Diffusion approximation of controlled branching processes using limit theorems for random step processes

Let $\left\{X_{n,j}:n=0,1,\ldots;j=1,2,\ldots\right\}$ be a sequence of independent and identically distributed (i.i.d.), non-negative and integer-valued random variables defined on a probability space $(\Omega,\cal{F},P)$. Let also $\{\phi_{n}(k):k=0,1,\ldots\}$, for $n=0,1,\ldots$, be a sequence of stochastic processes which consist of independent non-negative integer-valued random variables on $(\Omega,\mathcal{F},P)$ with the same one-dimensional distributions. Furthermore, let us assume that $\{X_{n,j}:n=0,1,\ldots;j=1,2,\ldots\}$ and $\{\phi_{n}(k):n=0,1,\ldots;k=0,1,\ldots\}$ are independent.

A controlled branching process (CBP) is defined recursively as

$${}Z_{n}=\sum^{\phi_{n-1}(Z_{n-1})}_{j=1}X_{n-1,j},\qquad n=1,2,\ldots,$$

(1)

where $\sum_{j=1}^{0}$ is defined as 0 and $Z_{0}$ is a non-negative, integer-valued, square-integrable random variable which is independent of $\left\{X_{n,j}:n=0,1,\ldots;j=1,2,\ldots\right\}$ and $\{\phi_{n}(k):n=0,1,\ldots;k=0,1,\ldots\}$.

Here, $Z_{n}$ denotes the size of the $n$-th generation of a population and $X_{n-1,j}$ is the offspring size of the $j$-th individual in the $(n-1)$-th generation. We will assume that the mean $m=E[X_{0,1}]$ and variance $\sigma^{2}=Var[X_{0,1}]$ are both finite.

The class of CBPs is a very general family of stochastic processes that collect as particular cases the simplest branching model, the standard Bienaymé–Galton–Watson (BGW) process, by considering $\phi_{n}(k)=k$ a.s. for each $k$, or a branching processes with immigration, by setting $\phi_{n}(k)=k+I_{n}$, where $\{I_{n}\}_{n\geq 0}$ are i.i.d. random variables (writing in this way the immigrants give rise to offspring at the same generation as their arrival and with the same probability law as $X_{0,1}$), among others. The monograph [7] provides an extensive description of its probabilistic theory.

The research of functional weak limit theorems for branching processes arises a lot of interest since many years ago. It was firstly formulated for a BGW process by [3] and proved by [10] and [12]. These results have been extended to another classes of branching processes. For instance, a wide literature exists around weak convergence results for branching processes with immigration (BPI) since the pioneer work by [15], see also [1] and references therein. In this paper we focus our attention on a weak convergence theorem for a critical CBP with a random initial number of individuals and assuming finite second order moment on the this initial value. A similar result was already established for a single CBP in [14], and for an array of CBPs in [6], by assuming fixed initial numbers of progenitors using infinitesimal generators results for their proofs. Inspired in the paper [1] on BPI we will use limit theorems for random step processes towards a diffusion process provided in [9] to obtain an alternative proof. The scheme of it follows similar steps to the ones in [1]. An important feature of a CBP is that the value of $Z_{n}$ conditioned on the knowledge of the previous generation, $Z_{n-1}=k$, is a random sum of random variables, namely $\sum_{j=1}^{\phi_{n}(k)}X_{n-1,j}$, instead of a non-random sum as in the case of a BPI. This leads to handle the proofs of each steps using conditioning arguments different from those used in [1].

Apart from this introduction, the paper is organized as follows. In Section 2 we provide the notation and some auxiliary results about the behaviour of the first and second moments of the process. Section 3 gathers the main theorem. For the ease of reading the paper, additional results are presented in the Appendix.

