The law of the iterated logarithm(abbr.LIL) of the classical Galton-Watson process was firstly proved by Heyde^[1] under the condition that the 2+δ moment of the process is finite. In the same year,Heyde and Leslie^[2] again obtained the LIL under the condition that the second moment is finite. Later,Asmussen^[3] gave another proof via a very delicate truncation procedure and Kronecker lemma. The proof of Huggins^[4] is based on the Skorohod embedding techniques and new properties of Brownian motion and stopping times.

Gao^[5] proved the LIL of the super-critical Galton-Watson processes in varying environment satisfied that there is a uniform upper bound for the 2+δ moment of the number of the offspring of each individual of each generation. In addition,the author pointed out a mistake in the proof of Theorem 1 in Heyde and Leslie^[2]. Enlightened by the proof of the LIL of the classical Galton-Watson process in Asmussen^[3],we obtain the LIL of the super-critical Galton-Watson processes in varying environment under the condition that the second moment has a uniform upper bound and a uniform lower bound.

1 Main result

Let Z₀≡1 and for all n≥1,define

${{Z}_{n+1}}=\left\{ \begin{array}{*{35}{l}} \sum\limits_{j=1}^{{{Z}_{n}}}{{{X}_{n,j}}}, & \text{if }{{Z}_{n}}\ne 0; \\ 0, & \text{if }{{Z}_{n}}=0, \\ \end{array} \right.$

where {X_n,j;n≥0,j≥1} are independent and for each n≥0,{X_n,j,j≥1} have the same distribution {p_n(k),k∈N }. p_n(k) denotes the probability of k offspring produced by an individual of the n’th generation and N is the set of non-negative integers. Then {Z_n,n≥0} is said to be a Galton-Watson process in varying environment(GWVE).

Let the generating functions of Z_n and X_n,j are respectively f_n(s) and $\phi $_n(s),and let m_n and μ_n are respectively their expectations. Then

$\begin{align} & {{f}_{n}}\left( s \right)={{\phi }_{0}}\left( {{\phi }_{1}}\left( \cdots {{\phi }_{n-1}}\left( s \right)\cdots \right) \right), \\ & {{m}_{n}}=f{{'}_{n}}\left( 1 \right)=\prod\limits_{i=0}^{n-1}{\phi {{'}_{i}}\left( 1 \right)}=\prod\limits_{i=0}^{n-1}{{{\mu }_{i}}.} \\ \end{align}$

From now on,we always assume that ∏_k=n^n-1μ_k=1,0＜m_n＜∞,$\forall $n≥0. It is known that {W_n:=Z_n/m_n,n≥0} is a nonnegative martingale and there exists a nonnegative random variable W so that lim_n→∞W_n=W a.s.. Moreover,if sup_nE (W_n²)＜∞,then E W=1 and σ²:=Var(W)=∑_n=0^∞δ_n²/(μ_n²m_n)＜∞. These results can be found in Fearn^[6].

Lemma 1.1 (Decomposition Lemma 1)

Let {Z_n,n≥0} be a GWVE,then $\forall $n≥0,r≥1 we have

${{Z}_{n+r}}-{{m}_{n,r}}{{Z}_{n}}=\left\{ \begin{array}{*{35}{l}} \sum\limits_{j=1}^{{{Z}_{n}}}{\left( Z_{n,r}^{\left( j \right)}-{{m}_{n,r}} \right)}, & \text{if }{{Z}_{n}}\ne 0; \\ 0, & \text{if }{{Z}_{n}}=0, \\ \end{array} \right.$

where Z_n,r^(j) represents the number of r’th generation offspring of the j’th of the Z_n individuals of the n’th generation,and {Z_n,r^(j),j≥1} are independent and identically distributed and independent of Z_n. Furthermore,

$\begin{align} & {{m}_{n,r}}:=\text{E}\left( Z_{n,r}^{\left( j \right)} \right)=\prod\limits_{j=n}^{n+r-1}{{{\mu }_{j}},} \\ & \sigma _{n,r}^{2}:=\text{Var}\left( Z_{n,r}^{\left( j \right)} \right)={{\left( {{m}_{n,r}} \right)}^{2}}\sum\limits_{j=n}^{n+r-1}{\frac{\delta _{j}^{2}}{\mu _{j}^{2}{{m}_{n,j-n}}}}. \\ \end{align}$

Proof See Ref.^[5].

