# GAUSSIAN STRUCTURE IN COALESCING STOCHASTIC FLOWS

A. A. DOROGOVTSEV
Institute of Mathematics,
National Academy of Science of Ukraine,
Kyiv, Ukraine<EMAIL_ADDRESS>E.V. GLINYANAYA
Institute of Mathematics,
National Academy of Science of Ukraine,
Kyiv, Ukraine<EMAIL_ADDRESS>
###### Abstract

In the paper we consider the point measure that corresponds to Arratia flow.
The central limit theorem of the multiple integrals with respect to this
measure was obtained.

00footnotetext: Keywords: central limit theorems; stationary point measure;
flow with coalescing. AMS Subject Classification: 60F05, 60H40, 60G57

##  1 Introduction.

Let $\\{x(u,t),\ u\in\mathbb{R},\ t\geq 0\\}$ be the Arratia flow [1]. It is
known [8] that for every $t>0$ the set $x(\mathbb{R},t)$ is countable and
locally finite. Denote $\\{u_{i}\\}_{i\in\mathbb{Z}}:=x(\mathbb{R},t)$ and
define the point measure

$N_{t}=\sum_{i\in\mathbb{Z}}\delta_{u_{i}}.$ (1)

In this paper we investigate limits in the central limit theorem for integral
functionals with respect to the factorial powers of the point measure $N_{t}.$

######  Definition 1.

([11]) n-th factorial power $N^{(n)}$ of the point measure
$N=\sum_{i\in\mathbb{Z}}\delta_{u_{i}}$ is defined by

$N^{(n)}(C)=\sum_{\begin{subarray}{c}i_{1}\neq\ldots\neq
i_{n}\end{subarray}}1\\!\\!\,\mathrm{I}{\\{(u_{i_{1}},\ldots,u_{i_{n}})\in
C\\}},$

where $C\in\mathcal{B}(\mathbb{R}^{n}).$

By $N^{\otimes n}$ we denote the usual power of measure $N,$ i.e. for
$\Delta_{i}\in\mathbb{R},$

$N^{\otimes n}(\Delta_{1}\times...\times\Delta_{n})=N(\Delta_{1})\ldots
N(\Delta_{n}).$

######  Definition 2.

A locally integrable function $\rho^{(n)}:\mathbb{R}^{n}\to\mathbb{R}_{+}$ is
called an $n-$point density function of a point measure $N$ if, for any subset
$A\in\mathcal{B}(\mathbb{R}^{n})$ the following formula holds

$\mathbf{E}N^{(n)}(A)=\int_{A}\rho^{(n)}(\vec{x})d\vec{x}$

For example, the Poisson process on the line with intensity measure
$\mu(dx)=p(x)dx$ has $n-$point density function that is equal to

$\rho^{(n)}(\vec{x})=p^{\otimes n}(\vec{x})=p(x_{1})p(x_{2})\ldots p(x_{n}).$

In the papers [12], [15] the authors proved existence of the $n-$point density
functions $\rho^{(n)}$ for the point measure $N_{t}.$ For instance,

$\displaystyle\rho_{t}^{(1)}(v)=\frac{1}{\sqrt{\pi t}}$ (2)
$\displaystyle\rho_{t}^{(2)}(v_{1},v_{2})=\dfrac{1}{\pi
t}\left(1+\dfrac{\left|v_{2}-v_{1}\right|}{2\sqrt{t}}\cdot
e^{-(v_{2}-v_{1})^{2}/4t}\cdot\int\limits_{\left|v_{2}-v_{1}\right|/\sqrt{t}}^{+\infty}e^{-v^{2}/4}\,dv-e^{-(v_{2}-v_{1})^{2}/2t}\right)$
(3)

Connection between integrals with respect to usual power of measure and
factorial power is given in the next lemma.

######  Lemma 3 ([3]).

For any symmetric non-negative function $f$ we have

$\int_{\mathbb{R}^{n}}f(\vec{x})N^{\otimes
n}(d\vec{x})=\sum_{k=1}^{n}\sum_{\begin{subarray}{c}1\leq l_{1}\leq\ldots\leq
l_{k}\\\
l_{1}+\ldots+l_{k}=n\end{subarray}}A_{l_{1},\ldots,l_{k}}^{n}\int_{\mathbb{R}^{k}}f(\underbrace{x_{1},\ldots,x_{1}}_{l_{1}},\ldots,\underbrace{x_{k},\ldots,x_{k}}_{l_{k}})N^{(k)}(d\vec{x}),$

$\int_{\mathbb{R}^{n}}f(\vec{x})N^{(n)}(d\vec{x})=\sum_{k=1}^{n}\sum_{\begin{subarray}{c}1\leq
l_{1}\leq\ldots\leq l_{k}\\\
l_{1}+\ldots+l_{k}=n\end{subarray}}a_{l_{1},\ldots,l_{k}}^{n}\int_{\mathbb{R}^{k}}f(\underbrace{x_{1},\ldots,x_{1}}_{l_{1}},\ldots,\underbrace{x_{k},\ldots,x_{k}}_{l_{k}})N^{\otimes
n}(d\vec{x}),$

where the constants $A_{l_{1},\ldots,l_{k}}^{n}$ and
$a_{l_{1},\ldots,l_{k}}^{n}$ do not depend on $f$ and $N$

Now we will define integral with respect to the point measure $N_{t}$ (1)
which corresponds to the Arratia flow at time $t$.

######  Theorem 4.

The operator $A_{t}$ which is defined for $\phi\in L_{2}([0,1])$ as

$A_{t}\phi=\int_{0}^{1}\phi(v)dN_{t}(v)$

is a continuous operator acting from $L_{2}([0,1])$ to
$L_{2}(\Omega,\mathcal{F},\mathbf{P})$ and

$\|A_{t}\|^{2}\leq\frac{1}{\pi t}+\frac{1}{\sqrt{\pi t}}$

###### Proof.

For non-negative function $\phi$, by Campbel formula [11]

$\mathbf{E}A_{t}\phi=\int_{0}^{1}\phi(v)\rho^{(1)}(v)dv$

and by the definition of $\rho^{(1)}$ and $\rho^{(2)}$

$\mathbf{E}\left(A_{t}\phi\right)^{2}=\int_{0}^{1}\int_{0}^{1}\phi(v_{1})\phi(v_{2})\rho^{(2)}(v_{1},v_{2})dv_{1}\,dv_{2}+\int_{0}^{1}\phi^{2}(v_{1})\rho^{(1)}(v_{1})dv_{1},$

For the point measure $N_{t}$ its $n-$point density functions $\rho_{t}^{(n)}$
satisfy inequality $\rho_{t}^{(n)}\leq\frac{1}{(\pi t)^{n/2}}$ for all $n\geq
1$, [12]. Using this we can write

$\mathbf{E}\left(A_{t}\phi\right)^{2}\leq\left(\frac{1}{\pi
t}+\frac{1}{\sqrt{\pi t}}\right)\|\phi\|^{2}_{L_{2}([0;1])}$

Using obtained inequality, operator $A_{t}$ can be extended from operator
defined on $C([0;1])$ to a continuous operator from $L_{2}([0;1])$ to
$L_{2}(\Omega,\mathcal{F},\mathbf{P}).$ ∎

######  Remark 5.

Note that the variance for the integral
$A_{t}\phi=\int_{0}^{1}\phi(v)dN_{t}(v)$ is given by formula

$\mathrm{Var}\,{A_{t}\phi}=\\\
\int_{0}^{1}\int_{0}^{1}\phi(v_{1})\phi(v_{2})\Big{(}\rho_{t}^{(2)}(v_{1},v_{2})-\frac{1}{{\pi
t}}\Big{)}dv_{1}\,dv_{2}+\frac{1}{\sqrt{\pi
t}}\int_{0}^{1}\phi^{2}(v_{1})dv_{1}$ (4)

From the lemma 3 we get the formulas for moments of the defined integral

######  Lemma 6.

$\mathbf{E}\left(\int_{0}^{1}\phi(u)dN_{t}(u)\right)^{k}=\sum_{j=1}^{k}\sum_{\begin{subarray}{c}l_{1},\ldots,l_{j}\geq
1,\\\
l_{1}+\ldots+l_{j}=k\end{subarray}}A^{k}_{l_{1},\ldots,l_{j}}\int_{\mathbb{R}^{j}}\prod_{i=1}^{j}\phi^{l_{i}}(x_{i})\rho_{t}^{(j)}(\vec{x})d\,\vec{x},$

with constants $A^{k}_{l_{1},\ldots,l_{j}}$ from Lemma 3. For example, for
$k=3,$

$\mathbf{E}\left(\int_{0}^{1}\phi(u)dN_{t}(u)\right)^{3}=\\\
=\int_{[0;1]^{3}}\phi^{\otimes
3}(\vec{u})\rho_{t}^{(3)}(\vec{u})d\vec{u}+3\int_{[0;1]^{2}}\phi^{\otimes
2}(u_{1},u_{2})\rho_{t}^{(2)}(u_{1},u_{2})\varphi(u_{1})d\vec{u}+\frac{1}{\sqrt{\pi
t}}\int_{[0;1]}\phi^{3}(u)du$

######  Remark 7.

This lemma shows that moments of integrals are finite if integrand has finite
moments.

Also we can define multiple integral with respect to factorial power of the
point measure $N_{t}$

######  Theorem 8.

Let $L_{2,{symm}}([0;1]^{k})$ be the space of square integrable symmetric
functions. The linear operator $A^{k}_{t}$ from $L_{2,symm}([0;1]^{k})$ to
$L_{2}(\Omega,\mathcal{F},\mathbf{P})$ which is defined as

$A^{k}_{t}\phi=\int_{[0;1]^{k}}\phi(\vec{v})N^{(k)}_{t}(d\,v_{1},\ldots,d\,v_{k}),$

is a continuous operator.

###### Proof.

By definition of $k-$point functions $\rho^{(k)}$ of the measure $N_{t}^{(k)}$

$\mathbf{E}\int_{[0;1]^{k}}\phi(\vec{v})N^{(k)}_{t}(d\,v_{1},\ldots,d\,v_{k})=\int_{[0;1]^{k}}\phi(\vec{v})\rho_{t}^{(k)}(\vec{v})d\,\vec{v}.$

Rewriting the square of integral with respect to factorial product as sum of
series indexed with non-equal indexes we get

$\mathbf{E}\left(\int_{[0;1]^{k}}\phi(\vec{v})N^{(k)}_{t}(d\,v_{1},\ldots,d\,v_{k})\right)^{2}=\\\
=\mathbf{E}\sum_{i_{1},j_{1}}\sum_{\begin{subarray}{c}i_{2}\neq i_{1}\\\
j_{2}\neq j_{1}\end{subarray}}\ldots\sum_{\begin{subarray}{c}i_{k}\neq
i_{1},\ldots,i_{k-1}\\\ j_{k}\neq
j_{1},\ldots,j_{k-1}\end{subarray}}\phi(u_{i_{1}},\ldots,u_{i_{k}})\phi(u_{j_{1}},\ldots,u_{j_{k}})=\\\
=\sum_{l=0}^{k}c_{l}\int_{[0;1]^{k}}\int_{[0;1]^{k-l}}\phi(\vec{u})\phi(u_{1},\ldots,u_{l},v_{1},\ldots,v_{k-l})\rho^{(2k-l)}(\vec{v},\vec{u})d\,\vec{v}d\,\vec{u}$

with some constants $c_{l}\in\mathbb{N}.$

Applying uniform estimation for the $n-$point densities
$\rho_{t}^{(n)}\leq\frac{1}{(\pi t)^{n/2}}$ [12] we get

$\mathbf{E}\left(\int_{[0;1]^{n}}\phi(\vec{v})N^{(n)}_{t}(d\,v_{1},\ldots,d\,v_{k})\right)^{2}\leq
C\int_{[0;1]^{n}}\phi^{2}(\vec{v})d\vec{v}.$

This proves the theorem. ∎

##  2  Finite-dimensional central limit theorem for $\mathbf{(A_{t})_{t>0}}$

Let $\mathcal{P}=\\{f:\mathbb{R}\to\mathbb{R}:\ f|_{[0;1]}\in L_{2}([0;1]),\
f(x)=f(x+1),\ x\in\mathbb{R}\\}$ be a class of functions $f$ such that:
1) $f$ is 1-periodic;
2) restriction of $f$ on the interval $[0;1]$ is square integrable.