We denote, for $k=0,1,\ldots,$

	$\displaystyle\varepsilon(k)$	$\displaystyle=$	$\displaystyle E[\phi_{n}(k)],$
	$\displaystyle\nu^{2}(k)$	$\displaystyle=$	$\displaystyle Var[\phi_{n}(k)],$

and assume all finite. It is easy to obtain that for $n=1,2,\ldots$,

	$\displaystyle E[Z_{n}|{\cal F}_{n-1}]$	$\displaystyle=$	$\displaystyle m\varepsilon(Z_{n-1}),$		(2)
	$\displaystyle Var[Z_{n}|{\cal F}_{n-1}]$	$\displaystyle=$	$\displaystyle\sigma^{2}\varepsilon(Z_{n-1})+m^{2}\nu^{2}(Z_{n-1}),$		(3)

where ${\cal F}_{n}$ is the $\sigma-$algebra generated by the random variables $Z_{0},Z_{1},\ldots,Z_{n}$, $n\geq 1$ (see Proposition 3.5 in [7]).

We introduce the quantities

$${}\tau_{m}(k)=E[Z_{n+1}Z^{-1}_{n}|Z_{n}=k]=m\varepsilon(k)k^{-1},\ k\geq 1.$$

(4)

The quantity $\tau_{m}(k)$ represents a mean growth rate. Intuitively, it can be interpreted as an average offspring per individual for a generation of size $k$.

Assuming that $\lim_{k\to\infty}\tau_{m}(k)=\tau_{m}$ exists, the process can be classified as:

$$\tau_{m}<1\quad\mbox{{subcritical}};\ \tau_{m}=1\quad\mbox{{critical}};\ \tau_{m}>1\quad\mbox{{supercritical}}.$$

We are interested in critical CBPs that satisfy the following hypotheses:

• A1)

  $\tau_{m}(k)=1+k^{-1}\alpha$, $\ k>0$, $\alpha>0$,

• A2)

  $\nu^{2}(k)=O(k^{\beta})$, $\beta<1$, as $k\to\infty$.

The behavior of critical CBPs was studied in [5]. Assuming that $P(\phi_{0}(0)=0)=1$, i.e. 0 is an absorbing state, and it is verified $P(X_{0,1}=0)>0$ or $P(\phi_{0}(k)=0)>0$, $k=1,2,\ldots$, it was established that under A1) and A2), if $\alpha>\sigma^{2}/(2m)$ and an assumption on conditional moments holds, then $P(Z_{n}\to\infty)>0.$ In the present paper we will consider critical CBPs, $\{Z_{n}\}_{n\geq 0}$, satisfying the above conditions, but with a reflecting barrier at zero, namely, $P(\phi_{n}(0)>0)>0$. Thus $\{Z_{n}\}_{n\geq 0}$ will have a finite number of returns to the sate zero till the explosion to infinity, i.e. $P(Z_{n}\to\infty)=1$.

Notice that under A1), $\varepsilon(k)=(k+\alpha)m^{-1}$, $k\geq 1$, and, for simplicity in the posterior calculations, we will also assume throughout the paper that $\varepsilon(0)=\alpha m^{-1}$.

In next result we calculate the first and second moments of a CBP which verifies A1) and A2).

$$E[Z_{k}]=E[m\varepsilon(Z_{k-1})]=E[Z_{0}]+k\alpha,\quad k\geq 0.$$

(5)

Using (3) we have

$\displaystyle E[Var[Z_{k}\mid\mathcal{F}_{k-1}]]$

$\displaystyle=$

$\displaystyle m^{-1}\sigma^{2}(E[Z_{0}]+(k-1)\alpha)+m^{2}E[\nu^{2}(Z_{k-1})].$

Now, from A2), we have that there exists $C>0$ such that $\nu^{2}(k)\leq Ck$ for all $k>0$, so that $E[\nu^{2}(Z_{k-1})]\leq CE[Z_{n-1}]+\nu^{2}(0)$. Consequently, letting $M_{1}=3\max\{m^{-1}\sigma^{2}E[Z_{0}],\$ $Cm^{2}E[Z_{0}],\ m^{2}\nu^{2}(0)\}$ and $M_{2}=2\max\{m^{-1}\sigma^{2}\alpha,m^{2}C\alpha\}$, we have

$$E[Var[Z_{k}\mid\mathcal{F}_{k-1}]]\leq M_{1}+M_{2}(k-1).$$

(6)

