Limit Theorems for a Galton–Watson Process with Immigration in Varying Environments

Abstract

In this paper, we obtain the central limit theorem and the law of the iterated logarithm for Galton–Watson branching processes with time-dependent immigration in varying environments.

1 Introduction

Let $\{T_{n};n\ge 0\}$ be a classical supercritical Galton–Watson branching process with offspring distribution $\{b_{n};n\ge 0\}$ and mean $m:=\sum _{n=0}^{\infty }nb_{n}, 1<m<\infty$. Define $S_{n}=T_{n}/m^{n}$, it is well known that there exists a nonnegative random variable $S$ such that $S=\lim _{n\rightarrow \infty }S_{n} ~ a.s.$. C. C. Heyde [10–12] derived the central limit theorem (CLT) and the law of the iterated logarithm (LIL) for $\{T_{n};n\ge 0\}$:

(I) Suppose that $\tau _{1}^{2}:=Var(T_{1})<\infty$ and $P(S>0)=1$. Set $\tau _{r}^{2}:=Var(T_{r}), r\ge 1$, then

$$\begin{aligned}&\tau _{r}^{-1}T_{n}^{-\frac{1}{2}}(T_{n+r}-m^{r}T_{n})\xrightarrow []{d} N(0,1)~(n\rightarrow \infty ),\\&(m^{2}-m)^{\frac{1}{2}}\tau _{1}^{-1}T_{n}^{-\frac{1}{2}}m^{n}(S-S_{n}) \xrightarrow []{d} N(0,1)~(n\rightarrow \infty ). \end{aligned}$$

(II) Suppose that $E(T_{1}^{3})<\infty$, then

$$\begin{aligned}&\limsup _{n\rightarrow \infty }(\liminf _{n\rightarrow \infty }) \frac{T_{n+r}-m^{r}T_{n}}{(2\tau _{r}^{2}T_{n}\log n)^{\frac{1}{2}}}=1(-1)~~a.s.~ \text {on} ~\{S>0\},\\&\limsup _{n\rightarrow \infty }(\liminf _{n\rightarrow \infty })\frac{m^{n}S-T_{n}}{(2\tau _{1}^{2}(m^{2}-m)^{-1}T_{n}\log n)^{\frac{1}{2}}}=1(-1)~~a.s.~ \text {on} ~\{S>0\}. \end{aligned}$$

Similar limit theorems for a classical Galton–Watson branching process with immigration were studied in [13].

A branching process in a random environment is a natural and important extension of the Galton–Watson process, it is a class of non-homegeneous Galton–Waltson process indexed by a time environment, which has been studied by many authors, see [1–3, 5, 6, 14, 15, 19–22].

In this article, we consider the Galton–Watson branching process with time-dependent immigration in the varying environments (IGWVE) defined as following:

Definition 1.1

Let $Z_{0}\equiv 1$ and for any $n\ge 0$,

$$\begin{aligned} Z_{n+1}=\sum _{j=1}^{Z_{n}}\xi _{n,j}+Y_{n+1} ~~a.s., \end{aligned}$$

where $\{\xi _{n,j}; n\ge 0,j\ge 1\}$ are independent and have the same distribution in row, $\{Y_{n},n\ge 1\}$ is a sequence of independent random variables taking values in ${\mathbb {N}}$ and independent of $\{\xi _{n,j}; n\ge 0,j\ge 1\}$, then $\{Z_{n}, n\ge 0\}$ is said to be a Galton–Watson branching process with time-dependent immigration in varying environments. Particularly, if for any $n\ge 1$, $Y_n\equiv 0$, then $\{Z_{n}, n\ge 0\}$ is said to be a Galton–Watson branching process in varying environments (GWVE).

Throughout this paper we assume that the variances of $\{\xi _{n,j}; n\ge 0,j\ge 1\}$ and $\{Y_{n},n\ge 1\}$ exist. Define $\mu _{n}=E(\xi _{n,j}), \delta _{n}^{2}=Var(\xi _{n,j})$ and $\nu _{n}=E(Y_{n})$, we assume that

$$\begin{aligned} P(\xi _{n,1}=0)\equiv 0,~~\infty >\mu _{n}>1, ~~\infty >\nu _{n}, \delta _{n}^{2}>0. \end{aligned}$$

Our first main result is the following growth rate of IGWVE.

Theorem 1.1

Let $\{Z_{n}, n\ge 0\}$ be an IGWVE. Define $m_{n}=\prod _{i=0}^{n-1}\mu _{i}, W_{n}=Z_{n}/m_{n}, n\ge 1,$ then $\{W_{n}, n\ge 1\}$ is a nonnegative submartingale. If

$$\begin{aligned} a:=\sum _{n=1}^{\infty }\frac{\nu _{n}}{m_{n}}<\infty , \end{aligned}$$

(1.1)

then there exists a nonnegative random variable $W$ with $E(W)<\infty$ such that $W_{n}\xrightarrow {a.s.} W.$ Furthermore, if

$$\begin{aligned} b:=\sup _{n\ge 1}\sum _{j=n}^{\infty }\frac{\delta _{j}^{2}}{\mu _{j}^{2}m_{n,j-n}}<\infty , \end{aligned}$$

(1.2)

where $m_{n,k}=\prod _{i=n}^{n+k-1}\mu _{i}$, then $W_{n}\xrightarrow {L^{2}}W$.

To prove the CLT and the LIL for IGWVE, we need the following two decomposition results.

(A) For any fixed $n\ge 0, r\ge 1$, one has

$$\begin{aligned} Z_{n+r}=\sum _{j=1}^{Z_{n}} X_{n,r}^{(j)} +Y_{n,r}~~ a.s., \end{aligned}$$

where $\{X_{n,r}^{(j)},j\ge 1\}$ are i.i.d. and independent of $Y_{n,r}$. The variances of $\{X_{n,r}^{(j)},j\ge 1\}$ and $Y_{n,r}$ exist. We define $m_{n,r}=E(X_{n,r}^{(1)}), \sigma _{n,r}^{2}=Var(X_{n,r}^{(1)}), \pi _{n,r}=E(Y_{n,r}), \theta _{n,r}^{2}=Var(Y_{n,r}).$

(B) For any fixed $n\ge 0$ one has

$$\begin{aligned} Z_{n}-m_{n}W=\sum _{j=1}^{Z_{n}}(1-V_{n}^{(j)})+I_{n}~~a.s., \end{aligned}$$

where $\{V_{n}^{(j)};j\ge 1\}$ are i.i.d. and independent of $I_{n}$. If (1.2) is satisfied, then the variances of $\{V_{n}^{(j)},j\ge 1\}$ exist. Furthermore, if

$$\begin{aligned} c:=\sup _{n\ge 0}\sum _{k=n+1}^{\infty }\frac{\nu _{k}}{m_{n,k-n}}<\infty , \end{aligned}$$

(1.3)

then the variances of $I_{n}$ exist. We define $\sigma _{n}^{2}\,=\,Var(V_{n}^{(1)}), \pi _{n}=E(I_{n}), \theta _{n}^{2}=Var(I_{n}).$

Using the above two decomposition results, we obtain the CLT and the LIL for IGWVE.

Theorem 1.2

Let $\{Z_{n}, n\ge 0\}$ be an IGWVE. If $Z_{n}\xrightarrow {P}\infty$, (1.2),(1.3) are satisfied and

$$\begin{aligned} 0<d=\inf _{n\ge 0}\frac{\delta _{n}^{2}}{\mu _{n}^{2}}, \end{aligned}$$

(1.4)

then for any fixed $r\ge 1$, when $n\rightarrow \infty$ one has

$$\begin{aligned} \frac{Z_{n+r}-m_{n,r}Z_{n}}{\sigma _{n,r}\sqrt{Z_{n}}}\xrightarrow {d}N(0,1), ~~~~\frac{Z_{n}-m_{n}W}{\sigma _{n}\sqrt{Z_{n}}}\xrightarrow {d}N(0,1). \end{aligned}$$

Remark 1.3

Note that $Z_n\ge 1$, a necessary and sufficient condition for $Z_{n}\xrightarrow {a.s.}\infty$ is

$$\begin{aligned} \sum _{n=0}^{\infty }(1-P(\xi _{n,1}=1))=+\infty , \end{aligned}$$

which imply $Z_{n}\xrightarrow {P}\infty$ [17].

