{E}_{v^{(-1)}(s_{i})}(\Delta_{i}Z_{n})\right)^{2}\right)$
$\displaystyle\leq$
$\displaystyle\sum_{i=0}^{k}\mathbb{E}_{v^{(-1)}(s)}(\Delta_{i}Z_{n})^{2}\mid{\mathcal{F}}_{n}(s)]=O\left(((k+1)h+n(k+1)\max_{0<i<k}\mathbb{P}_{v^{(-1)}(s)}(\mathcal{E}_{n}^{c}(s_{i}))\right).$

Putting together the estimates for $S_{1}$ and $S_{2}$ we have

$\mathbb{E}_{v^{(-1)}(s)}[(Z_{n}(t)-Z_{n}(s))^{2}]=O\left((t-s)+n(k+1)^{2}\max_{0<i<k}\mathbb{P}_{v^{(-1)}(s)}(\mathcal{E}_{n}^{c}(s_{i}))\right).$

Using this bound we have

$\mathbb{E}[(Z_{n}(s)-Z_{n}(r))^{2}(Z_{n}(t)-Z_{n}(s))^{2}]\leq
C(t-s)\mathbb{E}\left[(Z_{n}(s)-Z_{n}(r))^{2}\right]\\\
+C\mathbb{E}\left[(Z_{n}(s)-Z_{n}(r))^{2}n(k+1)^{2}\max_{0<i<k}\mathbb{P}_{v^{(-1)}(s)}(\mathcal{E}_{n}^{c}(s_{i}))\right].$

The second term is smaller than $n^{-10}$ using Lemma A.1. The first term is
bounded similarly as $\mathbb{E}_{v^{(-1)}(s)}[(Z_{n}(t)-Z_{n}(s))^{2}],$
therefore

$\mathbb{E}[(Z_{n}(s)-Z_{n}(r))^{2}(Z_{n}(t)-Z_{n}(s))^{2}]\leq
C(t-s)(s-r)\leq C(t-r)^{2},$

which finishes the proof of (27) for $t-s>n^{-5}$.

##  Appendix A Appendix

###  A.1 Rare events

In this section we give a bound on the probability of several events
introduced in the article. Recall that

  * •

$\mathcal{E}_{n}(t)$ is the event that, at time $t$, $|L_{n}(t)-n\rho(t)|\leq
n^{0.7}$ or equivalently $|X_{n}(t)|\leq n^{0.2}$;

  * •

$\mathcal{A}_{n}(t)$ is the event that no component of
$\tilde{\mathscr{G}}_{n}^{*}(t+h)$ has size greater than $\ln(n)^{2}$;

  * •

$\tilde{{\mathcal{A}}}_{n}(t)=\\{M_{i}\leq(\ln(n))^{2},\forall 1\leq i\leq
n^{2}\\}$ where the random variables $M_{i}$ are independent and distributed,
conditional on $\mathcal{F}_{n}(t)$, as the total progeny of a
$\mathrm{BGW}(n-L_{n}(t),\frac{t+h}{n})$.

We are going to show the following:

######  Lemma A.1.

Let $0<\gamma<0.5$.

  1. 1.

There exist positive constants $C$ and $C^{\prime}$ such that for every
$t\in[t_{0},t_{1}]$,

$\mathbb{P}(|X_{n}(t)|>n^{\gamma})\leq C^{\prime}\exp(-Cn^{2\gamma});$

  2. 2.

There exists $h_{0}>0$ such that, for $0\leq h\leq h_{0},$ uniformly in
$t\in[t_{0},t_{1}]$, on the event $\\{|X_{n}(t)|<n^{\gamma}\\}$, for all
$R>0$,

$\mathbb{P}(\mathcal{A}^{c}_{n}(t)\mid\mathcal{F}_{n}(t))=O(n^{-R})\text{ and
}\mathbb{P}(\tilde{\mathcal{A}^{c}}_{n}(t)\mid\mathcal{F}_{n}(t))=O(n^{-R}).$

#### Proof of the first assertion of Lemma A.1

Our main tool to prove the first part of Lemma A.1 is the following Theorem by
V.E. Stepanov:

######  Theorem (Theorem 1 in [15]).

Set $y_{0}>0$. Let $P_{n}(s)$ be the probability that a
$\mathrm{ER}(n,1-e^{-s})$ graph is connected.

