南晓杰
中国科学院大学数学科学学院, 北京 100049
摘要: 借助Berry-Esseen引理和Asmussen对条件Borel-Cantelli 引理的重要推广, 在变化环境中上临界分枝过程的每一代每一个个体的后代个体总数的 2 阶矩有一致上下界的情况下, 得到变化环境中分枝过程的重对数律, 从而改进了在相应的2+δ阶矩有限条件下的证明.
关键词: 变化环境分枝过程重对数律
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 resultLet 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 lemmasLemma 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 1.1Proof 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.
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