Assume that there exists a constant $0<\delta <1$ such that for any $r\ge 1$,

$$\begin{aligned} \sup _{n}E\left( \left| \frac{X_{n,r}^{(j)}-m_{n,r} }{\sigma _{n,r}}\right| ^{2+\delta }\right) <\infty ,~~~~ \sup _{n}E\left( \left| \frac{V_{n}^{(j)}-1 }{\sigma _{n}}\right| ^{2+\delta }\right) <\infty . \end{aligned}$$

(1.5)

For any $n\ge 1,~x\in R$ and $r\ge 1$, define

$$\begin{aligned} Q_{n,r}(x)=P\left( \frac{Z_{n+r}-m_{n,r}Z_{n}}{\sigma _{n,r}\sqrt{Z_{n}}}\le x\right) ,~~Q_{n}(x)=P\left( \frac{Z_{n}-m_{n}W}{\sigma _{n}\sqrt{Z_{n}}}\le x\right) .\quad \end{aligned}$$

(1.6)

Theorem 1.4

Let $\{Z_{n}, n\ge 0\}$ be an IGWVE. If $Z_n\xrightarrow {a.e.}+\infty$ and (1.2)–(1.5) are satisfied, then there exists constants $\{C_{r},r\ge 1\}$, $C$ and $D$ such that

$$\begin{aligned}&\sup _{x}|Q_{n,r}(x)-\Phi (x)|\le C_{r}E\left( Z_{n}^{-\frac{\delta }{2}}\right) + C\left[ E\left( Z_{n}^{-\frac{\delta }{2}}\right) \right] ^{\frac{1}{2}},\end{aligned}$$

(1.7)

$$\begin{aligned}&\sup _{x}|Q_{n}(x)-\Phi (x)|\le DE\left( Z_{n}^{-\frac{\delta }{2}}\right) +C\left[ E\left( Z_{n}^{-\frac{\delta }{2}}\right) \right] ^{\frac{1}{2}}. \end{aligned}$$

(1.8)

Theorem 1.5

Let $\{Z_{n};n\ge 0\}$ be an IGWVE. Suppose that there exist five constants $\alpha ,\beta ,\tau ,\gamma ,\delta$ with $\beta >\alpha >1,\tau >\gamma >0$ and $0<\delta <1$ such that for any $n\ge 0$,

$$\begin{aligned} \alpha \le \mu _{n}\le \beta , ~~\gamma ^{2}\le \delta _{n}^{2}\le \tau ^{2}, ~~\sum _{n=0}^{\infty }\left[ E\left( Z_{n}^{-\frac{\delta }{2}}\right) ^{\frac{1}{2}}\right] <\infty . \end{aligned}$$

(1.9)

If (1.3) and (1.5) are satisfied, then

$$\begin{aligned}&\limsup _{n\rightarrow \infty }(\liminf _{n\rightarrow \infty })\frac{Z_{n+r}-m_{n,r} Z_{n}}{(2\sigma _{n,r}^{2}Z_{n}\log n)^{\frac{1}{2}}}=1(-1)~~a.s.;\end{aligned}$$

(1.10)

$$\begin{aligned}&\limsup _{n\rightarrow \infty }(\liminf _{n\rightarrow \infty }) \frac{Z_{n}-m_{n}W}{(2\sigma _{n}^{2}Z_{n}\log n)^{\frac{1}{2}}}=1(-1) ~~a.s.. \end{aligned}$$

(1.11)

Remark 1.6

If (1.9) is satisfied, then $Z_n\xrightarrow {a.e.}+\infty$, (1.2) and (1.4) are true.

2 A Growth Rate for IGWVE

In order to prove Theorem 1.1, we need the following lemma:

Lemma 2.1

Let $\{X_{n},n\ge 0\}$ be a GWVE. For any fixed $n\ge 0, r\ge 1$ one has

$$\begin{aligned} X_{n+r}=\sum _{j=1}^{X_{n}}X_{n,r}^{(j)}, \end{aligned}$$

where $\{X_{n,r}^{(j)}; j\ge 1\}$ are i.i.d. and independent of $X_{n}$. Furthermore,

$$\begin{aligned} m_{n,r}=E(X_{n,r}^{(j)})=\prod _{j=n}^{n+r-1}\mu _{j},~~~ \sigma _{n,r}^{2}=Var(X_{n,r}^{(j)})=(m_{n,r})^{2}\sum _{j=n}^{n+r-1} \frac{\delta _{j}^{2}}{\mu _{j}^{2}m_{n,j-n}}. \end{aligned}$$

Note that $m_{n, 0}=1,~~ m_{n, 1}=\mu _{n},~~ X_{n,1}^{(j)}=\xi _{n,j}$ and $\sigma _{n,1}^{2}=\delta _{n}^{2}.$

Proof

Let $X_{n,r}^{(j)}$ be the size of the $r$th generation of GWVE starting with the $j$th particle at time $n$. The first result in Lemma 2.1 following from Definition 1.1. Similar to Proposition 4 and Proposition 6 in [8], the rest of Lemma 2.1 is true. $\square$

Proof of Theorem 1.1

By our basic assumption, it is obvious that $W_n$ is integrable for all $n\ge 1$. By the independence of $\{Y_n,n\ge 1\}$ and $\{\xi _{n,j},n\ge 0,j\ge 1\}$, one has

$$\begin{aligned} E(W_{n+1}|W_{n},W_{n-1},\cdots ,W_1)&= E\left( \frac{Z_{n+1}}{m_{n+1}} |Z_n,Z_{n-1},\cdots ,Z_1\right) \\&= \frac{1}{m_n}E\left[ \frac{1}{\mu _n}\left( \sum _{i=1}^{Z_n}\xi _{n,j} +Y_{n+1}\right) |Z_n\right] \\&= \frac{Z_n}{m_n}+\frac{v_{n+1}}{m_{n+1}}>\frac{Z_n}{m_n}=W_n, \end{aligned}$$

which means that $\{W_n, n\ge 1\}$ is a nonnegative submartingale.

Let $X_n$ be the size of the original GWVE at time $n$ and $U_{k,n}(k<n)$ be the number of descendants at time $n$ of the particles that immigrated in generation $k$, then

$$\begin{aligned} Z_n=X_n+\sum _{k=1}^nU_{k,n}, \end{aligned}$$

(2.1)

where $U_{n,n}=Y_n$. Let ${\mathcal {G}}$ be the $\sigma -$field generated by $\{Y_n,n\ge 1\}$, by the independence of $\{Y_n,n\ge 1\}$ and $\{\xi _{n,j},n\ge 0,j\ge 1\}$ one has

$$\begin{aligned} E(W_n|\mathcal {G})=E\left( \frac{Z_n}{m_n}|\mathcal {G}\right) = E\left( \frac{X_n}{m_n}\right) + E\left( \sum _{k=1}^{n}\frac{U_{k,n}}{m_n}|\mathcal {G}\right) . \end{aligned}$$

(2.2)

Now for $k\le n$, the random variable $U_{k,n}$ is the size of the $(n-k)$th generation of ordinary GWVE starting, however, with $Y_k$ particles at time $k$. Therefore, by the independence of $\{Y_n,n\ge 1\}$ and Lemma 2.1, its conditional expectation is just $Y_{k}m_{k,n-k}$. According to (2.2) and Lemma 2.1, one has

$$\begin{aligned} E(W_n|\mathcal {G})=1+\sum _{k=1}^{n}\frac{Y_{k}}{m_k}~~\text { and}~~ E(W_n)=1+\sum _{k=1}^{n}\frac{v_{k}}{m_k}. \end{aligned}$$

Since (1.1) is satisfied we know that $\{W_n,n\ge 1\}$ is $L^1$-bounded. By the convergence theorem of submartingale (see [9] Theorem 2.5), there exists a nonnegative random variable $W$ with $E(W)<\infty$ such that $W_n\xrightarrow {a.s.}W.$

In order to prove the last result in Theorem 1.1, we only need to prove that$\{W_n,n\ge 1\}$ is $L^2$-bounded (see Theorem 7.6.10 of [4]). In fact, by (2.1) and the independence of $\{Y_n,n\ge 1\}$ and $\{\xi _{n,j},n\ge 0,j\ge 1\}$ one has

$$\begin{aligned} E(W_n^2|\mathcal {G})&= E\left( \frac{1}{m_n^2} \left[ X_n+\sum _{k=1}^nU_{k,n}\right] ^2|\mathcal {G}\right) \\&= E\left\{ \frac{1}{m_n^2}\left[ X_n^2+2X_n\sum _{k=1}^nU_{k,n} +\left( \sum _{k=1}^nU_{k,n}\right) ^2\right] |\mathcal {G}\right\} \\&= \frac{1}{m_n^2}E(X_n^2)+2\sum _{k=1}^n\frac{Y_k}{m_k}+\frac{1}{m_n^2} E\left[ \left( \sum _{k=1}^n \left( \sum _{j=1+\sum _{i=1}^{k-1}Y_{i}}^{\sum _{i=1}^{k} Y_{i}}X_{k,n-k}^{(j)}\right) \right) ^{2}|\mathcal {G}\right] \\&= \frac{1}{m_n^2}E(X_n^2)+2\sum _{k=1}^n\frac{Y_k}{m_k}\\&+\,\frac{1}{m_n^2}\left[ \sum _{k=1}^n(Y_kVar(X_{k,n-k}^{(1)}))+ \sum _{k=1}^nY_k\left( E(X_{k,n-k}^{(1)})\right) ^{2}\right] , \end{aligned}$$

where $X_{k,n-k}^{(j)}$ is defined in Lemma 2.1, it is the size of the $(n-k)$th generation of GWVE starting with the $j$th particle immigrated at time $k$. By the Lemma 2.1 one has

$$\begin{aligned} E(W_n^2)&= \frac{1}{m_n^2}E(X_n^2)+2\sum _{k=1}^n\frac{v_k}{m_k}+ \frac{1}{m_n^2}\left\{ \sum _{k=1}^n (v_k \sigma _{k,n-k}^{2})+\sum _{k=1}^n (v_k m_{k,n-k}^{2})\right\} \\&= \frac{1}{m_n^2}E(X_n^2)+2\sum _{k=1}^n\frac{v_k}{m_k}+\sum _{k=1}^n \left[ \frac{v_k}{m_k^2}\cdot \left( \frac{\sigma _{k,n-k}^{2}}{m_{k,n-k}^{2}} +1\right) \right] \\&\le b+2a +ab+a<\infty , \end{aligned}$$

which means that $\{W_n,n\ge 1\}$ is $L^2$-bounded. We complete the proof. $\square$

3 The CLT for IGWVE

In order to prove Theorem 1.2, we need the following lemmas.