Remark 1.1 Z_n,1^(j)=X_n,j,m_0,r=m_r,m_n,1=μ_n,m_n,0=1,σ_0,r²=Var(Z_r)and σ_n,1²=δ_n².

Lemma 1.2(Decomposition Lemma 2)

Let {Z_n,n≥0} be a GWVE,then $\forall $n≥0

${{m}_{n}}W-{{Z}_{n}}=\left\{ \begin{array}{*{35}{l}} \sum\limits_{j=1}^{{{Z}_{n}}}{\left( W_{n}^{\left( j \right)}-1 \right)}, & \text{if }{{Z}_{n}}\ne 0; \\ 0, & \text{if }{{Z}_{n}}=0, \\ \end{array} \right.$

where {W_n^(j),j≥1} are independent and identically distributed and independent of Z_n. If

$\sum\limits_{j=n}^{\infty }{\frac{\delta _{j}^{2}}{\mu _{j}^{2}\prod\nolimits_{k=n}^{j-1}{{{\mu }_{k}}}}}＜\infty ,\forall n\ge 0,$

(1)

then

$\text{E}\left( W_{n}^{\left( j \right)} \right)=1\text{ and }\sigma _{j}^{2}:=\text{Var}\left( W_{n}^{\left( j \right)} \right)=\sum\limits_{j=n}^{\infty }{\frac{\delta _{j}^{2}}{\mu _{j}^{2}{{m}_{n,j-n}}}}.$

Proof See Ref.^[5].

Remark 1.2 W₀⁽¹⁾=W,σ₀²=σ²=Var(W)=$\sum\limits_{j=0}^{\infty }{\delta _{j}^{2}/\left( \mu _{j}^{2}{{m}_{j}} \right)}$.

Now assume that there exist four constants α,β,τ,γ with β>α>1 and τ>γ>0 such that $\forall $n≥0

$\alpha \le {{\mu }_{n}}\le \beta ,{{\gamma }^{2}}\le \delta _{n}^{2}\le {{\tau }^{2}},$

(2)

$\sum\limits_{n=0}^{\infty }{\int_{\left| y \right|>{{\zeta }^{n}}}{{{y}^{2}}\text{d}{{F}_{n}}\left( y \right)}}＜\infty ,$

(3)

$\sum\limits_{n=0}^{\infty }{\int_{\left| y \right|>{{\zeta }^{n}}}{{{y}^{2}}\text{d}{{G}_{n}}\left( y \right)}}＜\infty ,$

(4)

where 1＜ζ＜α^1/4,and F_n is the distribution of Z_n,r^(j)-m_n,r in Decomposition Lemma 1,and G_n is the distribution of W_n^(j)-1 in Decomposition Lemma 2.

For any given r≥1,define

Y_n,j:=Z_n,r^(j)-m_n,r,

Y′_n,j:=Y_n,jI(|Y_n,j|≤$\sqrt{{{m}_{n}}}$),

V_n,j:=W_n^(j)-1,

V′_n,j:=V_n,jI(|V_n,j|≤$\sqrt{{{m}_{n}}}$).

Theorem 1.1 Let {Z_n,n≥0} be a GWVE. Suppose that p_n(0)=0,$\forall $n≥0. If the conditions (2),(3),and (4) are satisfied,and

$\text{Var}(Y{{\prime }_{n,j}})/\text{Var}({{Y}_{n,j}})\to 1,\text{as }n\to \infty ,$

(5)

$\text{Var}(V{{\prime }_{n,j}})/\text{Var}({{V}_{n,j}})\to 1,\text{as }n\to \infty ,$

(6)

then for all r≥1,with probability one we have

$\underset{n\to \infty }{\mathop{\lim \sup }}\,(\underset{n\to \infty }{\mathop{\text{lim }\!\!~\!\!\text{ inf}}}\,)\frac{{{Z}_{n+r}}-{{m}_{n,r}}{{Z}_{n}}}{{{(2\sigma _{n,r}^{2}{{Z}_{n}}\text{log}n)}^{1/2}}}=1\left( -1 \right),$

(7)

$\underset{n\to \infty }{\mathop{\lim \sup }}\,(\underset{n\to \infty }{\mathop{\text{lim }\!\!~\!\!\text{ inf}}}\,)\frac{{{m}_{n}}W-{{Z}_{n}}}{{{(2\sigma _{n}^{2}{{Z}_{n}}\text{log}n)}^{1/2}}}=1\left( -1 \right).$

(8)

Remark 1.3 Since p_n(0)>0,$\forall $n≥0,one has W>0 a.s.,hence Z_n=O(m_n)a.s..