Hence,

	$\displaystyle Var[Z_{k}]$	$\displaystyle=$	$\displaystyle E[Var[Z_{k}\mid\mathcal{F}_{k-1}]]+Var[E[Z_{k}\mid\mathcal{F}_{k-1}]]$
		$\displaystyle\leq$	$\displaystyle M_{1}+M_{2}(k-1)+Var[Z_{k-1}]$
		$\displaystyle\leq$	$\displaystyle kM_{1}+2^{-1}M_{2}k(k-1)+Var[Z_{0}].$

The latter inequality proves that $E[Z_{k}^{2}]=O(k^{2})$.

Next Lemma presents certain relationships among the random variables $\{X_{n,j}:\ n=0,1,\ldots;\ j=1,2,\ldots\}$ which can be easily verified.

We introduce for each $n\in\mathbb{N}$, a stochastic process $W_{n}(t)=n^{-1}Z_{\lfloor nt\rfloor}$, for $t\geq 0$, $t\in\mathbb{R}$, $\lfloor\cdot\rfloor$ denoting the integer part. It is easy to see that $\left\{W_{n}\right\}_{n\geq 1}$ is a sequence of random functions that take values in $D_{[0,\infty)}[0,\infty)$, which is the space of non-negative functions on $[0,\infty)$ that are right continuous and have left limits. We also denote by $C_{c}^{\infty}[0,\infty)$ the space of infinitely differentiable functions on $[0,\infty)$ which have a compact support. Throughout the paper “$\stackrel{{\scriptstyle\mathcal{D}}}{{\to}}$” denotes the convergence of random functions in the Skorokhod topology.

In order to prove Theorem 3.1, we will establish previously the weak convergence of random step processes defined from a martingale difference created from the CBP.

We introduce the following sequence of martingale differences $\{M_{k}\}_{k\geq 1}$ with respect the filtration $\{\mathcal{F}_{k}\}_{k\geq 0}$ as:

$$M_{k}=Z_{k}-E[Z_{k}\mid\mathcal{F}_{k-1}]=Z_{k}-Z_{k-1}-\alpha,\ k\geq 1.$$

Consider the random step processes:

$$\mathcal{M}_{n}(t)=\frac{1}{n}\left(Z_{0}+\sum_{k=1}^{\lfloor nt\rfloor}M_{k}\right)=\frac{1}{n}Z_{\lfloor nt\rfloor}-\frac{\lfloor nt\rfloor}{n}\alpha,\quad t\geq 0,\quad n\in\mathbb{N}.$$

(8)

Proof As was done in [1], we prove the result by applying Theorem A1 in Appendix with $\mathcal{U}=\mathcal{M},\ U_{n}(k)=n^{-1}M_{k},\ k\in\mathbb{N}$, $U_{n}(0)=n^{-1}Z_{0}$, $\mathcal{F}_{n}(k)=\mathcal{F}_{k},k\geq 0$, where $n\in\mathbb{N}$ (yielding $\mathcal{U}_{n}=\mathcal{M}_{n},n\in\mathbb{N}$, as well), and with coefficient functions $\beta:[0,\infty)\times\mathbb{R}\rightarrow\mathbb{R}$ and $\gamma:[0,\infty)\times\mathbb{R}\rightarrow\mathbb{R}$ given by

$$\beta(t,x)=0,\quad\gamma(t,x)=\sqrt{m^{-1}\sigma^{2}\left(x+\alpha t\right)^{+}},\quad t\geq 0,\quad x\in\mathbb{R}.$$