Lemma 3.1

Let $\{Z_n,n\ge 0\}$ be an IGWVE. For any fixed $n\ge 0, r\ge 1$ one has

$$\begin{aligned} Z_{n+r}=\sum _{j=1}^{Z_n}X_{n,r}^{(j)}+Y_{n,r}~~~~~~a.s., \end{aligned}$$

(3.1)

where $\{X_{n,r}^{(j)},j\ge 1\}$ are i.i.d. and independent of $Y_{n,r}$. Furthermore,

$$\begin{aligned} \pi _{n,r}&= E(Y_{n,r})=\sum _{i=n+1}^{n+r}\nu _im_{i,n+r-i},\end{aligned}$$

(3.2)

$$\begin{aligned} \theta _{n,r}^{2}&= Var (Y_{n,r})= \sum _{i=n+1}^{n+r}\nu _i\sigma _{i,n+r-i}^2. \end{aligned}$$

(3.3)

Proof

Similar to Lemma 2.1, let $X_{n,r}^{(j)}$ be the size of the $r$th generation of the ordinary GWVE staring with the $j$th particle at time $n$. Let $U_{k,n}$ be the number of descendants at time $n$ of the particles that were immigrated in generation $k$. Define $Y_{n,r}=\sum _{k=n+1}^{n+r}U_{ k,n+r}.$ Then

$$\begin{aligned} Z_{n+r}=\sum _{j=1}^{Z_n}X_{n,r}^{(j)}+Y_{n,r},~~~~a.s.. \end{aligned}$$

By the independence of $\{Y_n,n\ge 1\}$ and $\{\xi _{n,j},n\ge 0,j\ge 1\}$ we know that $\{X_{n,r}^{(j)},j\ge 1\}$ are i.i.d. and independent of $Y_{n,r}$.

Let ${\mathcal {G}}$ be the $\sigma -$field generated by $\{Y_n,n\ge 1\}$. Similar to the calculation of (2.2) we have

$$\begin{aligned} E(Y_{n,r}|{\mathcal {G}})=E\left( \sum _{k=n+1}^{n+r}U_{ k,n+r}|{\mathcal {G}}\right) =\sum _{k=n+1}^{n+r}Y_{k}E(X_{k,n+r-k}^{(1)}). \end{aligned}$$

So by Lemma 2.1, we have $E(Y_{n,r})=\sum _{k=n+1}^{n+r}v_{k}m_{k,n+r-k}.$ On the other hand, by the independence of $\{Y_n,n\ge 1\}$ and $\{\xi _{n,j},n\ge 0,j\ge 1\}$ one has

$$\begin{aligned} Var(Y_{n,r}|{\mathcal {G}})&= Var\left( \sum _{k=n+1}^{n+r}U_{ k,n+r}|{\mathcal {G}}\right) =\sum _{k=n+1}^{n+r}Var(U_{ k,n+r}|{\mathcal {G}})\\&= \!\sum _{k=n+1}^{n+r}Var\left( \sum _{j=1+\sum _{i=n+1}^{k-1} Y_{i}}^{\sum _{i=n+1}^{k}Y_{i}}X_{k,n+r-k}^{(j)}|{\mathcal {G}}\!\right) \!=\! \sum _{k=n+1}^{n+r}Y_{k}Var(X_{k,n+r-k}^{(1)}). \end{aligned}$$

So by Lemma 2.1 we have $Var(Y_{n,r})=\sum _{k=n+1}^{n+r}v_{k}\sigma _{k,n+r-k}^{2}.$ We complete the proof. $\square$

Lemma 3.2

[8] Let $\{X_{n},n\ge 0\}$ be a GWVE. Define $V_{n}=X_{n}/m_{n}$ for all $n\ge 1$, then $\{V_{n},n\ge 1\}$ is a nonnegative martingale and there exists a nonnegative random variable $V$ such that $V_{n}\xrightarrow {a.s.}V$ when $n\rightarrow \infty$.

Lemma 3.3

Let $\{X_{n},n\ge 0\}$ be a GWVE. Define $V_{n,r}^{(j)}=X_{n,r}^{(j)}/m_{n,r}$, where $X_{n,r}^{(j)}$ and $m_{n,r}$ are defined in Lemma 2.1, then $\{V_{n,r}^{(j)},r\ge 1\}$ is a nonnegative martingale and $V_{n,r}^{(j)}\xrightarrow {a.s}V_{n}^{(j)}$ when $n,j$ fixed and $r\rightarrow \infty$. Then for any fixed $n\ge 1$ one has

$$\begin{aligned} X_{n}-m_{n}V=\sum _{j=1}^{X_{n}}(1-V_{n}^{(j)})~~a.s., \end{aligned}$$

where $V$ is defined in Lemma 3.2, $\{V_{n}^{(j)};j\ge 1\}$ are i.i.d. and independent of $X_{n}$.

Furthermore if (1.2) is satisfied, then $V_{n,r}^{(j)}\xrightarrow {a.s}V_{n}^{(j)}$ when $n,j$ fixed and $r\rightarrow \infty$, $E(V_{n}^{(j)})\equiv 1$,

$$\begin{aligned} Var(V_{n}^{(j)})=\sigma _{n}^{2}= \sum _{j=n}^{\infty }\frac{\delta _{j}^{2}}{\mu _{j}^{2}m_{n,j-n}}. \end{aligned}$$

Proof

Similar to Proposition 5 and Corollary 2 of [8], we know the first result in Lemma 3.3 is true. Then the second result following from Lemma 2.1. Using the same idea of Theorem 1 in [8], we can prove the last result. $\square$

Lemma 3.4

Let $\{Z_{n},n\ge 0\}$ be an IGWVE. For any fixed $n\ge 0$, one has

$$\begin{aligned} Z_{n}-m_{n}W=\sum _{j=1}^{Z_{n}}(1-V_{n}^{(j)})- I_{n} ~~a.s., \end{aligned}$$

(3.4)

where $\{V_{n}^{(j)},j\ge 1\}$ are i.i.d. and independent of $I_{n}$. Furthermore, if (2) and (3) are satisfied,

$$\begin{aligned} \pi _{n}&= E(I_{n})=\sum _{k=n+1}^{\infty }\frac{\nu _{k}}{m_{n,k-n}},\end{aligned}$$

(3.5)

$$\begin{aligned} \theta _{n}^{2}&= Var(I_{n})=\sum _{k=n+1}^{\infty } \frac{\nu _{k}\sigma _{k}^{2}}{m_{n,k-n}}. \end{aligned}$$

(3.6)

Proof

Applying Theorem 1.1 one has that for each $n\ge 1$,

$$\begin{aligned} \lim _{r\rightarrow \infty }\frac{Z_{n+r}}{m_{n+r}}=\lim _{r\rightarrow \infty }W_{n+r} =W ~~a.s.. \end{aligned}$$

So by Lemma 3.1 one deduce that

$$\begin{aligned} Z_{n}-m_{n}W&= Z_{n}-m_{n}\lim _{r\rightarrow \infty }\frac{Z_{n+r}}{m_{n+r}}\nonumber \\&= \lim _{r\rightarrow \infty }\left[ Z_{n}-\frac{m_{n}}{m_{n+r}} \left( \sum _{j=1}^{Z_{n}}X_{n,r}^{(j)}+Y_{n,r}\right) \right] \nonumber \\&= \lim _{r\rightarrow \infty }\left\{ \sum _{j=1}^{Z_{n}}\left[ 1- \frac{X_{n,r}^{(j)}}{m_{n,r}}\right] -\frac{Y_{n,r}}{m_{n,r}}\right\} ~~a.s.. \end{aligned}$$