Remark 1.4 We can obtain αⁿ≤m_n≤βⁿ and Eq.(1) from the condition (2). According Remark 1.3,we know that log Z_n-nlogm→log W a.s., which means loglog Z_n/logn→1a.s., so logn can be substituted by log log Z_n in Eq.(7) and Eq.(8).

$\begin{align} & \int_{\left| y \right|>{{\zeta }^{n}}}{{{\left| y \right|}^{2}}\text{d}{{F}_{n}}\left( y \right)}\le \frac{1}{{{\left( \log {{\zeta }^{n}} \right)}^{1+\delta }}}\int_{\left| y \right|>{{\zeta }^{n}}}{{{\left| y \right|}^{2}}{{\left( \log \left| y \right| \right)}^{\text{1+}\delta }}\text{d}{{F}_{n}}\left( y \right)}, \\ & \int_{\left| y \right|>{{\zeta }^{n}}}{{{\left| y \right|}^{2}}\text{d}{{G}_{n}}\left( y \right)}\le \frac{1}{{{\left( \log {{\zeta }^{n}} \right)}^{1+\delta }}}\int_{\left| y \right|>{{\zeta }^{n}}}{{{\left| y \right|}^{2}}{{\left( \log \left| y \right| \right)}^{\text{1+}\delta }}\text{d}{{G}_{n}}\left( y \right)}, \\ \end{align}$

where 0＜δ＜1,we can get the conditions (3) and (4) under the following conditions (9) and (10) are satisfied:

$\underset{n\ge 0}{\mathop{\text{sup}}}\,\int_{y\in R}{{{\left| y \right|}^{2}}{{(log\left| y \right|)}^{1+\delta }}\text{d}{{F}_{n}}\left( y \right)＜\infty },$

(9)

$\underset{n\ge 0}{\mathop{\text{sup}}}\,\int_{y\in R}{{{\left| y \right|}^{2}}{{(log\left| y \right|)}^{1+\delta }}\text{d}{{G}_{n}}\left( y \right)＜\infty },$

(10)

However (9) and (10) are weaker than (1.14) in Ref.^[5].

Remark 1.6 The condition (5) holds naturally for a classical super-critical Galton-Watson branching process {Z_n,n≥0} with E(Z₁log Z₁)＜∞. Moreover,if there exists a random variable Y∈L²(Ω,F , P) so that |Y_n,1|≤Y,then Eq.(5) can be deduced. Since ∑_nP(|Y_n,1|>$\sqrt{{{m}_{n}}}$)＜∞,we almost surely have

$Y{{\prime }_{n,1}}-{{Y}_{n,1}}\to 0~\text{and}~{{(Y{{\prime }_{n,1}})}^{2}}-{{({{Y}_{n,1}})}^{2}}\to 0.$

By the dominated convergence theorem,we have

$\text{Var}(Y{{\prime }_{n,1}})-\text{Var}({{Y}_{n,1}})=\text{Var}(Y{{\prime }_{n,j}})-\text{Var}({{Y}_{n,j}})\to 0~\text{a}\text{.s}.$

hence the condition (5) holds. For Eq.(6) we have similar results.

2 Basic lemmas

Lemma 2.1 Let {F_n,n≥0} be an increasing sequence of σ-algebras and {T_n,n≥0} a (not necessarily adapted) random variable sequence such that

$\sum\limits_{n=0}^{\infty }{{{\Delta }_{n}}:=}\sum\limits_{n=0}^{\infty }{\underset{y\in R}{\mathop{\sup }}\,\left| \text{P}\left( {{T}_{n}}\le y|{{F}_{n}} \right)-\Phi \left( y \right) \right|}＜\infty ,$

where Φ(y) is the distribution function of N(0,1). Then

$\underset{n\to \infty }{\mathop{\lim \sup }}\,\frac{{{T}_{n}}}{{{\left( 2\log n \right)}^{1/2}}}\le 1\text{a}\text{.s}.,$

with the inequality replaced by equality if T_n is measurable with respect to F_n+k for some 1≤k＜∞.