Firstly, we check that the SDE (9) has a pathwise unique strong solution $\left\{\mathcal{M}(t)^{(x)}\right\}_{t\geq 0}$ for all initial values $\mathcal{M}(0)^{(x)}=x\in\mathbb{R}$. In fact, notice that if $\left\{\mathcal{M}(t)^{(x)}\right\}_{t\geq 0}$ is a strong solution of the SDE (9) with initial value $\mathcal{M}(0)^{(x)}=x\in\mathbb{R}$, then, by Itô’s formula, the process $\mathcal{P}(t)=\mathcal{M}(t)^{(x)}+\alpha t,$ $t\geq 0$, is a solution of the SDE

$$\mathrm{d}\mathcal{P}(t)=\alpha\mathrm{d}t+\sqrt{m^{-1}\sigma^{2}\mathcal{P}(t)^{+}}\mathrm{d}\mathcal{W}(t),\quad t\geq 0,\ \mbox{with initial value }\mathcal{P}(0)=x.$$

(10)

Conversely, if $\{\mathcal{P}(t)^{(x)}\}_{t\geq 0}$ is a strong solution of the SDE (10) with initial value $\mathcal{P}^{(x)}(0)=x\in\mathbb{R}$, then, by Itô’s formula, the process $\mathcal{M}(t)=\mathcal{P}(t)^{(x)}-\alpha t,$ $t\geq 0$, is a strong solution of the SDE (9) with initial value $\mathcal{M}(0)=x$. Notice that SDE (10) is the same as SDE (7), consequently the SDE (10) and therefore the SDE (9) as well admit a pathwise unique strong solution with arbitrary initial value, and

$$\{\mathcal{M}(t)+\alpha t\}_{t\geq 0}\stackrel{{\scriptstyle\mathcal{D}}}{{=}}\{W(t)\}_{t\geq 0}.$$

(11)

Let us see that $E\left[\left(U_{n}(k)\right)^{2}\right]<\infty$ for all $n=1,2\ldots$ and $k=0,1,2,\ldots$. Indeed, taking into account (6) in Proposition 2.1,

$$E\left[\left(U_{n}(k)\right)^{2}\right]=n^{-2}E\left[M_{k}^{2}\right]=E[Var[Z_{k}\mid\mathcal{F}_{k-1}]]\leq\frac{M_{1}+M_{2}(k-1)}{n^{2}}<\infty$$

(12)

and, by the assumption in the statement of the theorem, $E\left[\left(U_{n}(0)\right)^{2}\right]=n^{-2}E\left[Z_{0}^{2}\right]<\infty,$ for $n=1,2,\ldots$ Moreover, $U_{n}(0)=n^{-1}Z_{0}\stackrel{{\scriptstyle\text{ a.s. }}}{{\longrightarrow}}0$ as $n\rightarrow\infty$, especially $U_{n}(0)\stackrel{{\scriptstyle\mathcal{D}}}{{\longrightarrow}}0$ as $n\rightarrow\infty$.

For conditions (i), (ii) and (iii) of Theorem A1 in Appendix, we have to check that for each $T>0$, $T\in\mathbb{R}$, as $n\to\infty$:

• a)

  $\sup_{t\in[0,T]}\left|\frac{1}{n}\sum_{k=1}^{\lfloor nt\rfloor}E\left[M_{k}\mid\mathcal{F}_{k-1}\right]-0\right|\stackrel{{\scriptstyle\mathrm{P}}}{{\longrightarrow}}0.$

• b)

  $\sup_{t\in[0,T]}\left|\frac{1}{n^{2}}\sum_{k=1}^{\lfloor nt\rfloor}E\left[M_{k}^{2}\mid\mathcal{F}_{k-1}\right]-\int_{0}^{t}\frac{\sigma^{2}}{m}\left(\mathcal{M}_{n}(s)+\alpha s\right)^{+}\mathrm{d}s\right|\stackrel{{\scriptstyle\mathrm{P}}}{{\longrightarrow}}0.$

• c)

  For all $\theta>0,\theta\in\mathbb{R}$, $\frac{1}{n^{2}}\sum_{k=1}^{\lfloor nT\rfloor}E\left[M_{k}^{2}\mathbb{I}_{\left\{\left|M_{k}\right|>n\theta\right\}}\mid\mathcal{F}_{k-1}\right]\stackrel{{\scriptstyle\mathrm{P}}}{{\longrightarrow}}0.$

Since $E\left[M_{k}\mid\mathcal{F}_{k-1}\right]=0$, $n\in\mathbb{N}$, $k\in\mathbb{N}$, a) holds.