(3.7)

By Lemma 3.3 one has

$$\begin{aligned} \lim _{r\rightarrow \infty }\sum _{j=1}^{Z_{n}}\left[ 1-\frac{X_{n,r}^{(j)}}{m_{n,r}}\right] =\sum _{j=1}^{Z_{n}}\left[ 1-V_{n}^{(j)}\right] ~~a.s.. \end{aligned}$$

(3.8)

One the other hand, define $T_{n,r}=\frac{Y_{n,r}}{m_{n,r}}=\sum _{k=n+1}^{n+r}U_{ k,n+r}/m_{n,r}$ for each $r\ge 1$. Let ${\mathcal {F}}_{n,r}$ be the $\sigma -$field generated by $\{Y_{n},\cdots ,Y_{n+r}, \xi _{i,j}, n-1\le i\le n+r-1,j\ge 1 \}$. Note that by Lemma 3.1 one has

$$\begin{aligned} E(T_{n,r})=\frac{1}{m_{n,r}}\sum _{k=n+1}^{n+r}v_{k}m_{k,n+r-k}= \sum _{k=n+1}^{n+r}\frac{v_{k}}{m_{n,k-n}}\le c, \end{aligned}$$

which means that $\{T_{n,r},r\ge 1\}$ is $L^{1}-$bounded. Now

$$\begin{aligned} E(T_{n,r+1}|{\mathcal {F}}_{n,r})&= E\bigg (\frac{Y_{n,r+1}}{m_{n,r+1}} |{\mathcal {F}}_{n,r}\bigg )=E\bigg (\frac{\sum _{k=n+1}^{n+r+1}U_{ k,n+r+1}}{m_{n,r+1}}|{\mathcal {F}}_{n,r}\bigg )\\&= \frac{1}{m_{n,r+1}}E\bigg \{\!Y_{n+r+1}+\sum _{k=n+1}^{n+r} \bigg [\sum _{j=1+\sum _{i=n+1}^{k-1}Y_{i}}^{\sum _{i=n+1}^{k} Y_{i}}X_{k,n+r+1-k}^{(j)}\bigg ]|{\mathcal {F}}_{n,r}\bigg \}\\&= \frac{1}{m_{n,r+1}}\bigg \{E(Y_{n+r+1})\!+\!\sum _{k=n+1}^{n+r} \bigg [\sum _{j=1+\sum _{i=n+1}^{k-1}Y_{i}}^{\sum _{i=n+1}^{k} Y_{i}}X_{k,n+r-k}^{(j)}E(\xi _{n+r,1})\!\bigg ]\!\bigg \}\\&= \frac{Y_{n,r}}{m_{n,r}}+\frac{v_{n+r+1}}{m_{n,r+1}}>T_{n,r}, \end{aligned}$$

which means that $\{T_{n,r},r\ge 1\}$ is a nonnegative submartingale with respect to ${\mathcal {F}}_{n,r},r\ge 1$, then there exists a nonnegative random variable $I_{n}$ with $E(I_{n})<\infty$ such that

$$\begin{aligned} \lim _{r\rightarrow \infty }T_{n,r}=I_{n}~~a.s.. \end{aligned}$$

(3.9)

By (3.7), (3.8)and (3.9) we obtain (3.4).

Now by Lemma 3.1 one has

$$\begin{aligned} E(T_{n,r})^{2}&= Var(T_{n,r})+(E(T_{n,r}))^{2}\\ {}&= \frac{1}{m_{n,r}^{2}} \left[ \sum _{k=n+1}^{n+r}\nu _k\sigma _{k,n+r-k}^2+\left( \sum _{k=n+1}^{n+r} \nu _km_{k,n+r-k}\right) ^{2}\right] \\&\le \sum _{k=n+1}^{n+r}\left[ \frac{v_{k}}{m_{n,k-n}^{2}}\cdot \frac{\sigma _{k,n+r-k}^2}{m_{k,n+r-k}^2}\right] +\left( \sum _{k=n+1}^{n+r} \frac{v_{k}}{m_{n,k-n}}\right) ^{2}\\&\le bc+c^{2}<\infty , \end{aligned}$$

which means that $\{T_{n,r},r\ge 1\}$ is $L^{2}-$bounded, so $T_{n,r}\xrightarrow {L^{2}}I_{n}$ when $r\rightarrow \infty$. Thus,

$$\begin{aligned} E(I_{n})&= \lim _{r\rightarrow \infty }E(T_{n,r})=\sum _{k=n+1}^{\infty } \frac{v_{k}}{m_{n,k-n}},\\ Var(I_{n})&= \lim _{r\rightarrow \infty }Var(T_{n,r})=\lim _{r\rightarrow \infty } \sum _{k=n+1}^{n+r}\left[ \frac{v_{k}}{m_{n,k-n}}\cdot \frac{\sigma _{k,n+r-k}^2}{m_{k,n+r-k}^2}\right] \\&= \sum _{k=n+1}^{\infty } \left[ \frac{v_{k}\sigma _{k}^{2}}{m_{n,k-n}}\right] . \end{aligned}$$

We complete the proof of Lemma 3.4. $\square$

Now we consider a double sequence of random variables $\{\zeta _{n,j}, j\ge 1,n\ge 1\}$, where for any $n\ge 1$, $\{\zeta _{n,j}, j\ge 1\}$ are i.i.d.. Let $\{N_{n},n\ge 1\}$ be a sequence of random variables taking values in ${\mathbb {Z}}_{+}:=\{1,2,\cdots \}$, where $N_{n}\xrightarrow {P}\infty$ and for any $n\ge 1$, $\{\zeta _{n,j}, j\ge 1\}$ and $N_{n}$ are independent. Define

$$\begin{aligned} S_{N_{n}}=\sum _{j=1}^{N_{n}}\zeta _{n,j}, \end{aligned}$$

then we have the following result:

Lemma 3.5

If $E(\zeta _{n,j})\equiv 0$ and $Var(\zeta _{n,j})\equiv 1$, then one has

$$\begin{aligned} \frac{S_{N_{n}}}{\sqrt{N_{n}}}\xrightarrow []{d}N(0,1),n\rightarrow \infty . \end{aligned}$$

Proof

Since $\{\zeta _{n,j},j\ge 1\}$ have the same distribution, we can set

$$\begin{aligned} \varphi _{n}(t)=E\left( exp\left( it\zeta _{n,j}\right) \right) . \end{aligned}$$

Note $E(\zeta _{n,j})\equiv 0$ and $Var(\zeta _{n,j})\equiv 1$, according to (3.8) of [18] P101, one obtains

$$\begin{aligned} \varphi _{n}(s)=\varphi _{n}(0)+\varphi '_{n}(0)s+\frac{\varphi ''_{n}(0)}{2!} s^{2}+o(s^{2})=1-\frac{s^{2}}{2}+o\left( \frac{s^{2}}{2}\right) ,(s\rightarrow 0). \end{aligned}$$

For any fixed $t$ and $k$ large enough

$$\begin{aligned} \varphi _{n}\left( \frac{t}{\sqrt{k}}\right) =1-\frac{t^{2}}{2k}+ o\left( \frac{1}{2k}\right) . \end{aligned}$$

(3.10)

Since for any $n\ge 1$, $\{N_{n},\zeta _{n,j},j\ge 1\}$ are independent, we have

$$\begin{aligned} E\left( \exp \left( it\frac{S_{N_{n}}}{\sqrt{N_{n}}}\right) \right)&= \sum _{k=1}^{\infty }E\left( \exp \left( it\frac{S_{N_{n}}}{\sqrt{N_{n}}}\right) |N_{n}=k\right) P(N_{n}=k)\nonumber \\&= \sum _{k=1}^{\infty }E\left( \exp \left( it\frac{S_{k}}{\sqrt{k}}\right) \right) P(N_{n}=k)\nonumber \\&= \sum _{k=1}^{\infty }\left( E\left( \exp \left( it\frac{\zeta _{n,1}}{\sqrt{k}}\right) \right) \right) ^{k}P(N_{n}=k). \end{aligned}$$

(3.11)

Fix $t\in {\mathbb {R}}:=(-\infty ,+\infty )$, note that $(1-\frac{x}{n}+o(\frac{1}{n}))^{n}\rightarrow e^{-x}~(n\rightarrow \infty ),$ so there exists a constant $M=M(\varepsilon )>0$ such that for any $k\ge M$ one has

$$\begin{aligned} \left| \left( 1-\frac{t^{2}}{2k}+o\left( \frac{1}{2k}\right) \right) ^{k}- \exp \left( -\frac{t^{2}}{2}\right) \right| \le \frac{\varepsilon }{4}. \end{aligned}$$

(3.12)