Proof See Ref.^[3].

Lemma 2.2 (Berry-Esseen Lemma)

Let{X_n,n≥1} be an independent and identically distributed random variable sequence such that E X_n=0,E X_n²=σ²>0 and E |X_n| ³＜∞. Denote S_n:=∑_k=1ⁿX_k. Then

$\underset{x\in \text{R}}{\mathop{\text{sup}}}\,\left| \text{P}\left( \frac{{{S}_{n}}}{\sigma \sqrt{n}}＜x \right)-\Phi \left( x \right) \right|\le A\frac{\text{E}{{\left| {{X}_{1}} \right|}^{3}}}{{{\sigma }^{3}}\sqrt{n}},$

where Φ(x) is the standard normal distribution and A is a positive constant that is called the Berry-Esseen constant.

Proof See Ref.^[7],P124.

Lemma 2.3 (Kronecker Lemma)

Let {b_n} be an increasing sequence of positive real numbers with b_n→∞,and let {x_n} be a sequence of real numbers with ∑_n=1^∞x_n=x(finite). Then

$\frac{1}{{{b}_{n}}}\sum\limits_{j=1}^{n}{{{b}_{j}}{{x}_{j}}}\to 0,\text{as }n\to \infty .$

Proof See Ref.^[7],P63.

3 Proof of Theorem 1.1

Proof Denote F₀:=σ(Z₀) and F_n:=σ{X_k,j;0≤k≤n-1,j≥1}. Then $\forall $n≥1,F_n is the σ-algebra generated by the individuals of previous n-1 generations. First prove Eq.(7). We only need to show

$\underset{n\to \infty }{\mathop{\lim ~\sup }}\,\frac{\overline{{{Z}_{n+r}}}-{{m}_{n,r}}\overline{{{Z}_{n}}}}{{{(2{{\sigma }^{2}}_{n,r}{{Z}_{n}}logn)}^{1/2}}}=1~a.s..$

(11)

In fact,if Eq.(11) is true,let Z_n=-Z_n,n≥0,then we have

$\underset{n\to \infty }{\mathop{\lim ~\sup }}\,\frac{{{Z}_{n+r}}-{{m}_{n,r}}{{Z}_{n}}}{{{(2{{\sigma }^{2}}_{n,r}{{Z}_{n}}logn)}^{1/2}}}=1~a.s.,$

(12)

which in fact is

$\underset{n\to \infty }{\mathop{\lim \inf }}\,\frac{{{Z}_{n+r}}-{{m}_{n,r}}{{Z}_{n}}}{{{\left( 2\sigma _{n,r}^{2}{{Z}_{n}}\log n \right)}^{1/2}}}=-1\text{ a}\text{.s}..$

Define

$\begin{align} & \widetilde{{{Y}_{n,j}}}:=Y{{'}_{n,j}}-\text{E}Y{{'}_{n,j}}, \\ & \widetilde{{{S}_{n}}}:=\sum\limits_{j=1}^{{{Z}_{n}}}{\widetilde{{{Y}_{n,j}}}}, \\ & {{\widetilde{{{\omega }_{n}}}}^{2}}:\text{=Var}\left( \widetilde{{{S}_{n}}}|{{F}_{n}} \right)={{Z}_{n}}\text{Var}\left( \widetilde{{{Y}_{n,j}}} \right), \\ & {{T}_{n}}:=\widetilde{{{S}_{n}}}/\widetilde{{{\omega }_{n}}}. \\ \end{align}$

By a standard moment inequality,

$\begin{align} & \text{E}\left( {{\left| \text{ }\widetilde{{{Y}_{n,j}}^{\prime }} \right|}^{3}} \right)\le \\ & \text{E}\left( {{\left| {{Y}^{\prime }}_{n,j} \right|}^{3}} \right)+3\text{E}\left( \left| {{Y}^{\prime }}_{n,j} \right| \right){{\left( \text{E}\left| {{Y}^{\prime }}_{n,j} \right| \right)}^{2}}+ \\ & 3\text{E}\left( \left| {{Y}^{\prime }}_{n,j} \right| \right)\text{E}\left( {{Y}^{'}}_{n,j}^{2} \right)+\text{E}\left( {{\left| {{Y}^{\prime }}_{n,j} \right|}^{3}} \right) \\ & \le 8\text{E}\left( {{\left| {{Y}^{\prime }}_{n,j} \right|}^{3}} \right)=8\int_{\left| y \right|\le \sqrt{{{m}_{n}}}}{{{\left| y \right|}^{\text{3}}}\text{d}{{F}_{n}}\left( y \right)}. \\ \end{align}$