Similarly to definition of the operator $A_{t}$ let us define the integral
operator

$A_{k,t}f=\int_{k}^{k+1}f(u)N_{t}(du).$

Such operators are well-defined for functions $f\in\mathcal{P}.$ In this
section we will prove that the sequence $\\{A_{k,t}f\\}_{k\geq 1}$ satisfies
central limit theorem.

The main reason why we can expect the validity of the central limit theorem is
the weak dependence of the sequence $\\{A_{k,t}f\\}_{k\geq 1}$. It can be
checked that this sequence satisfies a mixing condition. Recall that for a
stationary sequence $\\{\xi_{k}\\}_{k\in K}$ (here $K=\mathbb{R}\text{ or
}K=\mathbb{N}$) its $\alpha-$mixing coefficient is defined as

$\alpha_{\xi}(h)=\sup\\{|\mathbf{P}(AB)-\mathbf{P}(A)\mathbf{P}(B)|,\
A\in\mathcal{F}(\xi)_{-\infty}^{0},\ B\in\mathcal{F}(\xi)_{h}^{\infty}\\}$

where $\mathcal{F}(\xi)_{a}^{b}=\sigma\\{\xi_{k},\ a<k<b\\}.$

######  Lemma 9.

Let $f\in\mathcal{P}$. For fixed $t>0$ consider sequence $\xi_{k}=A_{k,t}f,$
$k\in\mathbb{Z}$. Then

$\alpha_{\xi}(h)\leq 4\int_{h/3}^{\infty}\frac{1}{\sqrt{2\pi
t}}e^{-x^{2}/2}dx\leq\frac{12}{h\sqrt{2\pi t}}\exp(-h^{2}/18).$

###### Proof.

For arbitrary $n,m\in\mathbb{N}$, denote by $D_{n,m}$ the set of vectors
$\vec{u},\vec{a}\in\mathbb{R}^{n}$, $\vec{v},\vec{b}\in\mathbb{R}^{m}$, such
that $u_{1}<u_{2}<\ldots u_{n}<v_{1}<v_{2}<\ldots<v_{m}$. Consider

$\Delta:=\sup_{D_{n,m}}|\mathbf{E}e^{i(\vec{a},x(\vec{u},t))}e^{i(\vec{a},x(\vec{v},t))}1\\!\\!\,\mathrm{I}_{\\{x(u_{n},t)<0\\}}1\\!\\!\,\mathrm{I}_{\\{x(v_{1},t)>h\\}}-\\\
-\mathbf{E}e^{i(\vec{a},x(\vec{u},t))}1\\!\\!\,\mathrm{I}_{\\{x(u_{n},t)<0\\}}\mathbf{E}e^{i(\vec{a},x(\vec{v},t))}1\\!\\!\,\mathrm{I}_{\\{x(v_{1},t)>h\\}}|$

where we denoted by $x(\vec{u},t)=(x(u_{1},t),\ldots,x(u_{n},t)).$ For $L<h/2$
let $B$ be the random event $B=\\{x(L,t)>0,x(h-L,t)<h\\}$. Denoting by
$\tilde{x}$ an independent copy of the Arratia flow we get

$\Delta\leq\sup_{D_{n,m}}|\mathbf{E}e^{i(\vec{a},x(\vec{u},t))}e^{i(\vec{a},x(\vec{v},t))}1\\!\\!\,\mathrm{I}_{\\{x(u_{n},t)<0\\}}1\\!\\!\,\mathrm{I}_{\\{x(v_{1},t)>h\\}}1\\!\\!\,\mathrm{I}_{B}-\\\
-\mathbf{E}e^{i(\vec{a},\tilde{x}(\vec{u},t))}1\\!\\!\,\mathrm{I}_{\\{\tilde{x}(u_{n},t)<0\\}}e^{i(\vec{a},x(\vec{v},t))}1\\!\\!\,\mathrm{I}_{\\{x(v_{1},t)>h\\}}|+\mathbf{P}(\bar{B})\leq\\\
\leq\sup_{D_{n,m}}|\mathbf{E}e^{i(\vec{a},\tilde{x}(\vec{u},t))}e^{i(\vec{a},x(\vec{v},t))}1\\!\\!\,\mathrm{I}_{\\{\tilde{x}(u_{n},t)<0\\}}1\\!\\!\,\mathrm{I}_{\\{x(v_{1},t)>h\\}}1\\!\\!\,\mathrm{I}_{B}-\\\
-\mathbf{E}e^{i(\vec{a},\tilde{x}(\vec{u},t))}1\\!\\!\,\mathrm{I}_{\\{\tilde{x}(v_{n},t)<0\\}}e^{i(\vec{a},\tilde{x}(\vec{v},t))}1\\!\\!\,\mathrm{I}_{\\{\tilde{x}(v_{1},t)>h\\}}1\\!\\!\,\mathrm{I}_{B}|+2\mathbf{P}(\bar{B}),$

where we use that the processes $x(u_{i},\cdot)$ and $x(v_{j},\cdot)$ are
independent up to the meeting moment, i.e. $d\langle
x(u_{i},\cdot),x(v_{j},\cdot)\rangle(t)=1\\!\\!\,\mathrm{I}_{\\{x(u_{i},t)=x(v_{j},t)\\}}dt.$
Now, one can get that for $L=h/3$ the upper bound $P(\bar{B})\leq
2\int_{h/3}^{\infty}\frac{1}{\sqrt{2\pi t}}e^{-x^{2}/2}dx$. Since the family
of functions
$\\{e^{i(\vec{a},x(\vec{u},t))}1\\!\\!\,\mathrm{I}_{x(u_{n},t)<0},\
u_{1}<u_{2}<\ldots<u_{n},\ n\in\mathbb{N}\\}$ generates the $\sigma-$field
$\sigma\\{\xi_{k},k<0\\}$ the statement of the lemma follows from obtained
inequality. ∎

It is known that if the $\alpha-$mixing coefficient for a stationary sequence
decrease fast enough then under suitable moments conditions this sequence
satisfies the central limit theorem [9].

######  Theorem 10.

  1. 1.

Let $f\in\mathcal{P}$ such that $f|_{[0;1]}\in L_{3}([0;1])$. For any $t>0$ we
have

$X_{t}^{n}(f)=\frac{\sum_{k=0}^{n-1}\left(A_{k,t}f-\mathbf{E}A_{k,t}f\right)}{\sqrt{n}}\Rightarrow\zeta_{f}(t),\
n\to\infty,$

where $\zeta_{f}(t)$ is a Gaussian random variable with zero mean and variance

$\sigma^{2}_{t}(f)=\int_{0}^{1}\int_{0}^{1}f(v_{1})f(v_{2})G_{t}(v_{1},v_{2})dv_{1}\,dv_{2}+\frac{1}{\sqrt{\pi
t}}\int_{0}^{1}f^{2}(v_{1})dv_{1},$

with
$G_{t}(v_{1},v_{2})=g_{t}(v_{1}-v_{2})+2\sum_{l=1}^{\infty}g_{t}(v_{1}-v_{2}+l),$
and $g_{t}(v_{1}-v_{2})=\rho_{t}^{(2)}(v_{1},v_{2})-\frac{1}{{\pi t}}.$

  2. 2.

For any $0<t_{1}<t_{2}<\ldots<t_{m}<T$ and $f\in\mathcal{P}$

$\Big{(}X_{t_{1}}^{n}(f),\ldots,X_{t_{m}}^{n}(f)\Big{)}\Rightarrow\left(\zeta_{f}(t_{1}),\ldots\zeta_{f}(t_{m})\right)$

where $\left(\zeta_{f}(t_{1}),\ldots\zeta_{f}(t_{m})\right)$ is a Gaussian
vector with zero mean and covariance matrix
$\Sigma_{t_{1},\ldots,t_{m}}(f)=(c_{ij})_{i,j=1}^{m}$ with

$c_{ij}=\lim_{n\to\infty}\frac{1}{n}\mathrm{cov}\,\left(X_{t_{i}}^{n}(f),X_{t_{j}}^{n}(f)\right),\
c_{ii}=\sigma^{2}_{t_{i}}(f),$

  3. 3.

For any functions $f^{(1)},\ldots,f^{(m)}\in\mathcal{P}$ and $t>0$ we have

$\Big{(}X_{t}^{n}\left((f^{(1)}\right),\ldots,X_{t}^{n}\left(f^{(m)}\right)\Big{)}\Rightarrow\left(\zeta_{f^{(1)}}(t),\ldots,\zeta_{f^{(m)}}(t)\right),\
n\to\infty,$

where $\left(\zeta_{t}(f^{(1)}),\ldots,\zeta_{t}(f^{(m)})\right)$ is a
centered Gaussian vector with covariance

$\mathrm{cov}\,\left(\zeta_{t}(f^{(i)}),{\zeta_{t}(f^{(j)})}\right)=\\\
=\frac{1}{2}\int_{0}^{1}\int_{0}^{1}\left[f^{(i)}(u)f^{(j)}(v)+f^{(i)}(v)f^{(j)}(u)\right]G_{t}(u,v)dudv+\\\
+\frac{1}{\sqrt{\pi t}}\int_{0}^{1}f^{(i)}(u)f^{(j)}(u)du.$

######  Remark 11.

The expression for the covariances $c_{i,l}$ in the second statement of the
theorem will be discussed in the next section.