Since $N_{n}\xrightarrow {P}\infty$, for any $\varepsilon >0$, there exists a constant $N=N(\varepsilon )>1$ such that for any $n\ge N$ one has $P(N_{n}\le M)<\frac{\varepsilon }{4}.$ Thus, when $n\ge N$, by (3.10),(3.11) and (3.12) one has

$$\begin{aligned} \Delta&:= \left| E\left( \exp \left( it\frac{S_{N_{n}}}{\sqrt{N_{n}}}\right) \right) -\exp \left( -\frac{t^{2}}{2}\right) \right| \\&= \left| \sum _{k=1}^{\infty }\left( E\left( \exp \left( it\frac{\zeta _{n,1}}{\sqrt{k}}\right) \right) \right) ^{k}P(N_{n}=k) -\exp \left( -\frac{t^{2}}{2}\right) \right| \nonumber \\&\le \left| \sum _{k=1}^{M}\left[ \left( E\left( \exp \left( it \frac{\zeta _{n,1}}{\sqrt{k}}\right) \right) \right) ^{k} -\exp \left( -\frac{t^{2}}{2}\right) \right] P(N_{n}=k)\right| \nonumber \\&+\left| \sum _{k=M+1}^{\infty }\left[ \left( 1-\frac{t^{2}}{2k} +o\left( \frac{1}{2k}\right) \right) ^{k}- \exp \left( -\frac{t^{2}}{2}\right) \right] P(N_{n}=k)\right| \nonumber \\&\le 2\cdot \frac{\varepsilon }{4}+ \frac{\varepsilon }{4}\cdot \sum _{k=M+1}^{\infty }P(N_{n}=k) <\varepsilon . \end{aligned}$$

This means the characteristic function of $S_{N_{n}}/\sqrt{N_{n}}$ convergent to that of a standard normal random variable, so by Lévy continuous theorem we complete the proof of this lemma. $\square$

Proof of Theorem 1.2

By Lemma 3.1, for any $n\ge 0,r\ge 1$ we have

$$\begin{aligned} Z_{n+r}-m_{n,r}Z_{n}=\sum _{j=1}^{Z_n}(X_{n,r}^{(j)}-m_{n,r})+Y_{n,r}~~a.s., \end{aligned}$$

where $\{X_{n,r}^{(j)}, j\ge 1\}$ are i.i.d. and independent of $Z_{n}$, $E(X_{n,r})^{(j)}\equiv m_{n,r}$, $Var(Z_{n,r}^{(j)})\equiv \sigma _{n,r}^{2}$. According to Lemma 3.5, if we can prove that $Y_{n,r}/\sigma _{n,r}\sqrt{Z_{n}}\xrightarrow {P}0$ when $r$ fixed and $n\rightarrow \infty$, then we complete the proof of the first part of Theorem 1.2. In fact, for any $\epsilon >0$,

$$\begin{aligned} P\left( \frac{Y_{n,r}}{\sigma _{n,r}\sqrt{Z_{n}}}\ge \epsilon \right) =E\left( P\left( \frac{Y_{n,r}}{\sigma _{n,r}\sqrt{Z_{n}}}\ge \epsilon |Z_{n}\right) \right) \le \frac{\theta _{n,r}^{2}}{\epsilon ^{2}\sigma _{n,r}^{2}}\cdot E\left( \frac{1}{Z_{n}}\right) . \end{aligned}$$

Note that $Z_{n}\xrightarrow {P}\infty$ and $Z_n\ge 1$, for any $\epsilon >0$, there exists two constants $N,M>0$ such that $1/M<\epsilon /2$ and $P(Z_n<M)<\epsilon /2$ for all $n\ge N.$ When $n\ge N,$

$$\begin{aligned} E\left( \frac{1}{Z_{n}}\right)&= E\left( \frac{1}{Z_{n}}I_{[Z_n<M]}\right) +E\left( \frac{1}{Z_{n}}I_{[Z_n\ge M]}\right) \\&\le 1\cdot P(Z_n<M) +\frac{1}{M}P(Z_n\ge M)< \epsilon . \end{aligned}$$

Since (1.2),(1.3) and (1.4) are satisfied, then

$$\begin{aligned} P\left( \frac{Y_{n,r}}{\sigma _{n,r}\sqrt{Z_{n}}}\ge \epsilon \right) \le \frac{bc}{\epsilon ^{2}d}\cdot E\left( \frac{1}{Z_{n}}\right) \rightarrow 0,~~n\rightarrow \infty . \end{aligned}$$

By Lemma 3.4, for any $n\ge 1$ one has

$$\begin{aligned} Z_{n}-m_{n}W=\sum _{j=1}^{Z_{n}}\left( 1-V_{n}^{(j)}\right) -I_{n}~~a.s., \end{aligned}$$

where $\{V_{n}^{(j)};j\ge 1\}$ are i.i.d.and independent of $Z_{n}$, $E(V_{n}^{(j)})\equiv 1$, $Var(V_{n}^{(j)})\equiv \sigma _{n}^{2}$. Similarly, according to Lemma 3.5, if we can prove that $I_{n}/\sigma _{n}\sqrt{Z_{n}}\xrightarrow {P}0$ when $n\rightarrow \infty$, then we complete the proof of the second part of Theorem 1.2. In fact, for any $\epsilon >0$,

$$\begin{aligned} P\left( \frac{I_{n}}{\sigma _{n}\sqrt{Z_{n}}}\ge \epsilon \right) =E\left( P\left( \frac{I_{n}}{\sigma _{n}\sqrt{Z_{n}}}\ge \epsilon |Z_{n}\right) \right) \le \frac{\theta _{n}^{2}}{\epsilon ^{2}\sigma _{n}^{2}}\cdot E\left( \frac{1}{Z_{n}}\right) . \end{aligned}$$

Note that $Z_{n}\xrightarrow {P}\infty$, (1.2)–(1.4) are satisfied, then

$$\begin{aligned} P\left( \frac{I_{n}}{\sigma _{n}\sqrt{Z_{n}}}\ge \epsilon \right) \le \frac{bc}{\epsilon ^{2}d}\cdot E\left( \frac{1}{Z_{n}}\right) \rightarrow 0,~~n\rightarrow \infty . \end{aligned}$$

We complete the proof of Theorem 1.2. $\square$

4 Convergence Rate in the CLT for IGWVE

In order to prove Theorem 1.4, we need following lemmas.

Lemma 4.1

([7] P322) Let $\{X_n,n\ge 1\}$ be a sequence of independent random variables with $E(X_{n})=0,n=1,2,\cdots$. Define

$$\begin{aligned} L_{n}=\sum _{j=1}^{n}X_{j},~~~~ G_{n}(x)=P\left( \frac{L_{n}}{\sqrt{Var(L_{n})}}\le x\right) , n\ge 1,x\in {\mathbb {R}}. \end{aligned}$$

If there exists an arbitrary small constant $1>\delta >0$ such that

$$\begin{aligned} \Gamma _{n}^{2+\delta }:=\sum _{j=1}^{n}E|X_{j}|^{2+\delta }<\infty , \end{aligned}$$

then there exists a constant $A$ such that

$$\begin{aligned} \sup _{x}|G_{n}(x)-\Phi (x)|\le A\Gamma _{n}^{2+\delta }[Var(L_{n})]^{-(1+\delta /2)}, \end{aligned}$$

where $\Phi (x)$ is the standard normal distribution.

Lemma 4.2

Let $\{\zeta _{n,j}, j\ge 1,n\ge 1\}$ be a double sequence of random variables with mean zero and variance 1, where for any $n\ge 1$, $\{\zeta _{n,j}, j\ge 1\}$ are i.i.d.. $\{N_{n},n\ge 1\}$ is a sequence of integer valued random variables with $P(N_{n}\rightarrow \infty )=1,$ as $n\rightarrow \infty$. For any $n\ge 1$, $N_{n}$ and $\{\zeta _{n,j}, j\ge 1\}$ are independent. $\{k_n,n\ge 1\}$ is a sequence of strictly increasing positive integers. Define

$$\begin{aligned} \tilde{L}_{n}&= \sum _{j=1}^{k_{n}}\zeta _{n,j},~~~~ \tilde{G}_{n}(x)=P\left( \frac{\tilde{L}_{n}}{\sqrt{k_{n}}}\le x\right) .\\ \widehat{L}_{n}&= \sum _{j=1}^{N_{n}}\zeta _{n,j},~~~~ \widehat{G}_{n}(x)=P\left( \frac{\widehat{L}_{n}}{\sqrt{N_{n}}}\le x\right) . \end{aligned}$$

If there exist two constants $\delta$ and $M>0$ such that for any $n\ge 1$,

$$\begin{aligned} \gamma _{n}:=E(|\zeta _{n,j}|^{2+\delta })\le M, \end{aligned}$$

then there exists a constants $C$ (not depend on $\{k_{n}\}$) such that

$$\begin{aligned} \sup _{x}|\tilde{G}_{n}(x)-\Phi (x)|\le Ck_{n}^{-\frac{\delta }{2}}, ~~~~\sup _{x}|\widehat{G}_{n}(x)-\Phi (x)|\le CE(N_{n}^{-\frac{\delta }{2}}). \end{aligned}$$