Letting A be the Berry-Essen constant,by the Berry-Esseen Lemma we have

$\begin{align} & {{\Delta }_{n}}:=\underset{y\in \mathbb{R}}{\mathop{\text{sup}}}\,\left| \mathbb{P}({{T}_{n}}\le y \right|{{F}_{n}})-\Phi \left( y \right)| \\ & \le 8A\frac{{{Z}_{n}}}{\widetilde{{{\omega }_{n}}^{3}}}\int_{\left| y \right|\le {{m}_{n}}}{{{\left| y \right|}^{3}}}d{{F}_{n}}\left( y \right). \\ \end{align}$

(13)

By the condition (2) we can deduce that there exist a uniform upper and a uniform lower bound only dependent on r for σ_n,r². So there exist positive and finite constants C₁ and C₂ which are only dependent on r such that

${{C}_{1}}\le \underset{n\to \infty }{\mathop{\lim ~ \text{inf}}}\,\frac{{{Z}_{n}}}{\widetilde{{{\omega }_{n}}^{2}}}\le \underset{n\to \infty }{\mathop{\lim ~\sup }}\,\frac{{{Z}_{n}}}{\widetilde{{{\omega }_{n}}^{2}}}\le {{C}_{2}}.$

(14)

From Remark 1.3 we know that Z_n=O(m_n)a.s.,hence ${{\widetilde{{{\omega }_{n}}}}^{2}}$=O(m_n)a.s..Combining with Eq.(13) and Eq.(14),we have

$\begin{align} & \sum\limits_{n=0}^{\infty }{\frac{1}{\sqrt{{{m}_{n}}}}}{{\int }_{\left| y \right|\le \sqrt{{{m}_{n}}}}}{{\left| y \right|}^{3}}d{{F}_{n}}\left( y \right) \\ & =\sum\limits_{n=0}^{\infty }{\frac{1}{\sqrt{{{m}_{n}}}}}\left( \sum\limits_{j=1}^{{{\xi }^{n}}}{{{\int }_{j-1 ＜\left| y \right|\le j}}}{{\left| y \right|}^{3}}d{{F}_{n}}{{\left( y \right)}^{n}} \right)+ \\ & \sum\limits_{n=0}^{\infty }{\frac{1}{\sqrt{{{m}_{n}}}}}\left( \sum\limits_{j={{\zeta }^{n}}+1}^{\sqrt{{{m}_{n}}}}{{{\int }_{j-1＜\left| y \right|\le j}}}{{\left| y \right|}^{3}}d{{F}_{n}}\left( y \right) \right) \\ & \le \sum\limits_{n=0}^{\infty }{\sum\limits_{j=1}^{{{\xi }^{n}}}{{{\int }_{j-1＜\left| y \right|\le j}}}}{{\left| y \right|}^{2}}d{{F}_{n}}\left( y \right)+ \\ & \sum\limits_{n=0}^{\infty }{\sum\limits_{j={{\zeta }^{n}}+1}^{\sqrt{{{m}_{n}}}}{{{\int }_{j-1＜\left| y \right|\le j}}}}{{\left| y \right|}^{2}}d{{F}_{n}}\left( y \right)) \\ & \le \sum\limits_{n=0}^{\infty }{{{C}_{3}}}{{({{\zeta }^{2}}/\sqrt{\alpha })}^{n}}+\sum\limits_{n=0}^{\infty }{{{C}_{3}}}{{\int }_{\left| y \right|>{{\zeta }^{n}}}}{{\left| y \right|}^{2}}d{{F}_{n}}\left( y \right), \\ \end{align}$

(15)

where C₃>0 is a constant. By using the condition (3),Eq.(13) and Eq.(15) we have ∑△_n＜∞ a.s.. Again applying Lemma 2.1 one eventually has

$\underset{n\to \infty }{\mathop{\lim \sup }}\,\frac{{{T}_{n}}}{{{\left( 2\log n \right)}^{1/2}}}\le 1,\text{a}\text{.s}..$

In addition,since T_n is measurable with respect to F_n+r,the above inequality should be replaced by equality.