Let us check b).

It is verified that, for $t>0$ and $n\in\mathbb{N}$,

	$\displaystyle\frac{1}{n^{2}}\sum_{k=1}^{\lfloor nt\rfloor}E\left[M_{k}^{2}\mid\mathcal{F}_{k-1}\right]$	$\displaystyle=$	$\displaystyle\frac{1}{n^{2}}\sum_{k=1}^{\lfloor nt\rfloor}Var[Z_{k}\mid\mathcal{F}_{k-1}]$
		$\displaystyle=$	$\displaystyle\frac{1}{n^{2}}\sum_{k=1}^{\lfloor nt\rfloor}\left(m^{2}\nu^{2}(Z_{k-1})+\frac{\sigma^{2}}{m}(Z_{k-1}+\alpha)\right)$
		$\displaystyle=$	$\displaystyle\frac{m^{2}}{n^{2}}\sum_{k=1}^{\lfloor nt\rfloor}\nu^{2}(Z_{k-1})+\frac{\lfloor nt\rfloor\alpha\sigma^{2}}{n^{2}m}+\frac{\sigma^{2}}{n^{2}m}\sum_{k=1}^{\lfloor nt\rfloor}Z_{k-1}.$

Consequently,

	$\displaystyle\frac{1}{n^{2}}\sum_{k=1}^{\lfloor nt\rfloor}E\left[M_{k}^{2}\mid\mathcal{F}_{k-1}\right]$	$\displaystyle-$	$\displaystyle\int_{0}^{t}\frac{\sigma^{2}}{m}\left(\mathcal{M}_{n}(s)+\alpha s\right)^{+}\mathrm{d}s=\frac{m^{2}}{n^{2}}\sum_{k=1}^{\lfloor nt\rfloor}\nu^{2}(Z_{k-1})+\frac{\lfloor nt\rfloor\alpha\sigma^{2}}{n^{2}m}$
			$\displaystyle-\frac{\sigma^{2}(nt-\lfloor nt\rfloor)}{mn^{2}}Z_{\lfloor nt\rfloor}-\frac{\sigma^{2}}{m}\frac{\lfloor nt\rfloor+(nt-\lfloor nt\rfloor)^{2}}{2n^{2}}\alpha.$

Since for each $T>0,T\in\mathbb{R}$,

$$\begin{array}[]{l}\sup_{t\in[0,T]}\frac{\lfloor nt\rfloor}{n^{2}}\leqslant\frac{T}{n}\rightarrow 0,\quad\text{ as }n\rightarrow\infty\\[7.22743pt] \sup_{t\in[0,T]}\frac{\lfloor nt\rfloor+(nt-\lfloor nt\rfloor)^{2}}{2n^{2}}\leqslant\frac{T}{2n}+\frac{1}{2n^{2}}\rightarrow 0,\quad\text{ as }n\rightarrow\infty,\end{array}$$

in order to show $b)$, it suffices to prove that for each $T>0,T\in\mathbb{R}$,

$$\frac{1}{n^{2}}\sup_{t\in[0,T]}\left((nt-\lfloor nt\rfloor)Z_{\lfloor nt\rfloor}\right)\leq\frac{1}{n^{2}}\sup_{t\in[0,T]}Z_{\lfloor nt\rfloor}\stackrel{{\scriptstyle{P}}}{{\longrightarrow}}0\quad\mbox{ as }n\rightarrow\infty.$$

(13)

and

$$\frac{m^{2}}{n^{2}}\sup_{t\in[0,T]}\sum_{k=1}^{\lfloor nt\rfloor}\nu^{2}(Z_{k-1})\stackrel{{\scriptstyle{P}}}{{\longrightarrow}}0\quad\mbox{ as }n\rightarrow\infty.$$

(14)