(4.1)

Proof

Let $C=AM$, where $A$ is chosen in Lemma 4.1, then using Lemma 4.1 we have the first inequality of (4.1). As for the second inequality, since for any $n\ge 1$, $N_{n}$ and $\{\xi _{n,j}, j\ge 1\}$ are independent, so

$$\begin{aligned} \widehat{G}_{n}(x)-\Phi (x)&= P\left( \frac{\sum _{i=1}^{N_{n}}\xi _{n,i}}{\sqrt{N_{n}}} \le x\right) -\Phi (x)\nonumber \\&= \sum _{j=1}^{\infty }\left[ P\left( \frac{\sum _{i=1}^{j}\xi _{n,i}}{\sqrt{j}}\le x\right) -\Phi (x)\right] P(N_{n}=j). \end{aligned}$$

By the first inequality of (4.1), one has

$$\begin{aligned} -CE\left( N_{n}^{-\frac{\delta }{2}}\right)&= -C\sum _{j=1}^{\infty } j^{-\frac{\delta }{2}}P(N_{n}=j)=\sum _{j=1}^{\infty }\left( -C\cdot j^{-\frac{\delta }{2}}\right) P(N_{n}=j)\nonumber \\&\le \sum _{j=1}^{\infty }\left( P\left( \frac{\sum _{i=1}^{j}\xi _{n,i}}{\sqrt{j}}\le x\right) -\Phi (x)\right) P(N_{n}=j)\le CE\left( N_{n}^{-\frac{\delta }{2}}\right) . \end{aligned}$$

We complete the proof. $\square$

Lemma 4.3

Assume that $\{\zeta _{n,j}, j\ge 1,n\ge 1\}, \{N_{n},n\ge 1\}$ satisfy the conditions of Lemma 4.2. Let $\{\eta _{n},n\ge 1\}$ be a sequence of independent random variables with $E(|\eta _{n}|)<\infty$ and for any $n\ge 1$, $\eta _{n}$ is independent of $\{\zeta _{n,j}, j\ge 1\}$ and $N_{n}$. Define

then for any sequence $\epsilon _{n}$ of positive constants one has

(4.2)

Proof

Let $\Phi (x)$ be the distribution function of standard normal distribution. By Lemma 4.2 and the independence of $\eta _{n}$ and $\{\zeta _{n,j}, j\ge 1\}$ one has

$$\begin{aligned} -CE(N_{n}^{-\frac{\delta }{2}})&= -C\sum _{j=1}^{\infty } j^{-\frac{\delta }{2}}P(N_{n}=j)\nonumber \\&= -C\sum _{j=1}^{\infty }j^{-\frac{\delta }{2}}\left[ \int _{-\infty }^{\infty } dP(j^{-\frac{1}{2}}\eta _{n}\le y)\right] P(N_{n}=j)\nonumber \\&\le \sum _{j=1}^{\infty }\bigg [\int _{-\infty }^{\infty }\bigg \{P\bigg (j^{-\frac{1}{2}} \sum _{i=1}^{j}\zeta _{n,i}\le x-y\bigg )\nonumber \\&-\Phi (x-y)\}dP(j^{-\frac{1}{2}} \eta _{n}\le y)\bigg ] P(N_{n}=j) \le CE(N_{n}^{-\frac{\delta }{2}}).\qquad \end{aligned}$$

(4.3)

But,

(4.4)

where $\zeta$ has the standard normal distribution and is independent of $\eta _{n}$. By (4.3) and (4.4), one has

(4.5)

Now for any $\epsilon _{n}>0$

$$\begin{aligned} P(\zeta +N_{n}^{-\frac{1}{2}}\eta _{n}\le x)&= P(\zeta +N_{n}^{-\frac{1}{2}}\eta _{n}\le x, N_{n}^{-\frac{1}{2}}|\eta _{n}|\le \epsilon _{n})\nonumber \\&+P(\zeta +N_{n}^{-\frac{1}{2}}\eta _{n}\le x, N_{n}^{-\frac{1}{2}}|\eta _{n}|> \epsilon _{n})\nonumber \\&\le P(\zeta \le x+\epsilon _{n})+P(N_{n}^{-\frac{1}{2}}|\eta _{n}|> \epsilon _{n}). \end{aligned}$$

(4.6)

Similarly, we have

$$\begin{aligned} P(\zeta +N_{n}^{-\frac{1}{2}}\eta _{n}> x)\le P(\zeta > x-\epsilon _{n})+P(N_{n}^{-\frac{1}{2}}|\eta _{n}|> \epsilon _{n}), \end{aligned}$$

or equivalently,

$$\begin{aligned} P(\zeta +N_{n}^{-\frac{1}{2}}\eta _{n}\le x)\ge P(\zeta \le x-\epsilon _{n})-P(N_{n}^{-\frac{1}{2}}|\eta _{n}|> \epsilon _{n}). \end{aligned}$$

(4.7)

Also, by mean value theorem, we know that

$$\begin{aligned} \sup _{x}|P(\zeta \le x-\epsilon _{n})-P(\zeta \le x)|&= \sup _{x}|\Phi (x-\epsilon _{n})-\Phi (x)|\nonumber \\&= \sup _{x}\Phi '(\alpha (x))\epsilon _{n}<\frac{\epsilon _{n}}{2}. \end{aligned}$$

(4.8)

Similarly,

$$\begin{aligned} \sup _{x}|P(\zeta \le x+\epsilon _{n})-P(\zeta \le x)|<\frac{\epsilon _{n}}{2}. \end{aligned}$$

(4.9)

By (4.6)–(4.9) and Markov inequality one has

$$\begin{aligned} \Lambda _{n}&:= \sup _{x}|P(\zeta +N_{n}^{-\frac{1}{2}}\eta _{n}\le x)-\Phi (x)|\le P(N_{n}^{-\frac{1}{2}}|\eta _{n}|> \epsilon _{n})+\frac{\epsilon _{n}}{2}\nonumber \\&\le \epsilon _{n}^{-1}E(N_{n}^{-\frac{1}{2}}| \eta _{n}|)+\frac{\epsilon _{n}}{2}=\epsilon _{n}^{-1} E(N_{n}^{-\frac{1}{2}})E(|\eta _{n}|)+\frac{\epsilon _{n}}{2}. \end{aligned}$$

(4.10)

By (4.5) and (4.10) one obtain (4.2). We complete the proof of Lemma 4.3. $\square$

Lemma 4.4

Let $\{Z_{n}, n\ge 0\}$ be an IGWVE. If (1.2)–(1.5) are satisfied, then there exists constants $\{C_{r},r\ge 1\}$ and $D$ such that for any sequence $\{\epsilon _{n},n\ge 1\}$ of positive constants,

$$\begin{aligned}&\sup _{x}|Q_{n,r}(x)-\Phi (x)|\le C_{r}E\left( Z_{n}^{-\frac{\delta }{2}}\right) +(\epsilon _{n}\sigma _{n,r})^{-1}\pi _{n,r}E\left( Z_{n}^{-\frac{\delta }{2}}\right) +\frac{\epsilon _{n}}{2},\end{aligned}$$

(4.11)

$$\begin{aligned}&\sup _{x}|Q_{n}(x)-\Phi (x)|\le DE(Z_{n}^{-\frac{\delta }{2}})+(\epsilon _{n} \sigma _{n})^{-1}\pi _{n}E\left( Z_{n}^{-\frac{\delta }{2}}\right) +\frac{\epsilon _{n}}{2}. \end{aligned}$$

(4.12)

Proof

By Lemma 3.1 one has

$$\begin{aligned} \frac{Z_{n+r}-m_{n,r}Z_{n}}{\sigma _{n,r}}= \sum _{j=1}^{Z_{n}} \frac{X_{n,r}^{(j)}-m_{n,r}}{\sigma _{n,r}}+\frac{Y_{n,r}}{\sigma _{n,r}}~~a.s., \end{aligned}$$

where $\{X_{n,r}^{(j)}; j\ge 1\}$ are i.i.d. and are independent of $Z_{n}$ and $Y_{n,r}$. Furthermore,

$$\begin{aligned} E\left( \frac{X_{n,r}^{(j)}-m_{n,r}}{\sigma _{n,r}}\right) \equiv 0,~~~~ Var\left( \frac{X_{n,r}^{(j)}-m_{n,r}}{\sigma _{n,r}}\right) \equiv 1. \end{aligned}$$

Note that $Y_{n,r}$ is independent of $Z_{n}$, by Lemma 4.3 and (1.5) we obtain (4.11).