${{S}_{n}}=\sum\limits_{j=1}^{{{Z}_{n}}}{{{Y}_{n,j}}}=\sum\limits_{j=1}^{{{Z}_{n}}}{\left\{ \widetilde{{{Y}_{n,j}}}+{{Y}_{n,j}}-Y{{'}_{n,j}}+\text{E}Y{{'}_{n,j}} \right\}}.$

Thus it suffices to verify

$\underset{n\to \infty }{\mathop{\lim \sup }}\,\frac{{{S}_{n}}}{{{\left( 2\sigma _{n,r}^{2}{{Z}_{n}}\log n \right)}^{1/2}}}=\underset{n\to \infty }{\mathop{\lim \sup }}\,\frac{{{T}_{n}}}{{{\left( 2\log n \right)}^{1/2}}}.$

Noting that

$\begin{align} & \frac{{{S}_{n}}}{{{\left( 2\sigma _{n,r}^{2}{{Z}_{n}}\log n \right)}^{1/2}}}=\frac{{{T}_{n}}}{{{\left( 2\log n \right)}^{1/2}}}{{\left( \frac{\text{Var}(Y{{\prime }_{n,j}})}{\sigma _{n,r}^{2}} \right)}^{1/2}}+ \\ & \frac{\sum\nolimits_{j=1}^{{{Z}_{n}}}{\left\{ {{Y}_{n,j}}-Y{{\prime }_{n,j}} \right\}}}{{{\left( 2\sigma _{n,r}^{2}{{Z}_{n}}\log n \right)}^{1/2}}}+\frac{\sum\nolimits_{j=1}^{{{Z}_{n}}}{\text{E}Y{{\prime }_{n,j}}}}{{{\left( 2\sigma _{n,r}^{2}{{Z}_{n}}\log n \right)}^{1/2}}}, \\ \end{align}$

it suffices to verify that

$Var(Y{{\prime }_{n,j}})/{{\sigma }^{2}}_{n,r}\to 1,$

(16)

$\frac{\sum _{j=1}^{{{Z}_{n}}}\{{{Y}_{n,j}}-Y{{\prime }_{n,j}}\}}{{{({{Z}_{n}}logn)}^{1/2}}}\to 0,$

(17)

$\frac{\sum _{j=1}^{{{Z}_{n}}}\mathbb{E}Y{{\prime }_{n,j}}}{{{({{Z}_{n}}logn)}^{1/2}}}\to 0.$

(18)

By the condition (5) we know that Eq.(16) holds. For Eq.(17) and Eq.(18),by Kronecer Lemma it only needs to prove that

$\sum\limits_{n=0}^{\infty }{\frac{1}{\sqrt{{{m}_{n}}logn}}}\sum\limits_{j=1}^{{{Z}_{n}}}{\left| {{Y}_{n,j}}-Y{{\prime }_{n,j}} \right|}＜\infty ,$

(19)

$\sum\limits_{n=0}^{\infty }{\frac{1}{\sqrt{{{m}_{n}}logn}}}\sum\limits_{j=1}^{{{Z}_{n}}}{\left| \mathbb{E}Y{{\prime }_{n,j}} \right|}＜\infty .$

(20)

Since

$\begin{align} & \left| \text{E}Y{{'}_{n,j}} \right|=\left| \text{E}\left( Y{{'}_{n,j}}-{{Y}_{n,j}} \right) \right|\le \text{E}\left| Y{{'}_{n,j}}-{{Y}_{n,j}} \right| \\ & =\text{E}\left| {{Y}_{n,j}} \right|I\left( \left| {{Y}_{n,j}} \right| \right)>\sqrt{{{m}_{n}}}=\int_{\left| y \right|\sqrt{{{m}_{n}}}}{\left| y \right|}\text{d}{{F}_{n}}\left( y \right), \\ \end{align}$

and noting that Z_n=O(m_n)a.s., it suffices for Eq.(19) and Eq.(20) that

$\sum\limits_{n=0}^{\infty }{\frac{{{m}_{n}}}{\sqrt{{{m}_{n}}logn}}}{{\int }_{\left| y \right|>\sqrt{{{m}_{n}}}}}\left| y \right|d{{F}_{n}}\left( y \right)＜\infty ,$