###### Proof.

Proof of the first statement of the theorem.
For each $t>0$ we denote

$X_{t}^{n}(f)=\frac{\sum_{k=0}^{n-1}\left(A_{k,t}f-\mathbf{E}A_{k,t}f\right)}{\sqrt{n}}=\frac{1}{\sqrt{n}}\sum_{k=0}^{n-1}Y_{k}$

Due to periodicity of the function $f$ and stationarity with respect to
spatial variable of the Arratia flow, the sequence of random variables
$\\{Y_{k}\\}_{k\in\mathbb{Z}}$ is strictly stationary. By lemma 9 its mixing
coefficient

$\alpha_{Y}(h)\leq c_{T}\int_{h}^{\infty}e^{-y^{2}/2}dy.$

By lemma 6 $\mathbf{E}|Y_{k}|^{3}<\infty.$ Now we can apply central limit
theorem for weakly dependent random variables ([9])

$X_{t}^{n}(f)\Rightarrow\mathcal{N}(0,\sigma^{2}_{t}(f)),\ \text{as
}{n\to\infty,}$

where

$\sigma^{2}_{t}(f)=\lim_{n\to\infty}\frac{1}{n}\mathrm{Var}\,{\sum_{k=0}^{n-1}Y_{k}}=\\\
=\lim_{n\to\infty}\frac{1}{n}\int_{0}^{n}\int_{0}^{n}f(v_{1})f(v_{2})\Big{(}\rho_{t}^{(2)}(v_{1},v_{2})-\frac{1}{{\pi
t}}\Big{)}dv_{1}\,dv_{2}+\frac{1}{\sqrt{\pi t}}\int_{0}^{n}f^{2}(v_{1})dv_{1}$

Note that the function $\Big{(}\rho_{t}^{(2)}(v_{1},v_{2})-\frac{1}{{\pi
t}}\Big{)}$ depends on $v_{1},v_{2}$ only via $|v_{1}-v_{2}|$ (see formula
(3)). Then we can define

$g_{t}(v_{1}-v_{2}):=\rho_{t}^{(2)}(v_{1},v_{2})-\frac{1}{{\pi t}}.$

Using periodicity of $f$

$\sigma^{2}_{t}(f)=\lim_{n\to\infty}\frac{1}{n}\Big{[}\int_{0}^{n}\int_{0}^{n}f(v_{1})f(v_{2})g_{t}(v_{1}-v_{2})dv_{1}\,dv_{2}+\frac{1}{\sqrt{\pi
t}}\int_{0}^{n}f^{2}(v_{1})dv_{1}\Big{]}=\\\
=\lim_{n\to\infty}\frac{1}{n}\Big{[}\sum_{k_{1},k_{2}=0}^{n-1}\int_{k_{1}}^{k_{1}+1}\int_{k_{2}}^{k_{2}+1}\\!\\!\\!f(v_{1})f(v_{2})g_{t}(v_{1}-v_{2})dv_{1}\,dv_{2}+n\frac{1}{\sqrt{\pi
t}}\int_{0}^{1}f^{2}(v_{1})dv_{1}\Big{]}=\\\
=\lim_{n\to\infty}\frac{1}{n}\Big{[}\sum_{k_{1},k_{2}=0}^{n-1}\int_{0}^{1}\int_{0}^{1}f(v_{1})f(v_{2})g_{t}(v_{1}+k_{1}-v_{2}-k_{2})dv_{1}\,dv_{2}\Big{]}+\frac{1}{\sqrt{\pi
t}}\int_{0}^{1}f^{2}(v_{1})dv_{1}=\\\
=\lim_{n\to\infty}\frac{1}{n}\int_{0}^{1}\int_{0}^{1}f(v_{1})f(v_{2})\Big{[}2\sum_{l=1}^{n-1}(n-l)g_{t}(v_{1}-v_{2}+l)+ng_{t}(v_{1}-v_{2})\Big{]}dv_{1}\,dv_{2}+\\\
+\frac{1}{\sqrt{\pi t}}\int_{0}^{1}f^{2}(v_{1})dv_{1}$

For the function
$g_{t}(v_{1}-v_{2}):=\rho_{t}^{(2)}(v_{1},v_{2})-\frac{1}{{\pi t}}$ one can
write the precise formula using (3) :

$\displaystyle g_{t}(v_{1}-v_{2})=\dfrac{1}{\pi
t}\left(\dfrac{\left|v_{2}-v_{1}\right|}{2\sqrt{t}}\cdot
e^{-(v_{2}-v_{1})^{2}/4t}\cdot\int\limits_{\left|v_{2}-v_{1}\right|/\sqrt{t}}^{+\infty}e^{-v^{2}/4t}\,dv-e^{-(v_{2}-v_{1})^{2}/2t}\right).$
(5)

From this it is easy to see that the series
$\sum_{l=1}^{\infty}lg_{t}(v_{1}-v_{2}+l)$ converges uniformly on
$v_{1},v_{2}\in[0,1]$, so we can pass to the limit with respect to $n$ in the
formula for $\sigma^{2}_{t}(f)$

$\sigma^{2}_{t}(f)=\int_{0}^{1}\int_{0}^{1}f(v_{1})f(v_{2})\Big{[}g_{t}(v_{1}-v_{2})+2\sum_{l=1}^{\infty}g_{t}(v_{1}-v_{2}+l)\Big{]}dv_{1}\,dv_{2}+\\\
+\frac{1}{\sqrt{\pi t}}\int_{0}^{1}f^{2}(v_{1})dv_{1}.$

Proof of the second and third statements of the theorem are similar to the
first one. ∎

##  3 Central limit theorem for conditional expectations.

Attempt to find covariances in the statement (2) of theorem 10 naturally leads
to the question of calculation conditional expectations
$\mathbf{E}\left(\int_{0}^{1}f(u)N_{t_{2}}(du)\big{|}\mathcal{F}_{t_{1}}^{x}\right)$.
In this section we study limit behaviour of the conditional expectations for
$t_{1}<t_{2}$

$\frac{1}{\sqrt{n}}\mathbf{E}\left(\int_{0}^{n}f(u)N_{t_{2}}(du)\big{|}\mathcal{F}_{t_{1}}^{x}\right),\text{
as }n\to\infty$

where, as before, $N_{t}$ is the point measure associated to the Arratia flow
$x$ and $\mathcal{F}_{t}^{x}=\sigma\\{x(u,s),\ u\in\mathbb{R},\ s\leq t\\}.$

To this aim we need representation of the Arratia flow in term of Brownian
web. Recall that Brownian web is defined as a family of random processes
$\\{\varphi_{s,\cdot}(u)\in C([s,\infty))\\}$ such that, given
$(u_{1},t_{1}),...,(u_{m},t_{m})$ the processes
$\varphi_{t_{1},\cdot}(u_{1}),...,\varphi_{t_{m},\cdot}(u_{m})$ are coalescing
Brownian motions (see, for instance, [6]). In this section we consider Arratia
flow $x(u,t)=\varphi_{0,t}(u).$ We construct the point measure $N_{t_{2}}$
using measure $N_{t_{1}}$ and the map $\varphi_{t_{1},t_{2}}$ from Brownian
web.

To describe general properties of such construction let us introduce the
family of random measures on $\mathbb{R}$ related to the point measure $\mu$
and non-decreasing function $\varphi:\mathbb{R}\to\mathbb{R}$ such that

$\displaystyle\lim_{x\to+\infty}\varphi(x)=+\infty,$
$\displaystyle\lim_{x\to-\infty}\varphi(x)=-\infty.$

For every $k\geq 2$ and vector $\vec{v}=(v_{1},...,v_{k})$ denote

$\displaystyle v_{*}=\min\\{v_{1},...,v_{k}\\},$ $\displaystyle
v^{*}=\max\\{v_{1},...,v_{k}\\}.$

Define the set

$M_{k}^{\varphi}=\\{\vec{v}:\varphi(v_{*})=\varphi(v^{*})\\}\subset\mathbb{R}^{k}$

Note, that $M_{k}^{\varphi}$ is a Borel subset of $\mathbb{R}^{k}.$ Denote by
$\mu_{\varphi}^{(k)}$ the restriction of the measure $\mu^{(k)}$ on the set
$M_{k}^{\varphi}.$ Finally, define the measure $\mu_{k,\varphi}$ as an image
of measure $\mu_{\varphi}^{(k)}$ under the mapping

$\mathbb{R}^{k}\ni\vec{v}=(v_{1},\ldots,v_{k})\mapsto\varphi(v_{*}).$

Such measures can be used for the expression of the counting measure obtained
from the measure $\mu$ under the action of the function $\phi.$ Define the
counting measure $\nu=(\mu\circ\varphi^{-1})_{*}$ by

$\nu(\Delta)=|\\{u\in\Delta:\ \mu(\varphi^{-1}\\{u\\})>0\\}|.$

The integrals with respect to measure $\nu$ can be expressed in terms of
integrals with respect to $\mu_{k,\varphi},\ k\geq 1.$

######  Lemma 12.

For a continuous function $f$ on $\mathbb{R}$ with compact support

$\int_{\mathbb{R}}fd\nu=\int_{\mathbb{R}}fd\mu_{1,\varphi}-\frac{1}{2!}\int_{\mathbb{R}}fd\mu_{2,\varphi}+\frac{1}{3!}\int_{\mathbb{R}}fd\mu_{3,\varphi}-\ldots+(-1)^{n+1}\frac{1}{n!}\int_{\mathbb{R}}fd\mu_{n,\varphi}+\ldots$

###### Proof.

Since the measure $\mu$ is locally finite and due to the the condition on the
function $\varphi$ the series from the right hand side contains only finite
number of summands with probability 1. To prove the lemma it is enough to
consider point $y$ such that $\nu(\\{y\\})=1.$ Then for some ordered points
$x_{1}<\ldots<x_{n}$

$\varphi(x_{1})=\ldots=\varphi(x_{n})=y,\ \
\mu(\\{x_{1}\\})=\ldots=\mu(\\{x_{n}\\})=1.$

Consequently,

$\nu(\\{y\\})=\Big{|}\bigcup_{k=1}^{n}\\{\varphi(x_{k})\\}\Big{|}=\sum_{k=1}^{n}|\\{\varphi(x_{k})\\}|-\sum_{1\leq
k_{1}<k_{2}\leq
n}|\\{\varphi(x_{k_{1}})\\}\cap\\{\varphi(x_{k_{2}})\\}|+\ldots\\\
+(-1)^{n}\Big{|}\bigcap_{k=1}^{n}\\{\varphi(x_{k})\\}\Big{|}=\mu_{1,\varphi}(\\{y\\})-\frac{1}{2!}\mu_{2,\varphi}(\\{y\\})+\ldots+\frac{1}{n!}\mu_{n,\varphi}(\\{y\\}).$

This proves the lemma. ∎

######  Remark 13.

Note, that in case when $\mu$ is generated by the Arratia flow, i.e.
$\mu=N_{t}$, one can check that series from the lemma absolutely converges for
any measurable and bounded function $f.$

In view of the lemma 12 we need to prove central limit theorem for the
integrals with respect to the measures $\mu_{k,\varphi}$ for the case when
$\mu=N_{t_{1}}$ and $\varphi_{t_{1},t_{2}}$ is the mapping from the Brownian
web. To calculate corresponding mean and variance let us find the expectation
of the integrals with respect to measures $\mu_{k,\varphi}.$ We will do this
in two cases. One when $\mu$ is random and $\varphi$ is deterministic and
another when $\mu$ is deterministic and $\varphi$ is random.