Similarly, by Lemma 3.4 one has

$$\begin{aligned} \frac{Z_{n}-m_{n}W}{\sigma _{n}}=\sum _{j=1}^{Z_{n}} \frac{1-V_{n}^{(j)}}{\sigma _{n}}+\frac{I_{n}}{\sigma _{n}}, \end{aligned}$$

where $\{V_{n}^{(j)}; j\ge 1\}$ are i.i.d. and are independent of $Z_{n}$ and $I_{n}$. Furthermore,

$$\begin{aligned} E\left( \frac{V_{n}^{(j)}-1}{\sigma _{n}}\right) \equiv 0,~~~~ Var\left( \frac{V_{n}^{(j)}-1}{\sigma _{n}}\right) \equiv 1. \end{aligned}$$

Note that $I_{n}$ is independent of $Z_{n}$, by Lemma 4.3 and (1.5) we obtain (4.12). $\square$

Proof of Theorem 1.4

Note that by Lemma 3.1, for any $n\ge 1,r\ge 1$, one has

$$\begin{aligned} \frac{\pi _{n,r}}{\sigma _{n,r}}=\frac{\sum _{i=n+1}^{n+r}\nu _im_{i,n+r-i}}{m_{n,r} \sqrt{\sum _{j=n}^{n+r-1}\frac{\delta _j^2}{\mu _j^2m_{n,j-n}}}}\le \frac{1}{d}\sum _{i=n+1}^{n+r}\frac{\nu _i}{m_{n,i-n}}\le \frac{c}{d} \end{aligned}$$

and by Lemma 3.4, for any $n\ge 1$ we have

$$\begin{aligned} \frac{\pi _{n}}{\sigma _{n}}=\frac{\sum _{i=n+1}^{\infty }\frac{\nu _i}{m_{i,i-n}}}{ \sqrt{\sum _{j=n}^{\infty }\frac{\delta _j^2}{\mu _j^2m_{n,j-n}}}}\le \frac{c}{d}. \end{aligned}$$

Taking $\epsilon _{n}=\left[ E\left( Z_{n}^{-1/2}\right) \right] ^{\frac{1}{2}}$ in (4.11) and (4.12), $C=c/d+1/2$, one obtains Theorem 1.4. $\square$

5 The LIL for IGWVE

In order to prove Theorem 1.5, we need the following lemmas.

Lemma 5.1

For any $\varepsilon >0$ one has

$$\begin{aligned}&\sum _{n=2}^{\infty }[1-\Phi ((1+\varepsilon )(2\log n)^{\frac{1}{2}})]<\infty ,\end{aligned}$$

(5.1)

$$\begin{aligned}&\sum _{n=2}^{\infty }[1-\Phi ((1-\varepsilon )(2\log n)^{\frac{1}{2}})]=\infty . \end{aligned}$$

(5.2)

Proof

For any $\varepsilon >0$, one has

$$\begin{aligned} 1-\Phi ((1+\varepsilon )(2\log n)^{\frac{1}{2}})&= \int _{(1+\varepsilon )(2\log n)^{\frac{1}{2}}}^{\infty }\frac{1}{\sqrt{2\pi }}e^{-\frac{t^{2}}{2}}dt \le \sum _{m=m_{0}}^{\infty }\frac{1}{\sqrt{2\pi }}e^{-\frac{m^{2}}{2}}\nonumber \\&\le \sum _{i=0}^{\infty }\frac{1}{\sqrt{2\pi }}e^{-\frac{m_{0}(m_{0}+i)}{2}} =N\sum _{i=0}^{\infty }\left( \frac{1}{\sqrt{e^{m_{0}}}}\right) ^{i}, \end{aligned}$$

(5.3)

where $m_{0}=\left[ (1+\varepsilon )(2\log n)^{\frac{1}{2}}\right] -1$ and $N=\frac{1}{\sqrt{2\pi }}e^{-\frac{m_{0}^{2}}{2}}$. When $n$ is large enough one has $\frac{1}{\sqrt{e^{m_{0}}}}<\frac{1}{2}$, so

$$\begin{aligned} 1-\Phi ((1+\varepsilon )(2\log n)^{\frac{1}{2}})\le N\sum _{i=0}^{\infty }\left( \frac{1}{\sqrt{e^{m_{0}}}}\right) ^{i}\le 2N =O(n^{-(1+\varepsilon )^{2}}), \end{aligned}$$

(5.4)

so we obtain the (5.1).

$$\begin{aligned} 1-\Phi ((1-\varepsilon )(2\log n)^{\frac{1}{2}})=\int _{(1-\varepsilon )(2\log n)^{\frac{1}{2}}}^{\infty }\frac{1}{\sqrt{2\pi }}e^{-\frac{t^{2}}{2}}dt \ge cn^{-(1-\varepsilon )^{2}}, \end{aligned}$$

(5.5)

where $c$ is some positive constant. We obtain the (5.2) by (5.5). $\square$

Lemma 5.2

Let $\{Z_{n}; n\ge 0\}$ be an IGWVE, then for any fixed $r\ge 1$, $\{Z_{rn}; n\ge 0\}$ is an IGWVE. If (1.2)-(1.5) are satisfied, then

$$\begin{aligned} \sup _{x}|Q_{rn,r}(x)-\Phi (x)|\le C_{r}E\left( Z_{rn}^{-\frac{\delta }{2}}\right) + C\left[ E\left( Z_{rn}^{-\frac{\delta }{2}}\right) \right] ^{\frac{1}{2}}, \end{aligned}$$

(5.6)

where $C_{r}$ and $C$ are defined in Theorem 1.3.

Proof

For any fixed $r\ge 1$, let $Z'_{n}=Z_{rn}$, by Lemma 3.1 one has

$$\begin{aligned} Z'_{n+1}=Z_{rn+r}=\sum _{j=1}^{Z_{rn}}X_{rn,r}^{(j)} +Y_{rn,r}=\sum _{j=1}^{Z'_{n}}X_{rn,r}^{(j)}+Y_{rn,r}, \end{aligned}$$

and for any $n\ge 0$, $\{X_{rn,r}^{(j)},j\ge 1\}$ are i.i.d. By the proof of Lemma 3.1 one may see that $\{X_{rn,r}^{(j)},j\ge 1\}$ are independent of $\{Z'_{1},Z'_{2},\cdots ,Z'_{n}\}$ and $Y_{rn,r}$, $\{Y_{rn,r},n\ge 0\}$ are independent and for any $n\ge 1$, $Y_{rn,r}$ is independent of $Z'_{n}$, so $\{Z_{rn}; n\ge 0\}$ is in fact an IGWVE.

(5.6) is derived from Theorem 1.4. $\square$

Proof of Theorem 1.5

We only need to prove the $\limsup$ part because of symmetry. Note that (1.2) and (1.4) are satisfied by (1.9). According to Theorem 1.4 and the assumption that $\sum _{n=0}^{\infty }\left[ E \left( Z_{n}^{-\frac{\delta }{2}}\right) \right] ^{\frac{1}{2}}<\infty$ one has

$$\begin{aligned} \sum _{n=1}^{\infty }\sup _{x}\left| Q_{n,r}(x)-\Phi (x)\right| <\infty . \end{aligned}$$

(5.7)

By Lemma 5.1, one has that for any $\varepsilon >0$,

$$\begin{aligned} \sum _{n=2}^{\infty }\left[ 1-\Phi ((1+\varepsilon )(2\log n)^{\frac{1}{2}})\right] <\infty , \end{aligned}$$

so applying (5.7) one obtains

$$\begin{aligned} \sum _{n=2}^{\infty }\left[ 1-Q_{n,r}\left( (1+\varepsilon )(2\log n)^{\frac{1}{2}} \right) \right] <\infty . \end{aligned}$$

(5.8)

Using (5.8) and Borel-Cantelli Lemma one has

$$\begin{aligned} \limsup _{n\rightarrow \infty }\frac{Z_{n+r}-m_{n,r}Z_{n}}{(2\sigma _{n,r}^{2}Z_{n}\log n)^{\frac{1}{2}}}\le 1~~a.s. \end{aligned}$$

(5.9)

and

$$\begin{aligned} \liminf _{n\rightarrow \infty }\frac{Z_{n+r}-m_{n,r}Z_{n}}{(2\sigma _{n,r}^{2}Z_{n}\log n)^{\frac{1}{2}}}\ge -1~~a.s.. \end{aligned}$$

(5.10)

Fix $r=1$. For any $0<\varepsilon <1, n\ge 2$, define

$$\begin{aligned} A_{n}=\left\{ Z_{n}-\mu _{n-1}Z_{n-1}>(1-\varepsilon ) \delta _{n-1}(2Z_{n-1}\log (n-1))^{\frac{1}{2}}\right\} , \end{aligned}$$

we know that $A_{n}\in \sigma (Z_{1},Z_{2},\cdots ,Z_{n})$. Observe that

$$\begin{aligned} P(A_{n+1}|Z_{1},Z_{2},\cdots ,Z_{n})&= \!P\left( \sum _{i=1}^{Z_{n}} (\xi _{n,i}\!-\!\mu _{n})\!+\!Y_{n+1}\!>\!(1\!-\!\varepsilon )\delta _{n}(2Z_{n}\log n)^{\frac{1}{2}}|Z_{n}\right) \nonumber \\&\ge P\left( \sum _{i=1}^{Z_{n}}(\xi _{n,i}-\mu _{n})>(1-\varepsilon ) \delta _{n}(2Z_{n}\log n)^{\frac{1}{2}}|Z_{n}\right) \nonumber \\&\ge 1-\Phi \left( (1-\varepsilon )(2\log n)^{\frac{1}{2}}\right) -CZ_{n}^{-\frac{\delta }{2}}, \end{aligned}$$