$P_{n}\left(\frac{y}{n}\right)=\left(1-\frac{y}{e^{y}-1}\right)(1-e^{-y})^{n}(1+o(1))$
(32)

where the $o(1)$ is uniform in $y\geq y_{0}$.

Note that the result of V.E. Stepanov uses a different time parametrization
(namely ${\frac{t}{n}=1-e^{-\frac{y}{n}}}$). For the sake of readability we
will use Stepanov’s parametrization, and prove the following lemma, which is
equivalent to the first part of Lemma A.1.

######  Lemma A.2.

Set $\hat{L}_{n}(y)=L_{n}(n(1-e^{-y/n}))$ the largest order of components of a
${\mathrm{ER}(n,1-e^{-\frac{y}{n}})}$ graph.
For $1<y_{0}<y_{1}$ and $0<\gamma<0.5$, there exist positive constants $C$ and
$C^{\prime}$ such that for every $y\in[y_{0},y_{1}]$ and $n\in\mathbb{N}$,

$\mathbb{P}(|\hat{L}_{n}(y)-n\rho(y)|\geq n^{0.5+\gamma})\leq
C^{\prime}e^{-Cn^{2\gamma}}.$

###### Proof.

The ideas come from the proof of Theorem 2 in Stepanov’s article [15], however
we will use them towards a different objective. We call $E_{n,k}(s)$ the
expected number of components with $k$ vertices in an
$\mathrm{ER}(n,1-e^{-s})$ graph. As pointed out in Stepanov’s article (Eq.
(27) in [15]),

$E_{n,k}\left(\frac{y}{n}\right)=\binom{n}{k}e^{-yk(1-k/n)}P_{k}\left(\frac{y}{n}\right).$

Our idea is to use the upper bound $\mathbb{P}(\hat{L}_{n}(y)=k)\leq
E_{n,k}(y/n)$. Obviously, this upper bound is extremely bad if $k$ is much
smaller than $n$. Therefore, we will treat these values separately. Set
$0<\epsilon<\rho(y_{0})\wedge(1-\rho(y_{1})).$ We now decompose

$\mathbb{P}(|\hat{L}_{n}(y)-n\rho(y)|\geq
n^{\gamma+0.5})\leq\mathbb{P}(\hat{L}_{n}(y)\leq\epsilon
n)+\sum_{\begin{subarray}{c}\epsilon n\leq k\leq(1-\epsilon)n:\\\
|\frac{k}{n}-\rho(y)|\geq
n^{\gamma-0.5}\end{subarray}}E_{n,k}\left(\frac{y}{n}\right)\\\
+\mathbb{P}(\hat{L}_{n}(y)\geq(1-\epsilon)n).$ (33)

Note that the first and third terms are monotonous in $y$. Therefore, we only
have to bound them for $y=y_{0}$ and $y=y_{1}$, respectively. This can be done
using a large deviation principle for $\hat{L}_{n}(y)$ at a fixed value of $y$
(see for example Theorem 3.1 of [11]). It yields bounds of order $\exp(-Cn)$.

To deal with the second term we use Stirling’s formula to get, for $\epsilon
n\leq k\leq(1-\epsilon)n$,

$E_{n,k}\left(\frac{y}{n}\right)=\frac{1}{\sqrt{2\pi
k(1-\frac{k}{n})}\left(\frac{k}{n}\right)^{k}(1-\frac{k}{n})^{n-k}}e^{-yk(1-\frac{k}{n})}P_{k}\left(\frac{y}{n}\right)(1+o(1)).$

Using (32), there exists a constant $C_{y_{0}}>0$ only dependent on $y_{0}$
such that for $y\geq y_{0}$,