First we check (13). For each $k\in\mathbb{N}$, we have $Z_{k}=Z_{k-1}+M_{k}+\alpha$, thus

$$Z_{k}=Z_{0}+\sum_{j=1}^{k}M_{j}+k\alpha,$$

and hence, for each $t>0,\ t\in\mathbb{R}$ and $n\in\mathbb{N}$, we get

$$Z_{\lfloor nt]}=\left|Z_{\lfloor nt\rfloor}\right|\leqslant Z_{0}+\sum_{j=1}^{\lfloor nt\rfloor}\left|M_{j}\right|+\lfloor nt\rfloor\alpha.$$

Consequently, in order to prove $(\ref{eq:28})$, it suffices to show

$$\frac{1}{n^{2}}\sup_{t\in[0,T]}\sum_{j=1}^{\lfloor nt\rfloor}\left|M_{j}\right|\leqslant\frac{1}{n^{2}}\sum_{j=1}^{\lfloor nT\rfloor}\left|M_{j}\right|\stackrel{{\scriptstyle{P}}}{{\longrightarrow}}0,\quad\text{ as }n\rightarrow\infty.$$

By (12), $E[M_{k}^{2}]=O(k),$ as $k\to\infty$, and therefore by Jensen’s inequality, $E[|M_{k}|]=O(k^{1/2}),$ as $k\to\infty$, and hence

$$E\left[\frac{1}{n^{2}}\sum_{j=1}^{\lfloor nT\rfloor}\left|M_{j}\right|\right]=\frac{1}{n^{2}}\sum_{j=1}^{\lfloor nT\rfloor}{O}\left(j^{1/2}\right)={O}\left(n^{-1/2}\right)\rightarrow 0,\quad\text{ as }n\rightarrow\infty.$$

Thus we obtain $n^{-2}\sum_{j=1}^{\lfloor nT\rfloor}\left|M_{j}\right|\stackrel{{\scriptstyle{P}}}{{\longrightarrow}}0$ as $n\rightarrow\infty$ implying $(\ref{eq:28})$.

Now, taking into account hypothesis A2)

$$E\left[\frac{1}{n^{2}}\sum_{k=1}^{\lfloor nT\rfloor}\nu^{2}(Z_{k-1})\right]=\frac{1}{n^{2}}\sum_{k=1}^{\lfloor nT\rfloor}O(k^{\beta})=O(n^{\beta-1}),$$

(15)

and hence (14).

Let us check c).

We write

$$M_{k}=\sum_{j=1}^{\phi_{k-1}(Z_{k-1})}(X_{k-1,j}-m)+m(\phi_{k-1}(Z_{k-1})-\varepsilon(Z_{k-1})).$$

Let denote $N_{k}=\sum_{j=1}^{\phi_{k-1}(Z_{k-1})}(X_{k-1,j}-m)$. It is verified for each $n,k\in\mathbb{N}$, and $\theta>0,\theta\in\mathbb{R}$ that

$$M_{k}^{2}\leq 2\left(N_{k}^{2}+m^{2}(\phi_{k-1}(Z_{k-1})-\varepsilon(Z_{k-1}))^{2}\right),$$

and

$$\mathbb{I}_{\left\{\left|M_{k}\right|>n\theta\right\}}\leq\mathbb{I}_{\left\{\left|N_{k}\right|>n\theta/2\right\}}+\mathbb{I}_{\left\{\left|\phi_{k-1}(Z_{k-1})-\varepsilon(Z_{k-1})\right|>n\theta/2m\right\}}.$$

Hence

$$M_{k}^{2}\mathbb{I}_{\left\{\left|M_{k}\right|>n\theta\right\}}\leq 2N_{k}^{2}\mathbb{I}_{\left\{\left|N_{k}\right|>n\theta/2\right\}}+2N_{k}^{2}\mathbb{I}_{\left\{\left|\phi_{k-1}(Z_{k-1})-\varepsilon(Z_{k-1})\right|>n\theta/2m\right\}}+2m^{2}(\phi_{k-1}(Z_{k-1})-\varepsilon(Z_{k-1}))^{2}.$$

In consequence, to check c) we will prove, as $n\to\infty$,

• c.1)

  $\frac{1}{n^{2}}\sum_{k=1}^{\lfloor nT\rfloor}E\left[N_{k}^{2}\mathbb{I}_{\left\{\left|N_{k}\right|>n\theta\right\}}\mid\mathcal{F}_{k-1}\right]\stackrel{{\scriptstyle{P}}}{{\longrightarrow}}0\quad$ for all $\theta>0,\theta\in\mathbb{R}$.