######  Lemma 14.

Suppose, that the random measure $\mu$ has the point densities $\rho^{(k)},$
$k\geq 1$ and the function $\varphi$ is deterministic. Then

$\mathbf{E}\int_{\mathbb{R}}fd\mu_{k,\varphi}=\int_{\mathbb{R}^{k}}f(\varphi(v_{*}))1\\!\\!\,\mathrm{I}_{M^{\varphi}_{k}}(\vec{v})\rho^{(k)}(\vec{v})d\vec{v}.$

###### Proof.

The proof follows directly from the definition of measures $\mu_{k,\varphi}.$
∎

To formulate the statement when $\varphi$ is random we need the following
notation and assumtion. Suppose that for arbitrary $a<b$ there exist density
$q(a,b,u)$ such that for every Borel subset $\Delta\subset\mathbb{R}$

$\mathbf{P}\\{\varphi(a)=\varphi(b),\
\varphi(a)\in\Delta\\}=\int_{\Delta}q(a,b,u)du.$

######  Lemma 15.

For the deterministic measure $\mu$ and random function $\varphi$ the
following relation holds

$\mathbf{E}\int_{\mathbb{R}}fd\nu=\int_{\mathbb{R}}\left(\int_{\mathbb{R}}f(v)p(u,v)dv)\right)\mu(du)-\\\
-\frac{1}{2!}\int_{\mathbb{R}^{2}}\left(\int_{\mathbb{R}}f(v)q(u_{*},u^{*},v)dv)\right)\mu^{(2)}(d\vec{u})+\ldots\\\
+(-1)^{n+1}\frac{1}{n!}\int_{\mathbb{R}^{n}}\left(\int_{\mathbb{R}}f(v)q(u_{*},u^{*},v)dv)\right)\mu^{(n)}(d\vec{u})+\ldots.$

Here $p(u,v)$ is the probability density of $\varphi(u),$ $u\in\mathbb{R}.$

###### Proof.

The statement of the lemma follows from lemma 12, definitions of measures
$\mu_{k,\varphi}$ and functions $q.$ ∎

Now we can consider summands from the series in the statement of lemma 15 with
random stationary measure $\mu$. Recall, that by definition, random measure
that is defined on $\mathcal{B}(\mathbb{R})$ is stationary if for any number
$m\in\mathbb{N},$ of Borel subsets
$\Delta_{1},\Delta_{2},...\Delta_{m}\in\mathcal{B}(\mathbb{R})$ and any real
nuber $h\in\mathbb{R}$ the following equality in distribuion holds

$(\mu(\Delta_{1}),\mu(\Delta_{2}),\ldots,\mu(\Delta_{m}))\overset{d}{=}(\mu(\Delta_{1}+h),\mu(\Delta_{2}+h),\ldots,\mu(\Delta_{m}+h))$

To prove the central limit theorem for integrals

$\frac{1}{n}\int_{\mathbb{R}^{n}}\left(\int_{0}^{n}f(v)q(u_{*},u^{*},v)dv)\right)\mu^{(k)}(d\vec{u})$

let us rewrite them.
For every $k\geq 2$

$\int_{\mathbb{R}^{k}}\left(\int_{0}^{n}f(v)q(u_{*},u^{*},v)dv\right)\mu^{(k)}(d\vec{u})=\int_{0}^{n}f(v)\left(\int_{\mathbb{R}^{k}}q(u_{*},u^{*},v)\mu^{(k)}(d\vec{u})\right)dv.$

Note that for a case when the random mappting $\varphi_{t_{1},t_{2}}$ is taken
from Arratia web it can be easily checked that

  1. 1.

$\forall z\in\mathbb{R},\ a\leq b,\ v\in\mathbb{R}:$

$q_{t_{2}-t_{1}}(a,b,v)=q_{t_{2}-t_{1}}(a+z,b+z,v+z),$

  2. 2.

$q_{t_{2}-t_{1}}(a,b,v)\leq p_{t_{2}-t_{1}}(a,v)\wedge p_{t_{2}-t_{1}}(b,v),$

where $p_{t}(a,\cdot)$ is the Gaussian density with mean $a$ and variance $t.$

######  Lemma 16.

Suppose that $\mu$ is stationary random point measure. Then for every $k\geq
2$ the random function

$\xi_{k}(v)=\int_{\mathbb{R}^{k}}q_{t_{2}-t_{1}}(u_{*},u^{*},v)\mu^{(k)}(d\vec{u}),\
v\in\mathbb{R}$

is a stationary random measure.

###### Proof.

Denote for any arbitrary $z\in\mathbb{R}$ the image of the measure $\mu^{(k)}$
under the transformation

$\mathbb{R}^{k}\ni(u_{1},\ldots,u_{k})\mapsto(u_{1}+z,\ldots,u_{k}+z)$

by $\mu_{z}^{(k)}.$ Since $\mu$ is a stationary random measure, then for every
$z_{1},\ldots,z_{n}$ and $h$ the sequences of random measures

$\left\\{\mu_{z_{1}}^{(k)},\ldots,\mu_{z_{n}}^{(k)}\right\\},\left\\{\mu_{z_{1}+h}^{(k)},\ldots,\mu_{z_{n}+h}^{(k)}\right\\}$

are equidistributed. Now the statement of the lemma follows from the
properties of the function $q_{t_{2}-t_{1}}.$ ∎

Similarly to mixing coefficients of the stationary process we can define
mixing coefficient of the point measure.

######  Definition 17.

A random stationary measure $\mu$ satisfies mixing condition with the function
$\alpha$ if for any $t\in\mathbb{R}$ and $h>0$

$\sup\Big{\\{}\mathbf{P}(A\cap B)-\mathbf{P}(A)\mathbf{P}(B)|,\
A\in\mathcal{F}_{-\infty}^{t},\
B\in\mathcal{F}_{t+h}^{\infty}\Big{\\}}\leq\alpha(h),$

where $\mathcal{F}_{s}^{t}=\sigma\\{\mu(\Delta):\Delta\subset[s,t]\\}$

Now we can prove the central limit theorem for the integrals with respect to
process $\xi_{k}$ that was defined in lemma 16.

######  Theorem 18.

Let random mapping $\varphi_{t_{1},t_{2}}$ be an Arratia web and measure
$\mu=N_{t_{1}}$ is the point measure that corresponds to the Arratia flow at
the moment of time $t_{1}$. Let $f\in C(\mathbb{R})$ be a 1-periodic function
such that $\int_{\mathbb{R}}f(u)du=0.$ Then there exists a Gaussian random
variable $\zeta_{k}(f)$ such that

$\frac{1}{\sqrt{n}}\int_{0}^{n}f(v)\xi_{k}(v)dv\Rightarrow\zeta_{k}(f),\
n\to\infty,$

where

$\xi_{k}(v)=\int_{\mathbb{R}^{k}}q_{t_{2}-t_{1}}(u_{*},u^{*},v)\mu^{(k)}(d\vec{u}),\
v\in\mathbb{R}.$

###### Proof.

Define random variables

$\eta_{l}=\int_{l}^{l+1}f(v)\xi_{k}(v)dv,\ l=0,1,\ldots.$

This sequence is stationary. Check that $\eta_{0}$ is centered.

$\mathbf{E}\eta_{0}=\int_{0}^{1}f(v)\mathbf{E}\xi_{k}(v)dv=\mathbf{E}\xi_{k}(0)\int_{0}^{1}f(v)dv=0.$

Now check the boundedness of moments. Consider

$\mathbf{E}|\int_{0}^{1}f(v)\xi_{k}(v)dv|^{n}\leq\\\
\leq\int_{0}^{1}\ldots\int_{0}^{1}|f(v_{1})|\ldots|f(v_{n})|\mathbf{E}\prod_{j=1}^{n}\int_{\mathbb{R}^{k}}q_{t_{k}-t_{1}}(u_{*},u^{*},v_{j})\mu^{(k)}(d\vec{u})d\vec{v}\leq\\\
\leq
c_{1}\mathbf{E}\prod_{j=1}^{n}\int_{\mathbb{R}^{k}}e^{-c_{1}\|\vec{u}\|^{2}}\mu^{(k)}(d\vec{u})<\infty.$

The boundedness of the last expression follows from the lemmas about the
moments of integrals with respect to factorial measures and estimations of the
$n-$point densities for the point measure of the Arratia flow.

To establish now the central limit theorem let us use approximations of the
random processes $\xi_{k}.$ For every $n\geq 1$ define

$\xi_{k}^{m}(v)=\int_{\mathbb{R}^{k}}q_{t_{2}-t_{1}}(u_{*},u^{*},v)1\\!\\!\,\mathrm{I}_{\\{v-m\leq
u_{*}\leq u^{*}\leq v+m\\}}\mu^{(k)}(d\vec{u}).$

Note that the sequence of random variables

$\eta_{l}^{m}=\int_{l}^{l+1}f(v)\xi_{k}^{m}(v)dv,\ l\geq 0$

satisfies mixing condition with the function $\alpha(\cdot-2m),$ where
$\alpha$ is the function from mixing condition for measure $\mu.$ Due to
estimation of this function for the point measure from the Arratia flow one
can conclude that the following convergence holds

$\frac{1}{\sqrt{n}}\sum_{l=0}^{n-1}\eta_{l}^{m}\Rightarrow\varkappa_{m},\ \
n\to\infty.$

Here $\varkappa_{m}$ is centered Gaussian variable with

$\mathbf{E}\varkappa_{m}^{2}=\int_{0}^{1}\int_{0}^{1}f(v_{1})f(v_{2})\mathbf{E}\prod_{j=1}^{2}\int_{\mathbb{R}^{k}}q_{t_{2}-t_{1}}(u_{*},u^{*},v_{j})1\\!\\!\,\mathrm{I}_{v_{j}-m\leq
u_{*}\leq u^{*}\leq v_{j}+m}\mu^{(k)}(d\vec{u}).$

For any $v\in\mathbb{R}$ and $m\geq 1$ denote

$B_{m}(v)=\\{\vec{u}\in\mathbb{R}^{k}:\ v-m\leq u_{*}\leq u^{*}\leq u+m\\}.$

Then for every $m\geq 1$

$\mathbf{E}\left(\frac{1}{\sqrt{n}}\sum_{l=0}^{n-1}\left(\eta_{l}-\eta^{m}_{l}\right)\right)^{2}=\frac{1}{n}\mathbf{E}\left(\int\limits_{0}^{n}\int\limits_{\mathbb{R}^{k}\setminus
B_{m}(v)}f(v)q_{t_{2}-t_{1}}(u_{*},u^{*},v)\mu^{(k)}(d\vec{u})dv\right)^{2}=\\\
=\frac{1}{n}\int\limits_{0}^{n}\int\limits_{0}^{n}f(v_{1})f(v_{2})\mathbf{E}\int\limits_{\mathbb{R}^{k}\setminus
B_{m}(v_{1})}\int\limits_{\mathbb{R}^{k}\setminus
B_{m}(v_{2})}q_{s}(u_{1*},u_{1}^{*},v_{1})q_{t_{2}-t_{1}}(u_{2*},u_{2}^{*},v_{2})\cdot\\\
\cdot\mu^{(k)}(d\vec{u}_{1})\mu^{(k)}(d\vec{u}_{2})dv_{1}dv_{2}.$

Consider mathematical expectation in the last expression. Using that

$q_{t_{2}-t_{1}}(a,b,v)\leq p_{s}(a,v)\wedge p_{s}(b,v),$

where $p_{t}(a,\cdot)$ is the Gaussian density with mean $a$ and variance $s,$
and the uniform upper bound on the $k-$point densities of the point measure
$\mu$ we get

$\mathbf{E}\int\limits_{\mathbb{R}^{k}\setminus
B_{m}(v_{1})}\int\limits_{\mathbb{R}^{k}\setminus
B_{m}(v_{2})}q_{t_{2}-t_{1}}(u_{1*},u_{1}^{*},v_{1})q_{t_{2}-t_{1}}(u_{2*},u_{2}^{*},v_{2})\mu^{(k)}(d\vec{u}_{1})\mu^{(k)}(d\vec{u}_{2})\leq\\\
\leq Ce^{-2\max(2m,|v_{1}-v_{2}|)}.$

Substitute this upper bound into the expression for
$\mathbf{E}\left(\frac{1}{\sqrt{n}}\sum_{l=0}^{n-1}\left(\eta_{l}-\eta^{m}_{l}\right)\right)^{2}$
we get

$\mathbf{E}\left(\frac{1}{\sqrt{n}}\sum_{l=0}^{n-1}\left(\eta_{l}-\eta^{m}_{l}\right)\right)^{2}\leq
C\frac{1}{n}\int\limits_{0}^{n}\int\limits_{0}^{n}e^{-\max(2m,|v_{1}-v_{2}|)}dv_{1}dv_{2}=\\\
=\frac{2C}{n}\int_{0}^{n}e^{-\max(2m,r)}dr\to 0,\text{ as }m\to\infty.$

From this we conclude that there exists $\sigma^{2}$ such that
$\mathbf{E}\varkappa_{m}^{2}\to\sigma^{2}$ as $m\to\infty$ and

$\varkappa_{m}\Rightarrow\varkappa,$

where $\varkappa$ is Gaussian random variable with zero mean and variance
$\sigma^{2}.$ Finally, from this follows