$P_{n}\left(\frac{y}{n}\right)\leq
C_{y_{0}}\left(1-\frac{y}{e^{y}-1}\right)(1-e^{-y})^{n}.$

In order to bound from above $E_{n,k}\left(\frac{y}{n}\right)$, we have to
deal with $P_{k}(\frac{y}{n})=P_{k}(\frac{k}{n}\frac{y}{k})$ where
$\frac{k}{n}y\geq\epsilon y_{0}$, hence

$E_{n,k}\left(\frac{y}{n}\right)\leq C_{\epsilon y_{0}}\frac{1}{\sqrt{2\pi
k(1-\frac{k}{n})}}\left(1-\frac{\frac{k}{n}y}{e^{\frac{k}{n}y}-1}\right)e^{n\phi(\frac{k}{n},y)},$

with

$\phi(x,y)=-xy(1-x)+x\ln(1-e^{-xy})-x\ln x-(1-x)\ln(1-x).$

This function cancels if and only if $x=0$ and $x=\rho(y)$. Indeed if we
express it as a function of $x$ and $\delta(x,y)=1-e^{-xy}-x$, it writes:

$\phi(x,y)=x\ln(1+\frac{\delta}{x})+(1-x)\ln(1-\frac{\delta}{1-x})=:\psi(x,\delta),$

and
$\dfrac{\partial\psi}{\partial\delta}(x,\delta)=-\dfrac{\delta}{(x+\delta)(1-x-\delta)}.$

We observe that $|\dfrac{\partial\psi}{\partial\delta}(x,\delta)|\geq
4|\delta|$, hence $\psi(x,\delta)\leq-2\delta^{2}$.

Therefore, $\psi$ vanishes only for $\delta=0$ (it increases before and
decreases after 0 since ${-x\leq\delta\leq 1-x}$).

Now note that $\delta(\cdot,y)$ is a concave function such that
$\delta(0,y)=\delta(\rho(y),y)=0.$ Furthermore
$\dfrac{\partial\delta(x,y)}{\partial x}|_{x=0}=y-1\geq y_{0}-1$ and
$\dfrac{\partial\delta(x,y)}{\partial
x}|_{x=\rho(y)}=y(1-\rho(y))-1=\lambda(y)-1<\lambda(y_{0})-1<0$.

Consequently for $k$ such that $\epsilon n\leq k\leq(1-\epsilon)n$ and
$|\frac{k}{n}-\rho(y)|\geq n^{\gamma-0.5}$, we get
${\left|\delta\left(\frac{k}{n},y\right)\right|>Cn^{\gamma-0.5}}$, which
implies $\left|\phi\left(\frac{k}{n},y\right)\right|>Cn^{2\gamma-1}$ and in
turn,

$E_{n,k}(\frac{y}{n})\leq C^{\prime}e^{-Cn^{2\gamma}}.$

Summing over those values of $k$ shows that the sum in the right hand side of
(33) is smaller than $C^{\prime}e^{-Cn^{2\gamma}}$.

∎

#### Proof of the second assertion of Lemma A.1

To prove the second part of Lemma A.1, let us start by showing that the event

$\mathcal{A}^{c}_{n}(t)=\\{\tilde{\mathscr{G}}_{n}^{*}(t+h)\text{ has
components of order greater than }\ln^{2}(n)\\}$

is a rare event uniformly for $t\in[t_{0},t_{1}]$ as $n$ tends to $+\infty$.
The idea is that on the event $\\{|X_{n}(t)|<n^{\gamma}\\}$,
$\tilde{\mathscr{G}}_{n}^{*}(t+h)$ is distributed as a subcritical ER graph
for $n$ large enough so we can apply the following result on subcritical ER
graph:

######  Theorem (Theorem 4.4 in [16]).

Set $I(c)=c-1-\ln(c)$ and $|\mathcal{C}_{max}|$ the order of the largest
components of an $\mathrm{ER}(n,\frac{c}{n})$ graph for $c>0$.
For every $0<c<1$ and $a>0$, $\mathbb{P}(|\mathcal{C}_{\max}|\geq a\ln(n))\leq
n^{1-aI(c)}$.