(5.11)

where the last inequality is derived from Lemma 4.3 when $N_{n}$ is a constant. By Lemma 5.1,

$$\begin{aligned} \sum _{n=2}^{\infty }\left[ 1-\Phi ((1-\varepsilon )(2\log n)^{\frac{1}{2}})\right] =\infty . \end{aligned}$$

(5.12)

According to the assumption that $\sum _{n=0}^{\infty }\left[ E \left( Z_{n}^{-\frac{\delta }{2}}\right) \right] ^{\frac{1}{2}}<\infty$, (5.11) and (5.12) one has

$$\begin{aligned} \sum _{n=1}^{\infty }P(A_{n+1}|Z_{1},Z_{2},\cdots ,Z_{n})=\infty , \end{aligned}$$

then by the extension of Borel-Cantelli lemma ([16] Corollary 7.20 ) one has $P(A_{n}~i.o.)=1$, which implies

$$\begin{aligned} \limsup _{n\rightarrow \infty }\frac{Z_{n+1}-\mu _{n}Z_{n}}{(2\delta _{n}^{2}Z_{n}\log n)^{\frac{1}{2}}}\ge 1~~a.s.. \end{aligned}$$

(5.13)

Given $r>1$, we use $Z'_{n}$ to denote $Z_{rn}$. Then by Lemma 5.2 one has that $\{Z'_{n},n\ge 0\}$ is an IGWVE. Applying the method for the proof of the case $r=1$ to the process $\{Z'_{n},n\ge 0\}$,

$$\begin{aligned} \limsup _{n\rightarrow \infty }\frac{Z'_{n+1}-m_{rn,r}Z'_{n}}{(2\sigma _{rn,r}^{2}Z'_{n}\log n)^{\frac{1}{2}}}\ge 1~~a.s., \end{aligned}$$

which means

$$\begin{aligned} \limsup _{n\rightarrow \infty }\frac{Z_{rn+r}-m_{rn,r}Z_{rn}}{(2\sigma _{rn,r}^{2}Z_{rn}\log rn)^{\frac{1}{2}}}\ge 1~~a.s.. \end{aligned}$$

But for any fixed $r>1,$

$$\begin{aligned} \limsup _{n\rightarrow \infty }\frac{Z_{n+r}-m_{n,r} Z_{n}}{(2\sigma _{n,r}^{2}Z_{n}\log n)^{\frac{1}{2}}} \ge \limsup _{n\rightarrow \infty }\frac{Z_{rn+r}-m_{rn,r} Z_{rn}}{(2\sigma _{rn,r}^{2}Z_{rn}\log rn)^{\frac{1}{2}}}\ge 1~~a.s.. \end{aligned}$$

(5.14)

Finally, we derive (1.10) from (5.9) and (5.14).

According to Theorem 1.4 one has

$$\begin{aligned} \sum _{n=1}^{\infty }\sup _{x}\left| Q_{n}(x)-\Phi (x)\right| <\infty , \end{aligned}$$

(5.15)

but Lemma 5.1 tells us for any $\varepsilon >0$

$$\begin{aligned} \sum _{n=2}^{\infty }\left[ 1-\Phi \left( (1+\varepsilon )(2\log n)^{\frac{1}{2}}\right) \right] <\infty , \end{aligned}$$

so by (5.15) one has

$$\begin{aligned} \sum _{n=2}^{\infty }\left[ 1-Q_{n}\left( (1+\delta )(2\log n)^{\frac{1}{2}} \right) \right] <\infty . \end{aligned}$$

Thus by the Borel-Cantelli Lemma,

$$\begin{aligned}&\limsup _{n\rightarrow \infty }\frac{Z_{n}-m_{n}W}{(2\sigma _{n}^{2}Z_{n}\log n)^{\frac{1}{2}}}\le 1 ~~~~a.s.,\end{aligned}$$

(5.16)

$$\begin{aligned}&\liminf _{n\rightarrow \infty }\frac{Z_{n}-m_{n}W}{(2\sigma _{n}^{2}Z_{n}\log n)^{\frac{1}{2}}}\ge -1 ~~~~a.s.. \end{aligned}$$

(5.17)

As for the lower bound, first note that

$$\begin{aligned} \liminf _{n\rightarrow \infty }\frac{m_{n}W-Z_{n}}{(2\sigma _{n}^{2}Z_{n}\log n)^{\frac{1}{2}}}&\le \liminf _{n\rightarrow \infty }\frac{m_{n,r}^{-1} Z_{n+r}-Z_{n}}{(2\sigma _{n}^{2}Z_{n}\log n)^{\frac{1}{2}}}\nonumber \\&-\liminf _{n\rightarrow \infty }\frac{m_{n,r}^{-1}Z_{n+r}-m_{n}W}{(2\sigma _{n}^{2}Z_{n}\log n)^{\frac{1}{2}}}=:I_{1}-I_{2}, \end{aligned}$$

(5.18)

but by (1.10) we have

$$\begin{aligned} I_{1}\le -\inf _{n}\frac{\sigma _{n,r}}{\sigma _{n}m_{n,r}}~~~a.s., \end{aligned}$$

(5.19)

recalling the assumptions that for any $k\ge 1$, $\alpha \le \mu _{k}\le \beta , ~~\gamma ^{2}\le \delta _{k}^{2}\le \tau ^{2}$ and the definition of $\sigma _{n,r}$ and $\sigma _{n}$ one derives

$$\begin{aligned} 1-K_{2}\beta ^{-r}\ge \frac{\sigma _{n,r}}{\sigma _{n}m_{n,r}}=1- \frac{\sum _{j=n+r}^{\infty }\frac{\delta _{j}^{2}}{\mu _{j}^{2}m_{n,j-n}}}{ \sum _{j=n}^{\infty }\frac{\delta _{j}^{2}}{\mu _{j}^{2}m_{n,j-n}}}\ge 1-K_{1}\alpha ^{-r}, \end{aligned}$$

(5.20)

where $K_1$ and $K_2$ are two positive constants. Now, we consider $I_{2}$. One sees that

$$\begin{aligned} \frac{m_{n,r}^{-1}Z_{n+r}-m_{n}W}{(2\sigma _{n}^{2}Z_{n}\log n)^{\frac{1}{2}}}&= \frac{Z_{n+r}-m_{n+r}W}{(2\sigma _{n+r}^{2}Z_{n+r}\log (n+r))^{\frac{1}{2}}}\nonumber \\&\cdot \left[ \frac{\frac{Z_{n+r}}{m_{n+r}}}{\frac{Z_{n}}{m_{n}}}\right] ^{\frac{1}{2}} \cdot \left[ \frac{\sigma _{n+r}^{2}}{m_{n,r}\sigma _{n}^{2}}\right] ^{\frac{1}{2}}, \end{aligned}$$

(5.21)

so according to (5.21) we have

$$\begin{aligned} I_{2}\ge -\sup _{n}\left[ \frac{\sigma _{n+r}^{2}}{m_{n,r} \sigma _{n}^{2}}\right] ^{\frac{1}{2}}~~a.s., \end{aligned}$$

(5.22)

but $\sup _{n}\frac{1}{m_{n,r}}\le \alpha ^{-r}$ and

$$\begin{aligned} \sup _{n}\frac{\sigma _{n+r}^{2}}{\sigma _{n}^{2}}\le \frac{\sum _{j=n+r}^{\infty }\frac{\tau ^{2}}{\alpha ^{j-n-r+2}}}{ \sum _{j=n}^{\infty }\frac{\gamma ^{2}}{\beta ^{j-n+2}}}\le \frac{\tau ^{2}}{\gamma ^{2}}=:K_{3}. \end{aligned}$$

(5.23)

Since $r$ is arbitrary, by (5.18)–(5.23) one has

$$\begin{aligned} \liminf _{n\rightarrow \infty }\frac{m_{n}W-Z_{n}}{(2\sigma _{n}^{2}Z_{n}\log n)^{\frac{1}{2}}}\le -1~~ a.s., \end{aligned}$$

which implies

$$\begin{aligned} \limsup _{n\rightarrow \infty }\frac{Z_{n}-m_{n}W}{(2\sigma _{n}^{2}Z_{n}\log n)^{\frac{1}{2}}}\ge 1 ~~~~a.s.. \end{aligned}$$

$\square$