$\frac{1}{\sqrt{n}}\sum_{l=0}^{n-1}\eta_{l}\Rightarrow\zeta_{k},\ \
n\to\infty$

and this proves the statement of the lemma. ∎

Putting together lemma (15) and theorem (18) we get

######  Theorem 19.

Let $f\in C(\mathbb{R})$ be a 1-periodic function such that
$\int_{\mathbb{R}}f(u)du=0$ and $(N_{t})_{t\geq 0}$ is the point measure that
corresponds to the Arratia flow $\\{x(u,t),\ u\in\mathbb{R},\ t\geq 0\\}.$
Then for $t_{1}<t_{2}$

$\frac{1}{\sqrt{n}}\mathbf{E}\left(\int_{0}^{n}f(u)N_{t_{2}}(du)\big{|}\mathcal{F}_{t_{1}}^{x}\right)\Rightarrow\sum_{k=1}^{\infty}(-1)^{k+1}\frac{1}{k!}\zeta_{k}(f),\text{
as }n\to\infty$

where $\zeta_{k}(f)$ is centered Gaussian variable with

$\mathbf{E}\zeta_{k}(f)^{2}=\int_{0}^{1}\int_{0}^{1}\int_{\mathbb{R}^{k}}f(v_{1})f(v_{2})\prod_{j=1}^{2}q_{t_{2}-t_{1}}(u_{*},u^{*},v_{j})\rho_{t_{1}}^{(k)}(\vec{u})(d\vec{u})$

##  4  Convergence of integrals in functional spaces.

Note that the second part of the theorem 10 states that for fixed function $f$
there exists the limiting Gaussian process indexed by time $t\geq 0$. This
process defined by it’s finite-dimensional distributions with covariance
matrix $\Sigma_{t_{1},\ldots,t_{m}}$. In this section we consider the sequence
of processes $\\{X_{t}^{n},\ t\in[t_{0},T]\\}$ and study its convergence in
the space of continuous functions. The second statement of the theorem 10,
gives us the convergence of finite-dimensional distributions of this sequence
to Gaussian distribution.

In the sequel, together with the Arratia flow we use notion of the dual flow
$\\{y(u,s),\ u\in\mathbb{R},s\in[0,t]\\}$ for the Arratia flow (see, for
example, [5, 14]). The dual flow moves in the backward time and its
trajectories do not cross trajectories of the Arratia flow. The Arratia flow
and its dual flow exist as weak limits of scaled random walks moving forward
and backward in time, respectively, whose trajectories do not intersect ([1]).
The dual Arratia flow can be constructed using a Brownian web [11]. Assume
that the Arratia flow $x(u,t)$ is a Brownian web that starts from time
$s=0,$i.e. $x(u,t)=\varphi_{0,t}(u).$ Then for fixed $t>0$ a dual flow for the
the Arratia flow can be defined as follows: for any $s\in[0;t],$

$y(u,t-s)=\inf\\{\varphi_{r,s}(v):\varphi_{r,t}(v),\ v\in\mathbb{Q},\
r\in\mathbb{Q}\cap[0;t]\\}\ \text{a.s.}$

Now we find out conditions on a function $f$ under which the processes
$\left\\{X_{t}^{n}(f)=\frac{1}{\sqrt{n}}\int_{0}^{n}f(u)N_{t}(du),\
t\in[t_{0},T]\right\\},$ $(n\geq 1)$ have continuous modification.

######  Lemma 20.

Let $f\in C^{1}(\mathbb{R})$ be a function such that
$\text{supp}f|_{[0;1]}\subset[\varepsilon,1-\varepsilon]$ and
$\int_{0}^{1}f(u)du=0.$ Then for every $0<t_{0}<t_{1}<t$ and $k=3,4,...$

$\mathbf{E}|X_{t}^{1}(f)-X_{t_{1}}^{1}(f)|^{k}\leq
C_{t_{0},\varepsilon,k,f}\left[(t-t_{1})^{2}+(t-t_{1})^{k}\right],$

where the constant $C$ depends on $t_{0},\ \varepsilon,\ k,\ f.$

###### Proof.

Now

$X_{t_{1}}^{1}(f)-X_{t}^{1}(f)=\int_{0}^{1}f(u)N_{t_{1}}(du)-\int_{0}^{1}f(u)N_{t}(du)=\\\
=\int_{0}^{1}\left[f(u)-f(\varphi_{t_{1},t}(u))\right]N_{t_{1}}(du)-\int_{\mathbb{R}\setminus[0;1]}f(\varphi_{t_{1},t}(u))N_{t_{1}}(du)$

The upper bound for $k-$th moment ($k\in\mathbb{Z}$) of the first summand

$\mathbf{E}\left|\int_{0}^{1}\left[f(u)-f(\varphi_{t_{1},t}(u))\right]N_{t_{1}}(du)\right|^{k}\leq\\\
\leq\mathbf{E}\int_{0}^{n}{\begin{subarray}{c}k\\\
\ldots\end{subarray}}\int_{0}^{n}\prod_{i=1}^{k}|f(u_{i})-f(\varphi_{t_{1},t}(u_{i}))|N_{t_{1}}(du_{1})\ldots
N_{t_{1}}(du_{k})=\\\ =\mathbf{E}\int_{0}^{n}{\begin{subarray}{c}k\\\
\ldots\end{subarray}}\int_{0}^{n}h(\vec{u};t_{1},t)N_{t_{1}}(du_{1})\ldots
N_{t_{1}}(du_{k}),$

where

$h(\vec{u};t_{1},t)=\mathbf{E}\prod_{i=1}^{k}|f(u_{i})-f(\varphi_{t_{1},t}(u_{i}))|\leq\prod_{i=1}^{k}\left(\mathbf{E}|f(u_{i})-f(\varphi_{t_{1},t}(u_{i}))|^{k}\right)^{1/k}.$

Denote by
$g(\vec{u})=\left(\mathbf{E}|f(u)-f(\varphi_{t_{1},t}(u))|^{k}\right)^{1/k}.$
Then we can continue the estimation using lemma 6 and uniform upper bound for
$\rho^{(k)}$

$\mathbf{E}\int_{0}^{n}{\begin{subarray}{c}k\\\
\ldots\end{subarray}}\int_{0}^{n}h(\vec{u};t_{1},t)N_{t_{1}}(du_{1})\ldots
N_{t_{1}}(du_{k})\leq\mathbf{E}\left(\int_{0}^{1}g(u;t_{1},t)N_{t_{1}}(du)\right)^{k}\leq\\\
\leq
C_{t_{1}}\int_{0}^{1}g^{k}(u;t_{1},t)du=C_{t_{1}}\int_{0}^{1}\mathbf{E}|f(u)-f(u+w(t-t_{1}))|^{k}du=\\\
=C_{t_{1}}\int_{0}^{1}\int_{\mathbb{R}}\left(f(u)-f(u+\sqrt{t-t_{1}}y)\right)^{k}\frac{1}{\sqrt{2\pi}}e^{-y^{2}/2}dydu\leq\\\
\leq
M{C}_{t_{1}}\int_{0}^{1}\int_{\mathbb{R}}(\sqrt{t-t_{1}})^{k}y^{k}\frac{1}{\sqrt{2\pi}}e^{-y^{2}/2}dydu=M\widetilde{C}_{t_{1}}(t-t_{1})^{k/2},$

where $M=\sup_{u\in[0,1]}f^{\prime}(u)$ and the constant
$C_{t_{1}}\leq\frac{\text{const}}{t_{0}^{k/2}}$ for $t_{0}<t_{1}<t.$ Using
this upper bound we get

$\mathbf{E}\left|\int_{0}^{1}\big{(}f(u)-f(\varphi_{t_{1},t}(u))\big{)}N_{t_{1}}(du)\right|^{k/2}\left|\int_{0}^{1}\big{(}f(u)-f(\varphi_{t,t_{2}}(u))\big{)}N_{t}(du)\right|^{k/2}\leq\\\
\leq\left(\mathbf{E}\left|\int_{0}^{1}\big{(}f(u)-f(\varphi_{t_{1},t}(u))\big{)}N_{t_{1}}(du)\right|^{k}\right)^{1/2}\left(\mathbf{E}\left|\int_{0}^{1}\big{(}f(u)-f(\varphi_{t,t_{2}}(u))\big{)}N_{t}(du)\right|^{k}\right)^{1/2}\leq\\\
\leq C\left((t-t_{1})^{k/2}(t_{2}-t)^{k/2}\right)^{1/2}\leq
C(t_{2}-t_{1})^{k/2},$

Consider now

$\int_{\mathbb{R}\setminus[0;1]}f(\varphi_{t_{1},t}(u)N_{t_{1}}(du)=\int_{1}^{\infty}f(\varphi_{t_{1},t}(u)N_{t_{1}}(du)+\int_{-\infty}^{0}f(\varphi_{t_{1},t}(u)N_{t_{1}}(du).$

Let us estimate the first summand in this expression. The second one can be
estimated similarly.

$\mathbf{E}\left|\int_{1}^{\infty}f\big{(}\varphi_{t_{1},t}(u)\big{)}N_{t_{1}}(du)\right|^{k}\leq\mathbf{E}\int_{1}^{\infty}{\begin{subarray}{c}k\\\
\ldots\end{subarray}}\int_{1}^{\infty}\prod_{i=1}^{k}|f\big{(}\varphi_{t_{1},t}(u_{i})\big{)}|N_{t_{1}}(du_{1})\ldots
N_{t_{1}}(du_{k})\leq\\\
\leq\mathbf{E}\int_{1}^{\infty}{\begin{subarray}{c}k\\\
\ldots\end{subarray}}\int_{1}^{\infty}r(\vec{u};t_{1},t)N_{t_{1}}(du_{1})\ldots
N_{t_{1}}(du_{k}),$

where
$r(\vec{u};t_{1},t)=\mathbf{E}\prod_{i=1}^{k}|f\big{(}\varphi_{t_{1},t}(u_{i})\big{)}|.$
Since $\text{supp}f\subset[\varepsilon,1-\varepsilon]$ and $f$ is bounded we
have

$r(\vec{u},t_{1},t)\leq\left(\prod_{i=1}^{k}\mathbf{E}|f\big{(}\varphi_{t_{1},t}(u_{i})\big{)}|^{k}\right)^{1/k}\leq
C\left(\prod_{i=1}^{k}\mathbf{P}\left\\{\varphi_{t_{1},t}(u_{i})\in[\varepsilon,1-\varepsilon]\right\\}\right)^{1/k}\leq\\\
\leq
C\left(\prod_{i=1}^{k}\mathbf{P}\left\\{\sup_{s\in[0;t-t_{1}]}w(s)>u_{i}-1+\varepsilon\right\\}\right)^{1/k},$

where $w$ is a Wiener process, $w(0)=0.$ Using estimation

$\mathbf{P}\left\\{\sup_{s\in[0;t-t_{1}]}w(s)>u-1+\varepsilon\right\\}\leq\sqrt{\frac{2}{\pi}}\exp\left\\{-\frac{(u-1+\varepsilon)^{2}}{2(t-t_{1})}\right\\}\frac{\sqrt{t-t_{1}}}{u-1+\varepsilon}$

and lemma 6

$\mathbf{E}\left|\int_{1}^{\infty}f(\varphi_{t_{1},t}(u))N_{t_{1}}(du)\right|^{k}\leq\mathbf{E}\int_{1}^{\infty}{\begin{subarray}{c}k\\\
\ldots\end{subarray}}\int_{1}^{\infty}r(\vec{u};t_{1},t)N_{t_{1}}(du_{1})\ldots
N_{t_{1}}(du_{k})\leq\\\
\leq\mathbf{E}\int_{1}^{\infty}{\begin{subarray}{c}k\\\
\ldots\end{subarray}}\int_{1}^{\infty}\left(\prod_{i=1}^{k}\sqrt{\frac{2}{\pi}}\exp\left\\{-\frac{(u_{i}-1+\varepsilon)^{2}}{2(t-t_{1})}\right\\}\frac{\sqrt{t-t_{1}}}{u_{i}-1+\varepsilon}\right)^{1/k}N_{t_{1}}(du_{1})\ldots
N_{t_{1}}(du_{k})\leq\\\ \leq
C_{t_{1}}\int_{1}^{\infty}\sqrt{\frac{2}{\pi}}\exp\left\\{-\frac{(u-1+\varepsilon)^{2}}{2(t-t_{1})}\right\\}\frac{\sqrt{t-t_{1}}}{u-1+\varepsilon}du\leq\text{const}C_{t_{1}}e^{-\varepsilon^{2}/2(t-t_{1})}(t-t_{1})/\varepsilon^{2}\leq\\\
\leq C_{t_{0},\varepsilon}(t-t_{1})^{2}$

∎

The estimation form the previous lemma allows to apply Kolmogorov theorem. So
$\int_{0}^{1}f(u)N_{t}(du)$ has continuous modification under some conditions
on $f.$ To prove that $\int_{0}^{k}f(u)N_{t}(du)$ has continuous modification
we need to get similar estimation for this process. To do this we need
following lemma.