In our case, the parameter $c$ depends on $X_{n}(t)$: the graph
$\tilde{\mathscr{G}}_{n}^{*}(t+h)$ is distributed as a
${\mathrm{ER}(n-L_{n}(t),\frac{c_{n}(t)}{n-L_{n}(t)})}$ graph with
$c_{n}(t)=\lambda(t)-\frac{X_{n}(t)}{\sqrt{n}}(t+h)+h(1-\rho(t))$. As $I(c)$
is decreasing on $[0,1],$ the result follows as soon as we show that there
exists $\epsilon\in]0,1[$ and $n_{0}$ not depending on $t$ such that, for
$n\geq n_{0},$ $c_{n}(t)<1-\epsilon$ on the event
$\\{|X_{n}(t)|<n^{\gamma}\\}$. This holds for $h$ small enough since, on the
event $\\{|X_{n}(t)|<n^{\gamma}\\}$,
$\frac{X_{n}(t)}{\sqrt{n}}(t+h)<2t_{1}n^{\gamma-0.5}$ and
$\sup_{[t_{0},t_{1}]}\lambda(t)<1$.

To deal with the event
$\tilde{{\mathcal{A}}}_{n}(t)=\\{M_{i}\leq(\ln(n))^{2},\forall 1\leq i\leq
n^{2}\\}$, we observe that, on the event $\\{|X_{n}(t)|<n^{\gamma}\\}$, the
random variables $M_{i}$, $1\leq i\leq n^{2}$, can be dominated by the total
progeny of a sub-critical Poisson Galton-Watson process with parameter
$1-\epsilon$ for every $t\in[t_{0},t_{1}].$ This follows a Borel distribution,
which admits an exponential moment.

###  A.2 Proof of Proposition 4.1

Let $k\in\mathbb{N}^{*}$. Consider $k$ positive reals $0<s_{1}<\ldots<s_{k}$
and $k$ functions $f_{1},\ldots f_{k}$ in $\mathcal{H}$. We shall prove that
$\mathbb{E}(\prod_{i=1}^{k}f_{i}(Z_{n}(s_{i})))$ converges towards
$\mathbb{E}(\prod_{i=1}^{k}f_{i}(Z(s_{i})))$ as $n$ tends to $+\infty$. To
study the difference of these two expectations, set

$I_{n,u,v}(f)=\mathbb{E}(f(Z_{n}(v))\mid\mathcal{F}_{n}(u))-T_{v-u}f(Z_{n}(u))\quad\forall
0\leq u<v\text{ and }\forall f\in\mathcal{H}.$

With this notation,

$\mathbb{E}(f_{1}(Z_{n}(s_{1})))=\mathbb{E}(I_{n,0,s_{1}}(f_{1}))+\mathbb{E}\left(T_{s_{1}}f_{1}(Z_{n}(0))\right)$

and for $k\geq 2$, by setting $s_{0}=0$

$\mathbb{E}\left(\prod_{i=1}^{k}f_{i}(Z_{n}(s_{i}))\right)=\mathbb{E}\left(\prod_{i=1}^{k-1}f_{i}(Z_{n}(s_{i}))I_{n,s_{k-1},s_{k}}(f_{k})\right)\\\
+\mathbb{E}\left(\prod_{i=1}^{k-1}f_{i}(Z_{n}(s_{i}))T_{s_{k}-s_{k-1}}f_{k}(Z_{n}(s_{k-1}))\right).$

Let us note that if $h_{1},h_{2}\in\mathcal{H}$ and $0<u<v$, the two functions
$T_{v-u}h_{1}$ and $h_{1}T_{u,v}h_{2}$ belong to $\mathcal{H}$. Therefore,
Proposition 4.1 follows by induction on $k$ if we show that
$\mathbb{E}(|I_{n,u,v}(f)|)$ converges to $0$ as $n$ tends to $+\infty$ for
every $0\leq u<v$ and $f\in\mathcal{H}$.
Set $h_{n}=\frac{v-u}{N_{n}}$, $u_{j}=u+jh_{n}$ for $j\in\\{0,\ldots,N_{n}\\}$
so that $v=u_{N_{n}}$ and $g_{j}=T_{(N-j)h_{n}}f$. With these notations,