• c.2)

  $\frac{1}{n^{2}}\sum_{k=1}^{\lfloor nT\rfloor}E\left[N_{k}^{2}\mathbb{I}_{\left\{\left|\phi_{k-1}(Z_{k-1})-\varepsilon(Z_{k-1})\right|>n\theta\right\}}\mid\mathcal{F}_{k-1}\right]\stackrel{{\scriptstyle{P}}}{{\longrightarrow}}0$ for all $\theta>0,\theta\in\mathbb{R}$.

• c.3)

  $\frac{1}{n^{2}}\sum_{k=1}^{\lfloor nT\rfloor}E\left[(\phi_{k-1}(Z_{k-1})-\varepsilon(Z_{k-1}))^{2}\mid\mathcal{F}_{k-1}\right]\stackrel{{\scriptstyle{P}}}{{\longrightarrow}}0.$

In what follows let $\theta>0,\theta\in\mathbb{R}$ be fixed.

Let us see c.3). It is verified that

	$\displaystyle\frac{1}{n^{2}}\sum_{k=1}^{\lfloor nT\rfloor}E\left[(\phi_{k-1}(Z_{k-1})-\varepsilon(Z_{k-1}))^{2}\mid\mathcal{F}_{k-1}\right]$	$\displaystyle=$	$\displaystyle\frac{1}{n^{2}}\sum_{k=1}^{\lfloor nT\rfloor}Var\left[\phi_{k-1}(Z_{k-1})\mid\mathcal{F}_{k-1}\right]$
		$\displaystyle=$	$\displaystyle\frac{1}{n^{2}}\sum_{k=1}^{\lfloor nT\rfloor}\nu^{2}(Z_{k-1})\stackrel{{\scriptstyle{P}}}{{\longrightarrow}}0,\mbox{ as }n\to\infty.$

This latter was proved by considering (15).

Now, we check c.1). By the properties of conditional expectation with respect to a $\sigma$ -algebra, we get for all $n,k\in\mathbb{N}$,

$$E\left[N_{k}^{2}\mathbb{I}_{\left\{\left|N_{k}\right|>n\theta\right\}}\mid\mathcal{F}_{k-1}\right]=F_{n,k}\left(Z_{k-1}\right),$$

where on $\{Z_{k-1}=z\}$, with $z=0,1,\ldots$

$$F_{n,k}(z)=E\left[S_{k}(z)^{2}\mathbb{I}_{\left\{\left|S_{k}(z)\right|>n\theta\right\}}\right],\mbox{ where }S_{k}(z)=\sum_{j=1}^{\phi_{k-1}(z)}\left(X_{k-1,j}-m\right).$$

Consider the decomposition $F_{n,k}(z)=A_{n,k}(z)+B_{n,k}(z)$ with

	$\displaystyle A_{n,k}(z)$	$\displaystyle=$	$\displaystyle E\left[\sum_{j=1}^{\phi_{k-1}(z)}(X_{k-1,j}-m)^{2}\mathbb{I}_{\{|S_{k}(z)|>n\theta\}}\right],$
	$\displaystyle B_{n,k}(z)$	$\displaystyle=$	$\displaystyle E\left[\sum_{j,j^{\prime},j\not=j^{\prime}}^{\phi_{k-1}(z)}(X_{k-1,j}-m)(X_{k-1,j^{\prime}}-m)\mathbb{I}_{\{\left|S_{k}(z)\right|>n\theta\}}\right].$