######  Lemma 21.

Let $F$ be a function that depends on set of random points $x([0;1],t)$ such
that

$\mathbf{E}F\left(x([0;1],t)\right)=0\ \text{ and }\
\mathbf{E}F^{8}\left(x([0;1],t)\right)<\infty$

Then for any $k_{1}<k_{2}<k_{3}<k_{4}$ the following upper bound holds

$\mathbf{E}\prod_{i=1}^{4}F\left(x([k_{i},k_{i}+1],t)\right)\leq\\\
\leq\left(\mathbf{E}F^{4}(x([0;1],t))\right)^{1/2}e^{-(k_{2}-k_{1}-1)^{2}/8t}e^{-(k_{4}-k_{3}-1)^{2}/8t}+\\\
+\left(\mathbf{E}F^{8}(x([0;1],t))\right)^{1/2}e^{-(k_{2}-k_{1}-1)^{2}/24t}e^{-(k_{3}-k_{2}-1)^{2}/24t}e^{-(k_{4}-k_{3}-1)^{2}/24t}.$

###### Proof.

At first, consider

$\mathbf{E}\prod_{i=1}^{2}F\left(x([k_{i},k_{i}+1],t)\right)=\mathbf{E}\prod_{i=1}^{2}F\left(x([k_{i},k_{i}+1],t)\right)1\\!\\!\,\mathrm{I}_{x(k_{1}+1,t)<x(k_{2},t)}+\\\
+\mathbf{E}\prod_{i=1}^{2}F\left(x([k_{i},k_{i}+1],t)\right)1\\!\\!\,\mathrm{I}_{x(k_{1}+1,t)=x(k_{2},t)}.$

Let us denote by $\tilde{x}$ the independent copy of the Arratia flow. Then

$\mathbf{E}\prod_{i=1}^{2}F\left(x([k_{i},k_{i}+1],t)\right)1\\!\\!\,\mathrm{I}_{x(k_{1}+1,t)<x(k_{2},t)}+\mathbf{E}\prod_{i=1}^{2}F\left(x([k_{i},k_{i}+1],t)\right)1\\!\\!\,\mathrm{I}_{x(k_{1}+1,t)=x(k_{2},t)}\leq\\\
\leq\mathbf{E}F\left(x([k_{1},k_{1}+1],t)\right)F\left(\tilde{x}([k_{2},k_{2}+1],t)\right)1\\!\\!\,\mathrm{I}_{x(k_{1}+1,t)<\tilde{x}(k_{2},t)}+\\\
+\left[\mathbf{E}\prod_{i=1}^{2}F^{p}\left(x([k_{i},k_{i}+1],t)\right)\right]^{1/p}\left[\mathbf{P}\\{x(k_{1}+1,t)=x(k_{2},t)\\}\right]^{1/q}\leq\\\
\leq\mathbf{E}F\left(x([k_{1},k_{1}+1],t)\right)\mathbf{E}F\left(\tilde{x}([k_{2},k_{2}+1],t)\right)+\\\
+2\left[\mathbf{E}\prod_{i=1}^{2}F^{p}\left(x([k_{i},k_{i}+1],t)\right)\right]^{1/p}\left[\mathbf{P}\left\\{\mathcal{N}(0,1)>\frac{k_{2}-k_{1}-1}{\sqrt{2}}\right\\}\right]^{1/q}\leq\\\
\leq\frac{2}{\sqrt{2\pi}}\left(\mathbf{E}F^{2p}(x([0;1],t))\right)^{1/p}\exp\left\\{-\frac{(k_{2}-k_{1}-1)^{2}}{4qt}\right\\}\left(\frac{\sqrt{2t}}{k_{2}-k_{1}-1}\right)^{1/q}$

We use similar consideration to obtain upper bound for
$\mathbf{E}\prod_{i=1}^{4}F\left(x([k_{i},k_{i}+1],t)\right).$

$\mathbf{E}\prod_{i=1}^{4}F\left(x([k_{i},k_{i}+1],t)\right)=\\\
=\mathbf{E}\prod_{i=1}^{4}F\left(x([k_{i},k_{i}+1],t)\right)1\\!\\!\,\mathrm{I}_{x(k_{2}+1,t)<x(k_{3},t)}+\\\
+\mathbf{E}\prod_{i=1}^{4}F\left(x([k_{i},k_{i}+1],t)\right)1\\!\\!\,\mathrm{I}_{x(k_{2}+1,t)=x(k_{3},t)}=\\\
=\mathbf{E}\prod_{i=1}^{2}F\left(x([k_{i},k_{i}+1],t)\right)\prod_{i=3}^{4}F\left(\tilde{x}([k_{i},k_{i}+1],t)\right)1\\!\\!\,\mathrm{I}_{x(k_{2}+1,t)<\tilde{x}(k_{3},t)}+\\\
+\mathbf{E}\prod_{i=1}^{4}F\left(x([k_{i},k_{i}+1],t)\right)1\\!\\!\,\mathrm{I}_{x(k_{2}+1,t)=x(k_{3},t)},$

where $\tilde{x}$ is an independent copy of the Arratia flow. Using
independency and the previous upper bound with $p=q=2$ we can continue

$\mathbf{E}\prod_{i=1}^{2}F\left(x([k_{i},k_{i}+1],t)\right)\prod_{i=3}^{4}F\left(\tilde{x}([k_{i},k_{i}+1],t)\right)1\\!\\!\,\mathrm{I}_{x(k_{2}+1,t)<\tilde{x}(k_{3},t)}+\\\
+\mathbf{E}\prod_{i=1}^{4}F\left(x([k_{i},k_{i}+1],t)\right)1\\!\\!\,\mathrm{I}_{x(k_{2}+1,t)=x(k_{3},t)}\leq$
$\leq\mathbf{E}\prod_{i=1}^{2}F\left(x([k_{i},k_{i}+1],t)\right)\mathbf{E}\prod_{i=3}^{4}F\left(\tilde{x}([k_{i},k_{i}+1],t)\right)+\\\
+\mathbf{E}\prod_{i=1}^{4}F\left(x([k_{i},k_{i}+1],t)\right)1\\!\\!\,\mathrm{I}_{x(k_{2}+1,t)=x(k_{3},t)}1\\!\\!\,\mathrm{I}_{x(k_{1}+1,t)<x(k_{2},t)}+\\\
+\mathbf{E}\prod_{i=1}^{4}F\left(x([k_{i},k_{i}+1],t)\right)1\\!\\!\,\mathrm{I}_{x(k_{2}+1,t)=x(k_{3},t)}1\\!\\!\,\mathrm{I}_{x(k_{1}+1,t)=x(k_{2},t)}\leq\\\
\leq\left(\mathbf{E}F^{4}(x([0;1],t))\right)^{1/2}e^{-(k_{2}-k_{1}-1)^{2}/8t}e^{-(k_{4}-k_{3}-1)^{2}/8t}+\\\
+\mathbf{E}F(\tilde{x}([k_{1},k_{1}+1],t)\mathbf{E}\prod_{i=2}^{4}F(x([k_{i},k_{i}+1],t))+\\\
+\mathbf{E}\prod_{i=1}^{4}F\left(x([k_{i},k_{i}+1],t)\right)1\\!\\!\,\mathrm{I}_{x(k_{2}+1,t)=x(k_{3},t)}1\\!\\!\,\mathrm{I}_{x(k_{1}+1,t)=x(k_{2},t)}1\\!\\!\,\mathrm{I}_{x(k_{3}+1,t)=x(k_{4},t)}\leq$
$\leq\left(\mathbf{E}F^{4}(x([0;1],t))\right)^{1/2}e^{-(k_{2}-k_{1}-1)^{2}/8t}e^{-(k_{4}-k_{3}-1)^{2}/8t}+\\\
+\left[\mathbf{E}\prod_{i=1}^{4}F^{2}\left(x([k_{i},k_{i}+1],t)\right)\right]^{1/2}\mathbf{P}\left\\{x(k_{i}+1,t)=x(k_{i+1},t),\
i=1,2,3\right\\}^{1/2}\leq\\\
\leq\left(\mathbf{E}F^{4}(x([0;1],t))\right)^{1/2}e^{-(k_{2}-k_{1}-1)^{2}/8t}e^{-(k_{4}-k_{3}-1)^{2}/8t}+\\\
+\left(\mathbf{E}F^{8}(x([0;1],t))\right)^{1/2}e^{-(k_{2}-k_{1}-1)^{2}/24t}e^{-(k_{3}-k_{2}-1)^{2}/24t}e^{-(k_{4}-k_{3}-1)^{2}/24t}.$

Lemma is proved. ∎

######  Theorem 22.

Let $f\in C^{1}(\mathbb{R})$ be a 1-periodic function such that
$\text{supp}f|_{[0,1]}\subset[\varepsilon,1-\varepsilon]$ and
$\int_{0}^{1}f(u)du=0.$ Fix $0<t_{0}<T<\infty.$ Then for any $s,t$ such that
$0<t_{0}\leq s<t\leq T<\infty$ and for any $n\geq 1$ there exists a constant
$C$ which does not depend on $s,t$ and $n$ such that

$\mathbf{E}|X^{n}_{t}(f)-X^{n}_{s}(f)|^{4}\leq C|t-s|^{2}.$

###### Proof.

Using representation
$X^{n}_{t}(f)=\frac{1}{\sqrt{n}}\sum_{k=0}^{n-1}\int_{k}^{k+1}f(u)N_{t}(du)$
we have for any $n\geq 1$ and $s<t$

$\mathbf{E}(X^{n}_{t}(f)-X^{n}_{s}(f))^{4}=\frac{1}{n^{2}}\sum_{k_{1},k_{2},k_{3},k_{4}=0}^{n-1}\mathbf{E}\prod_{i=1}^{4}\left(\int_{k_{i}}^{k_{i}+1}f(u)N_{t}(du)-\int_{k_{i}}^{k_{i}+1}f(u)N_{s}(du)\right).$

Using notion of the dual flow $\psi$ to the Arratia flow, the integral
$\int_{k}^{k+1}f(u)N_{t}(du)$ can be considered as functional from
$\psi_{t,0}([k,k+1]).$ Let us denote by