$\displaystyle I_{n,u,v}(f)$ $\displaystyle=$
$\displaystyle\mathbb{E}(g_{N_{n}}(Z_{n}(u_{N_{n}}))\mid\mathcal{F}_{n}(u_{0}))-g_{0}(Z_{n}(u_{0}))$
$\displaystyle=$
$\displaystyle\sum_{j=0}^{N_{n}-1}\mathbb{E}\left[\Big{(}\mathbb{E}(g_{j+1}(Z_{n}(u_{j+1}))\mid\mathcal{F}_{n}(u_{j}))-g_{j}(Z_{n}(u_{j}))\Big{)}\mid\mathcal{F}_{n}(u_{0})\right].$

Let us note that $g_{j}=T_{h_{n}}g_{j+1}$. Therefore,
$g_{j+1}(Z_{n}(u_{j+1}))-g_{j}(Z_{n}(u_{j})$ can be decomposed as the
difference of $g_{j+1}(Z_{n}(u_{j+1}))-g_{j+1}(Z_{n}(u_{j})))$ and
$T_{h_{n}}g_{j+1}(Z_{n}(u_{j}))-g_{j+1}(Z_{n}(u_{j})))$.

By applying Taylor expansion to
$g_{j+1}(Z_{n}(u_{j+1}))-g_{j+1}(Z_{n}(u_{j}))$ and by Assumptions A1,

$\mathbb{E}(g_{j+1}(Z_{n}(u_{j+1}))\mid\mathcal{F}_{n}(u_{j}))-g_{j+1}(Z_{n}(u_{j}))=\\\
h_{n}\left(g^{\prime}_{j+1}(Z_{n}(u_{j}))a(Z_{n}(u_{j}))+\frac{1}{2}g^{\prime\prime}_{j+1}(Z_{n}(u_{j}))b(Z_{n}(u_{j}))\right)+R_{n,j}$

with
$|R_{n,j}|\leq\sum_{i=1}^{3}||g^{(i)}_{j+1}||_{\infty}|e_{i,n}(u_{j},h_{n},Z_{n}(u_{j}))|$.
By the properties of the semigroup $T_{t}$, we have a similar expansion of
$T_{h_{n}}g_{j+1}(Z_{n}(u_{j}))-g_{j+1}(Z_{n}(u_{j}))$:

$T_{h_{n}}g_{j+1}(Z_{n}(u_{j}))-g_{j+1}(Z_{n}(u_{j}))=\\\
h_{n}\left(g^{\prime}_{j+1}(Z_{n}(u_{j}))a(Z_{n}(u_{j}))+\frac{1}{2}g^{\prime\prime}_{j+1}(Z_{n}(u_{j}))b(Z_{n}(u_{j}))\right)+\bar{R}_{n,j}$

with
$|\bar{R}_{n,j}|\leq\sum_{i=1}^{3}||g_{j+1}^{(i)}||_{\infty}|\epsilon_{i,}(h_{n},Z_{n}(u_{j}))|$.
By Assumption A2,
$\sup_{j\in\\{1,\ldots,N_{n}\\}}\sum_{k=1}^{3}||g_{j}^{(k)}||_{\infty}$ is
finite and depends only of $v-u$. By denoting it $M(v-u)$, we have:

$\mathbb{E}(|I_{n,u,v}(f)|)\leq
M(u,v)\sum_{j=0}^{N_{n}-1}\sum_{i=1}^{3}\left\\{\mathbb{E}\left(\left|e_{i,n}(u_{j},h_{n},Z_{n}(u_{j}))\right|\right)+\mathbb{E}\left(\left|\epsilon_{i,}(h_{n},Z_{n}(u_{j}))\right|\right)\right\\}.$

Finally, Assumptions A1 and A3 imply that the upper bound converges towards 0
as $n$ tends to $+\infty$.

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