(21)

(for the first,taking the mean). And Eq.(21) certainly holds since even

$\begin{align} & \sum\limits_{n=0}^{\infty }{\sqrt{{{m}_{n}}}}\int_{\left| y \right|\sqrt{{{m}_{n}}}}{\left| y \right|}\text{d}{{F}_{n}}\left( y \right)\le \sum\limits_{n=0}^{\infty }{\int_{\left| y \right|\sqrt{{{m}_{n}}}}{{{y}^{2}}}\text{d}{{F}_{n}}\left( y \right)} \\ & \le \sum\limits_{n=0}^{\infty }{\int_{\left| y \right|>{{\xi }^{2n}}}{{{\left| y \right|}^{2}}}\text{d}{{F}_{n}}\left( y \right)}＜\infty . \\ \end{align}$

Therefore,Eq.(17) and Eq.(18) hold.

In order to prove Eq.(8),recall V_n,j:=W_n^(j)-1. Repeating the proof of Eq.(11) and noting that there doesn’t exist 1≤r＜∞ so that T_n is measurable with respect to F_n+r,we have

$\underset{n\to \infty }{\mathop{\lim ~\sup }}\,$

(22)

Note that

$\begin{align} & \frac{{{m}_{n}}\left( W-{{W}_{n}} \right)}{{{\left( 2\sigma _{n}^{2}{{Z}_{n}}\log n \right)}^{1/2}}}=\frac{{{m}_{n+k}}\left( W-{{W}_{n+k}} \right)}{2\sigma _{n+k}^{2}{{Z}_{n+k}}\log \left( n+k \right)}\cdot \\ & {{\left( \frac{\sigma _{n+k}^{2}{{Z}_{n+k}}\log \left( n+k \right)}{\sigma _{n}^{2}{{Z}_{n}}\log n} \right)}^{1/2}}\frac{1}{{{m}_{n+k}}}+ \\ & \frac{{{m}_{n+k}}\left( {{W}_{n+k}}-{{W}_{n}} \right)}{{{\left( 2\sigma _{n,k}^{2}{{Z}_{n}}\log n \right)}^{1/2}}}{{\left( \frac{\sigma _{n,k}^{2}}{\sigma _{n}^{2}} \right)}^{1/2}}\frac{1}{{{m}_{n+k}}}. \\ \end{align}$

So the lim sup part is at least

$\begin{align} & -\underset{n\to \infty }{\mathop{\lim }}\,{{\left( \frac{\sigma _{n+k}^{2}{{Z}_{n+k}}\log \left( n+k \right)}{\sigma _{n}^{2}{{Z}_{n}}\log n} \right)}^{1/2}}\frac{1}{{{m}_{n+k}}}+ \\ & \underset{n\to \infty }{\mathop{\lim }}\,{{\left( \frac{\sigma _{n+k}^{2}}{\sigma _{n}^{2}} \right)}^{1/2}}\frac{1}{{{m}_{n,k}}}. \\ \end{align}$

Again since the first item of the above expression is smaller than Cα^-k/2 and the second item

$\frac{\sigma _{n+k}^{2}}{\sigma _{n}^{2}}\frac{1}{{{\left( {{m}_{n,k}} \right)}^{2}}}=\left( \sum\limits_{j=n}^{n+k-1}{\frac{\delta _{j}^{2}}{\mu _{j}^{2}{{m}_{n,j-n}}}} \right)/\left( \sum\limits_{j=n}^{\infty }{\frac{\delta _{j}^{2}}{\mu _{j}^{2}{{m}_{n,j-n}}}} \right)=\frac{1}{1+{{b}_{n,k}}},$

where b_n,k＜Cα^-k and C>0 is a constant,the lim sup part of Eq.(8) is at least 1 as k→∞. The lim inf part of Eq.(8) can be proved in the same way as for Eq.(12).□

I am very grateful to my supervisor Prof. HU for intensive discussion with me and also benefit much from the communications with Gao Zhenlong,Liang Longyue,Zhang Zhiyang,and Zhao Rongjie.