$F\left(\psi_{t,0}([k,k+1])\right)-F\left(\psi_{s,0}([k,k+1])\right)=\int_{k}^{k+1}f(u)N_{t}(du)-\int_{k}^{k+1}f(u)N_{s}(du).$

Since

$\mathbf{E}\prod_{i=1}^{4}\left(F\left(\psi_{t,0}([k_{i},k_{i}+1])\right)-F\left(\psi_{s,0}([k,k+1])\right)\right)=\\\
=\mathbf{E}\prod_{i=1}^{4}\left(F\left(x([k_{i},k_{i}+1],t)\right)-F\left(x([k_{i},k_{i}+1],s)\right)\right)$

for arbitrary $k_{1}<k_{2}<k_{3}<k_{4}$ lemma 21 gives the upper bound

$\mathbf{E}\prod_{i=1}^{4}\left(F\left(\psi_{t,0}([k_{i},k_{i}+1])\right)-F\left(\psi_{s,0}([k_{i},k_{i}+1])\right)\right)\leq
Ce^{-(k_{2}-k_{1}-1)^{2}/8t}e^{-(k_{4}-k_{3}-1)^{2}/8t}+\\\
+Ce^{-(k_{2}-k_{1}-1)^{2}/24t}e^{-(k_{3}-k_{2}-1)^{2}/24t}e^{-(k_{4}-k_{3}-1)^{2}/24t}.$

Now, using this upper bound and equality

$\sum_{1\leq k_{1}<k_{2}<k_{3}<k_{4}\leq
n}f(k_{2}-k_{1},k_{3}-k_{2},k_{4}-k_{3})=\\\
=\sum_{k_{1}=1}^{n-3}\sum_{h_{1}=1}^{n-k_{1}-2}\sum_{h_{2}=1}^{n-k_{1}-h_{1}}\sum_{h_{3}=1}^{n-k_{1}-h_{1}-h_{2}}f(h_{1},h_{2},h_{3}),$

we get

$\frac{1}{n^{2}}\sum_{1\leq k_{1}<k_{2}<k_{3}<k_{4}\leq
n}\mathbf{E}\prod_{i=1}^{4}\left(F\left(\psi_{t,0}([k_{i},k_{i}+1])\right)-F\left(\psi_{s,0}([k_{i},k_{i}+1])\right)\right)\leq\\\
\leq\left(\mathbf{E}F^{4}(\psi_{t,0}([0;1]))-F\left(\psi_{s,0}([k_{i},k_{i}+1])\right)\right)^{1/2}\cdot\\\
\cdot\frac{1}{n^{2}}\sum_{k_{1}=1}^{n-3}\sum_{h_{1}=1}^{n-k_{1}-2}\sum_{h_{2}=1}^{n-k_{1}-h_{1}}\sum_{h_{3}=1}^{n-k_{1}-h_{1}-h_{2}}\left(\exp\left\\{-(h_{1}-1)^{2}/24t-(h_{3}-1)^{2}/24t\right\\}\right)\leq\\\
\leq
C\left(\mathbf{E}F^{4}(\psi_{t,0}([0;1]))-F\left(\psi_{s,0}([k_{i},k_{i}+1])\right)\right)^{1/2}.$

From the Lemma 20 we get the upper bound

$\left(\mathbf{E}\left|F(\psi_{t,0}([0;1]))-F(\psi_{s,0}([0;1]))\right|^{4}\right)^{1/2}=\\\
=\left(\mathbf{E}\left[\int_{k}^{k+1}f(u)N_{t}(du)-\int_{k}^{k+1}f(u)N_{s}(du)\right]^{4}\right)^{1/2}\leq\\\
\leq C[(t-s)^{2}+(t-s)^{4}]^{1/2}$

∎

######  Corollary 23.

Let $f\in C^{1}(\mathbb{R})$ be a 1-periodic function such that
$\text{supp}f|_{[0,1]}\subset[\varepsilon,1-\varepsilon]$ and
$\int_{0}^{1}f(u)du=0.$ Then for $0<t_{0}<T<\infty$ and $n\geq 1$ the random
processes $\left\\{X_{t}^{n}(f)=\frac{1}{\sqrt{n}}\int_{0}^{n}f(u)N_{t}(du),\
t\in[t_{0},T]\right\\},$ $(n\geq 1)$ have continuous modification and

$\left\\{\frac{1}{\sqrt{n}}\int_{0}^{n}f(u)N_{t}(du),\
t\in[t_{0},T]\right\\}\Rightarrow\zeta_{f}(\cdot)\ n\to\infty\ \text{in
}C([t_{0},T]).$

###### Proof.

Existence of continuous modifications immediately follows from previous
theorem and Kolmogorov theorem.

For the proof of weak convergence we note that Theorem 10 stated the finite-
dimensional convergence. The tightness of the sequence $\\{X_{t}^{n}(f),\
t\in[t_{0};T]\\}$ follows from theorem 22 (see, for example, Corollary 16.9,
[10]) ∎

######  Corollary 24.

For every $f\in C^{1}([0;1])$ be a function such that
$\text{supp}f\subset[\varepsilon,1-\varepsilon]$ the limiting Gaussian process
$\\{\zeta_{f}(t),t\in[t_{0},T]\\}$ from the theorem 10 is continuous.

###### Proof.

In the proof of the previous theorem we obtained the inequality

$\mathbf{E}|X^{n}_{t}(f)-X^{n}_{s}(f)|^{4}\leq C|t-s|^{2}.$

By Theorem 10 weak convergence of finite-dimensional distributions of the
sequence $X_{n}$ holds. By Fatou’s lemma

$\mathbf{E}\liminf_{n\to\infty}|X^{n}_{t}(f)-X^{n}_{s}(f)|^{4}\leq\liminf_{n\to\infty}\mathbf{E}|X^{n}_{t}(f)-X^{n}_{s}(f)|^{4}\leq
C|t_{2}-t_{1}|^{2}.$

From this follows that for the limiting process $\zeta_{f}(\cdot)$ the same
inequality holds

$\mathbf{E}|\zeta_{f}(t)-\zeta_{f}(s)|^{4}\leq C|t-s|^{2}.$

∎

##  5 Limiting Gaussian process as a functional.

The statement 3 from the Theorem 10 give us possibility to define finite-
dimensional distributions of the process $\zeta$ as a process indexed by the
functions $\varphi\in L_{3}([0;1])$. Namely it was proved that for functions
$\varphi_{1},...,\varphi_{m}\in L_{3}([0;1])$ the weak convergence of the
random vector $\left(X_{t}^{n}(\varphi_{1}),...,X_{t}^{n}(\varphi_{m})\right)$
to the random vector
$\left(\zeta_{\varphi_{1}}(t),\ldots,\zeta_{\varphi_{m}}(t)\right)$ holds. By
Kolmogorov theorem one can define the Gaussian random field $\zeta$ on the
space of parameters $L_{3}([0;1])$. Note that the covariance of $\zeta$ is

$\mathrm{cov}\,\left(\zeta_{f}(t),{\zeta_{g}(t)}\right)=\\\
=\frac{1}{2}\int_{0}^{1}\int_{0}^{1}f(u)g(v)(G_{t}(u,v)+G_{t}(v,u))dudv+\\\
+\frac{1}{\sqrt{2\pi}}\int_{0}^{1}f(u)g(u)du=(f,(1+\tilde{G}_{t})g),$

where we denoted by $\tilde{G}_{t}$ the integral operator in $L_{2}$ with
kernel $\tilde{G}_{t}=\frac{1}{2}(G_{t}(u,v)+G_{t}(v,u)).$

######  Lemma 25.

The function $\tilde{G}_{t}$ is continuous.

###### Proof.

From the theorem 10, the function $G_{t}$ is equal to the series

$G_{t}(v_{1},v_{2})=g_{t}(v_{1}-v_{2})+2\sum_{l=1}^{\infty}g_{t}(v_{1}-v_{2}+l),\text{
where }g_{t}(v_{1}-v_{2})=\rho_{t}^{(2)}(v_{1},v_{2})-\frac{1}{{\pi t}}.$

Using formula (5) for $g_{t}$ by Weierstrass criterion for uniform convergence
with upper bound
$\dfrac{K+l}{2\sqrt{t}}\cdot
e^{-(l-K)^{2}/4t}\cdot\int\limits_{\left|K+l\right|/\sqrt{t}}^{+\infty}e^{-v^{2}/4}\,dv+e^{-(L-K)^{2}/2t},$
the series $\sum_{l=1}^{\infty}g_{t}(x+l)$ converges uniformly on the interval
$[-K,K]$. ∎

Since for every fixed $t\geq 0$ $\tilde{G}_{t}$ is continuous, $\zeta$ can be
uniquely expanded to Gaussian random field defined on set of the functions
$L_{2}([0;1])$.

We will consider $\zeta$ as a generalized random element in Hilbert space
$L_{2}([0;1])$ ([13, 4] ) with the covariance operator $1+\tilde{G}_{t}.$ One
of the most known examples of generalized Gaussian random element on $L_{2}$
is a formal derivative $dw$ of the 1-dimensional Wiener process $w(\cdot)$
which often called white noise. Note that for this formal derivative one can
consider stochastic integral and multiple stochastic integral for function
from $L_{2}([0;1]^{k})$. The definition of the integrals

$\int_{0}^{1}\int_{0}^{t_{1}}\ldots\int_{0}^{t_{k-1}}f(t_{1},t_{2},\ldots,t_{k})dw(t_{1})\ldots
dw(t_{k})$

is obvious. In the context of our paper one can expect that the multiple
integrals with respect to point measure $N_{t}$ after suitable normalization
will converge to some integrals defined by $\zeta.$ First of all let us
describe the construction of multiple integrals with respect to $\zeta.$ In
the articles by A. V. Skorokhod [13] and by A.A. Dorogovtsev [4] it was
proposed to describe such integrals in term of the Hilbert-Schmidt forms from
generalized Gaussian elements. Let us describe the suitable construction.

Now we will discuss the possibility to define the Hilbert-Schmidt form of the
process $\zeta$. Let $H$ be a real separable Hilbert space. Consider
generalized Gaussian random element $\xi$ in $H$ with zero mean and covariance
operator $A$. We denote by $H^{\otimes k}_{symm}$ the space of symmetric
$k-$linear Hilbert-Schmidt forms of $H$. The Wick product (see, for example,
[4]) of Gaussian random variables we denote by $*.$

######  Lemma 26.

Assume that covariance operator $A$ of a generalized Gaussian element $\xi$ is
invertible. Let $A_{k}\in H^{\otimes k}_{symm}$. Then for every orthonormal
basis $\\{e_{j}\\}_{j\geq 1}$ in $H$ the series

$A_{k}(\xi,\ldots,\xi):=\sum_{n_{1},\ldots,n_{k}=1}^{\infty}A(e_{n_{1}},\ldots,e_{n_{k}})\xi(e_{n_{1}})*\ldots*\xi(e_{n_{k}}))$

converges in $L_{2}$ and its value $A_{k}(\xi,\ldots,\xi)$ does not depend on
choice of a basis in $H.$

###### Proof.

Let $A_{k}$ be a finite-dimensional form, that is there exists a basis
$\\{e_{j}\\}_{j\geq 1}$ in $H$ such that
$A_{k}=\sum_{n_{1},\ldots,n_{k}=1}^{N}a_{n_{1},\ldots,n_{k}}e_{n_{1}}\otimes\ldots\otimes
e_{n_{k}}.$ Let us define $\xi_{0}=A^{-1/2}\xi$ by the rule

$\xi_{0}(e)=\xi(A^{-1/2}e),\ e\in H.$

Then for finite-dimensional symmetric form $A_{k}$

$A_{k}(\xi,\ldots,\xi)=A_{k}(A^{1/2}\xi_{0},\ldots A^{1/2}\xi_{0})$

and

$\mathbf{E}A_{k}(\xi,\ldots,\xi)^{2}=k!\|A_{k}(A^{1/2}\cdot,\ldots
A^{1/2}\cdot)\|_{k}^{2},$

where $\|\cdot\|_{k}$ is the Hilbert-Schmidt norm. Note that
$\|A_{k}(A^{1/2}\cdot,\ldots
A^{1/2}\cdot)\|_{k}\leq\|A^{1/2}\|^{k}\|A_{k}\|_{k}.$ From this upper bound
for the norm and the fact that the definition of $A_{k}(A^{1/2}\cdot,\ldots
A^{1/2}\cdot)$ does not depend on choice of basis the statement of the lemma
follows. ∎

From the lemma 25 it follows that the integral operator with kernel
$\tilde{G}$ is a Hilbert-Schmidt operator and so it has discrete spectrum with
eigenvalues of finite multiplicity. From this follows $\dim
Ker(1+\tilde{G})<+\infty.$ Using the lemma 26 we can define the action of
$k-$linear Hilbert-Schmidt form on limiting Gaussian element $\zeta$ since its
covariance operator $1+\tilde{G}$ is invertible on $Ker^{\bot}(1+\tilde{G})$.

Next theorem gives us the limit of multiple integrals. Before we formulate it
we need to define a $k-$linear form associated to a function $f.$ Let
$f:\mathbb{R}^{k}\to\mathbb{R}$ be a symmetric function from
$L_{2}([0;1]^{k}).$ Then for an orthogonal basis $\\{e_{j}\\}_{j\geq 1}$ in
$L_{2}([0;1])$ it can be written as the series

$f=\sum_{i_{1},\ldots,i_{k}}a_{i_{1},\ldots,i_{k}}e_{i_{1}}\otimes\ldots\otimes
e_{i_{k}}.$

We denote by $A_{f}$ the $k-$linear Hilbert-Schmidt form on $L_{2}([0;1])$
which is represented with the same series

$A_{f}=\sum_{i_{1},\ldots,i_{k}}a_{i_{1},\ldots,i_{k}}e_{i_{1}}\otimes\ldots\otimes
e_{i_{k}},$

that is
$A_{f}(h_{1},\ldots,h_{k})=\sum_{i_{1},\ldots,i_{k}}a_{i_{1},\ldots,i_{k}}(e_{i_{1}},h_{1})\ldots(e_{i_{k}},h_{k}).$
As was mentioned above the action of the form $A_{f}$ on the generalized
Gaussian element $\zeta$ is well defined. Also we denote by

$A_{f}(\zeta,\ldots,\zeta,\delta_{x_{j}},\ldots,\delta_{x_{k}})=\sum_{i_{1},\ldots,i_{k}}a_{i_{1},\ldots,i_{k}}\zeta(e_{i_{1}})*\ldots*\zeta(e_{i_{j-1}})e_{i_{j}}(x_{j})\ldots
e_{i_{k}}(x_{k}).$

In the term of multi-linear forms from Gaussian element $\zeta$ we can
describe the limit distribution of the multiple integral with respect to the
point measure of the Arratia flow $N_{t}.$

######  Theorem 27.

Let $f:\mathbb{R}^{k}\to\mathbb{R}$ be a symmetric periodic with period 1
function such that $f|_{[0;1]^{k}}\in L_{2}([0;1]^{k})$ and
$\int_{0}^{1}f(\vec{x})dx_{j}=0,$ $j=1,\ldots,k$. Then for $k\in 2\mathbb{Z}$

$\frac{1}{n^{k/2}}\int_{0}^{n}{\begin{subarray}{c}k\\\
\ldots\end{subarray}}\int_{0}^{n}f(\vec{x})N_{t}^{\otimes k}(dx_{1}\ldots
dx_{k})\Rightarrow A_{f}(\zeta,\ldots,\zeta)+\\\
+\int_{0}^{1}\int_{0}^{1}C_{k}^{2}A_{f}(\zeta,\ldots,\zeta,\delta_{x_{k-1}},\delta_{x_{k}})G_{t}(x_{k-1},x_{k})dx_{k}dx_{k-1}\ldots+\\\
+\ldots\int_{0}^{1}{\begin{subarray}{c}\\\
\ldots\end{subarray}}\int_{0}^{1}\frac{k!}{(k/2)!2^{k/2}}A_{f}(\delta_{x_{1}},\ldots\delta_{x_{k}})G_{t}(x_{1},x_{2})\ldots
G_{t}(x_{k-1},x_{k})d\vec{x},\ n\to\infty$

and for $k\in\mathbb{Z}\setminus 2\mathbb{Z}$

$\frac{1}{n^{k/2}}\int_{0}^{n}{\begin{subarray}{c}k\\\
\ldots\end{subarray}}\int_{0}^{n}f(\vec{x})N_{t}^{\otimes k}(dx_{1}\ldots
dx_{k})\Rightarrow A_{f}(\zeta,\ldots,\zeta)+\\\
+\int_{0}^{1}\int_{0}^{1}C_{k}^{2}A_{f}(\zeta,\ldots,\zeta,\delta_{x_{k-1}},\delta_{x_{k}})G_{t}(x_{k-1},x_{k})dx_{k}dx_{k-1}\ldots+\\\
+\ldots\int_{0}^{1}{\begin{subarray}{c}\\\
\ldots\end{subarray}}\int_{0}^{1}\frac{k!}{((k-1)/2)!2^{(k-1)/2}}A_{f}(\zeta,\delta_{x_{2}},\ldots\delta_{x_{k}})G_{t}(x_{2},x_{3})\ldots
G_{t}(x_{k-1},x_{k})d\vec{x},\ n\to\infty$

###### Proof.

We prove the statement using mathematical induction with respect to dimension
$k$. Consider $k=2$. Let $\\{e_{j}\\}_{j\geq 0}$ be a orthonormal basis in
$L_{2}([0;1])$ such that $\int_{0}^{1}e_{i}(x)dx=0$ for $i\geq 1$ and
$e_{0}=1.$ Then function can be represented as
$f|_{[0;1]^{2}}=\sum_{i,j=1}^{\infty}c_{ij}e_{i}\otimes e_{j}.$ Denote by
$\varphi_{j}$ the periodic extension on $\mathbb{R}$ of the function $e_{j}$.
Then we have

$\int_{0}^{n}\int_{0}^{n}f(x,y)N_{t}^{\otimes
2}(dxdy)=\int_{0}^{n}\int_{0}^{n}f(x,y)N_{t}(dx)N_{t}(dy)-\int_{0}^{n}f(x,x)N_{t}(dx)=\\\
=\sum_{i,j}^{\infty}c_{ij}\int_{0}^{n}\varphi_{i}(x)N_{t}(dx)\int_{0}^{n}\varphi_{j}(y)N_{t}(dy)-\int_{0}^{n}f(x,x)N_{t}(dx).$

Using theorem 10 we have

$\frac{1}{n}\int_{0}^{n}\varphi_{i}(x)N_{t}(dx)\int_{0}^{n}\varphi_{j}(y)N_{t}(dy)\Rightarrow\zeta(e_{i})\zeta(e_{j}),\
n\to\infty$

and $\frac{1}{n}\int_{0}^{n}f(x,x)N_{t}(dx)\Rightarrow\frac{1}{\sqrt{\pi
t}}\int_{0}^{1}f(x,x)dx,$ $n\to\infty.$ From this follows

$\frac{1}{n}\int_{0}^{n}\int_{0}^{n}f(x,y)N_{t}^{\otimes 2}(dxdy)=\\\
=\sum_{i,j}^{\infty}c_{ij}\frac{1}{n}\int_{0}^{n}\varphi_{i}(x)N_{t}(dx)\int_{0}^{n}\varphi_{j}(y)N_{t}(dy)-\frac{1}{n}\int_{0}^{n}f(x,x)N_{t}(dx)\Rightarrow\\\
\Rightarrow\sum_{i,j}^{\infty}c_{ij}\zeta(e_{i})\zeta(e_{j})-\frac{1}{\sqrt{\pi
t}}\int_{0}^{1}f(x,x)dx,\ n\to\infty.$

By definition of Wick product the sum in the last expression is written as

$\sum_{i,j}^{\infty}c_{ij}\zeta(e_{i})\zeta(e_{j})-\frac{1}{\sqrt{\pi
t}}\int_{0}^{1}f(x,x)dx=\\\
=\sum_{i,j}^{\infty}c_{ij}(\zeta(e_{i})*\zeta(e_{j})+\frac{1}{2}\int_{0}^{1}\int_{0}^{1}e_{i}(x)e_{j}(y)(G_{t}(x,y)+G_{t}(y,x))dxdy=\\\
=A_{f}(\zeta,\zeta)+\int_{0}^{1}\int_{0}^{1}f(x,y)G_{t}(x,y)dxdy,$

where in the last equality we used symmetry of function $f.$

Now assume that we know the statement of theorem for $k-1$. To do the step of
induction we rewrite the $k-$multiple integral with respect to $N^{\otimes k}$
in therm of $(k-1)-$multiple integral. Since a basis of $L_{2}$ can be taken
with step functions $e_{j}$, we can write

$\int_{0}^{1}{\begin{subarray}{c}k\\\
\ldots\end{subarray}}\int_{0}^{1}e_{1}(x_{1})\ldots e_{k}(x_{k})N^{\otimes
k}(dx_{1}\ldots dx_{k})=\\\ =\sum_{i_{1}}\sum_{i_{2}\neq
i_{1}}\ldots\sum_{i_{k}\neq i_{k-1},\ldots,i_{1}}e_{1}(u_{i_{1}})\ldots
e_{k}(u_{i_{k}})=\\\ =\sum_{i_{1}}\sum_{i_{2}\neq
i_{1}}\ldots\sum_{i_{k-1}\neq
i_{k-2},\ldots,i_{1}}\left(\sum_{i_{k}}e_{1}(u_{i_{1}})\ldots
e_{k}(u_{i_{k}})-\sum_{j=1}^{k-1}e_{1}(u_{i_{1}})\ldots
e_{k}(u_{i_{j}})\right)=\\\ =\int_{0}^{1}{\begin{subarray}{c}k-1\\\
\ldots\end{subarray}}\int_{0}^{1}e_{1}(x_{1})\ldots e_{k-1}(x_{k-1})N^{\otimes
k-1}(dx_{1}\ldots dx_{k-1})\int_{0}^{1}e_{k}(x)N(dx)-\\\
-\sum_{j=1}^{k-1}\int_{0}^{1}{\begin{subarray}{c}k-1\\\
\ldots\end{subarray}}\int_{0}^{1}e_{1}(x_{1})\ldots
e_{k-1}(x_{k-1})e_{k}(x_{j})N^{\otimes k-1}(dx_{1}\ldots dx_{k-1}).$

Using the limit behaviour for $(k-1)-$multiple integral we get the statement
of the theorem.

∎

######  Corollary 28.

For the function $f$ satisfying conditions of the previous theorem and off-
diagonal (i.e.$f(\vec{x})=0$ if $x_{i}=x_{j}$ for some $i\neq j$) we have

$\frac{1}{n^{k/2}}\int_{0}^{n}{\begin{subarray}{c}k\\\
\ldots\end{subarray}}\int_{0}^{n}f(\vec{x})N^{\otimes k}(dx_{1}\ldots
dx_{k})\Rightarrow A_{f}(\zeta,\ldots,\zeta),\ n\to\infty.$

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