Citation: Bernardino Romano, Andrea Agapito Ludovici, Francesco Zullo, Alessandro Marucci, Lorena Fiorini. Italian landscape macrosystem (ILM) from urban pressure to a National Wildway[J]. AIMS Environmental Science, 2020, 7(6): 505-525. doi: 10.3934/environsci.2020032
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Existing methods and algorithms appeared in some literatures assume that variables are independent, but it is not plausible. In many stochastic models and statistical applications, those variables involved are dependent. Hence, it is important and meaningful to extend the results of independent variables to dependent cases. One of these dependence structures is weakly dependent (i.e., $ {{\rho }^{*}} $-mixing or $ \tilde{\rho} $-mixing), which has attracted the concern by many researchers.
Definition 1.1. Let $ \left\{ {{X}_{n}}; n\ge 1 \right\} $ be a sequence of random variables defined on a probability space $ \left(\Omega, \mathcal{F}, P \right) $. For any $ S\subset \text{N = }\left\{ 1, 2, \ldots \right\} $, define $ {{\mathcal{F}}_{S}} = \sigma \left({{X}_{i}}, i\in S \right) $. The set $ {{L}_{2}}\left({{\mathcal{F}}_{S}} \right) $ is the class of all $ \mathcal{F} $-measureable random variables with the finite second moment. For some integer $ s\ge 1 $, denote the mixing coefficient by
| $ \begin{equation} {{\rho }^{*}}\left( s \right) = \sup \left\{ \rho \left( {{\mathcal{F}}_{S}}, {{\mathcal{F}}_{T}} \right):S, T\subset \text{N}, \text{dist}\left( S, T \right)\ge s \right\}, \end{equation} $ | (1.1) |
where
| $ \begin{equation} \rho \left( {{\mathcal{F}}_{S}}, {{\mathcal{F}}_{T}} \right) = \sup \left\{ \frac{\left| EXY-EXEY \right|}{\sqrt{\operatorname{Var}X}\cdot \sqrt{\operatorname{Var}Y}}:X\in {{L}_{2}}\left( {{\mathcal{F}}_{S}} \right), Y\in {{L}_{2}}\left( {{\mathcal{F}}_{T}} \right) \right\}. \end{equation} $ | (1.2) |
Noting that the above fact $ \text{dist}\left(S, T \right)\ge s $ denotes $ \text{dist}\left(S, T \right) = \inf \left\{ \left| i-j \right|:i\in S, j\in T \right\}\ge s $. Obviously, $ 0\le {{\rho }^{*}}\left(s+1 \right)\le {{\rho }^{*}}\left(s \right)\le 1 $ and $ {{\rho }^{*}}\left(0 \right) = 1 $. The sequence $ \left\{ {{X}_{n}}; n\ge 1 \right\} $ is called $ {{\rho }^{*}} $-mixing if there exists $ s\in \text{N} $ such that $ {{\rho }^{*}}\left(s \right) < 1 $. Clearly, if $ \left\{ {{X}_{n}}; n\ge 1 \right\} $ is a sequence of independent random variables, then $ {{\rho }^{*}}\left(s \right) = 0 $ for all $ s\ge 1 $.
$ {{\rho }^{*}} $-mixing seems similarly to another dependent structure: $ \rho $-mixing, but they are quite different from each other. $ {{\rho }^{*}} $-mixing is also a wide range class of dependent structures, which was firstly introduced to the limit theorems by Bradley [4]. From then on, many scholars investigated the limit theory for $ {{\rho }^{*}} $-mixing random variables, and a number of important applications for $ {{\rho }^{*}} $-mixing have been established. For more details, we refer to [12,16,18,19,21,23,24] among others.
The concept of complete convergence was firstly given by Hsu and Robbins[9] as follows: A sequence of random variables $ \left\{ {{X}_{n}}; n\ge 1 \right\} $ converges completely to a constant $ \lambda $ if $ \sum\limits_{n = 1}^{\infty }{P\left(\left| {{X}_{n}}-\lambda \right| > \varepsilon \right)} < \infty $ for all $ \varepsilon > 0 $. By the Borel-Cantelli lemma, the above result implies that $ {{X}_{n}}\to \lambda $ almost surely (a.s.). Thus, the complete convergence plays a crucial role in investigating the limit theory for summation of random variables as well as weighted sums.
Chow [8] introduced the following notion of complete moment convergence: Let $ \left\{ {{Z}_{n}}; n\ge 1 \right\} $ be a sequence of random variables, and $ {{a}_{n}} > 0 $, $ {{b}_{n}} > 0 $, $ q > 0 $. If $ \sum\limits_{n = 1}^{\infty }{{{a}_{n}}E\left(b_{n}^{-1}\left| {{Z}_{n}} \right|-\varepsilon \right)_{+}^{q}} < \infty $ for all $ \varepsilon \ge 0 $, then the sequence $ \left\{ {{Z}_{n}}; n\ge 1 \right\} $ is called to be the complete $ q $-th moment convergence. It will be shown that the complete moment convergence is the more general version of the complete convergence, and is also much stronger than the latter (see Remark 2.1).
According to the related statements of Rosalsky and Thành[14] as well as that of Thành[17], we recall the definition of stochastic domination as follows.
Definition 1.2. A sequence of random variables $ \left\{ {{X}_{n}}; n\ge 1 \right\} $ is said to be stochastically dominated by a random variable $ X $ if for all $ x\ge 0 $ and $ n\ge 1 $,
| $ \begin{equation*} {\mathop {\sup }\limits_{n \ge 1} }\, P\left( \left| {{X}_{n}} \right|\ge x \right)\le P\left( \left| X \right|\ge x \right). \end{equation*} $ |
The concept of stochastic domination is a slight generalization of identical distribution. It is clearly seen that stochastic dominance of $ \left\{ {{X}_{n}}; n\ge 1 \right\} $ by the random variable $ X $ implies $ E{{\left| {{X}_{n}} \right|}^{p}}\le E{{\left| X \right|}^{p}} $ if the $ p $-th moment of $ \left| X \right| $ exists, i.e. $ E{{\left| X \right|}^{p}} < \infty $.
As is known to us all, the weighted sums of random variables are used widely in some important linear statistics (such as least squares estimators, nonparametric regression function estimators and jackknife estimates). Based on this respect, many probability statisticians devote to investigate the probability limiting behaviors for weighted sums of random variables. For example, Bai and Cheng[3], Cai[5], Chen and Sung[6], Cheng et al.[7], Lang et al.[11], Peng et al.[13], Sung[15,16] and Wu[20] among others.
Recently, Li et al.[12] extended the corresponding result of Chen and Sung[6] from negatively associated random variables to $ {{\rho }^{*}} $-mixing cases by a total different method, and obtained the following theorem.
Theorem A. Let $ \left\{ X, {{X}_{n}}; n\ge 1 \right\} $ be a sequence of identically distributed $ {{\rho }^{*}} $-mixing random variables with $ E{{X}_{n}} = 0 $, and let $ \left\{ {{a}_{ni}}; 1\le i\le n, n\ge 1 \right\} $ be an array of real constants such that $ \sum\limits_{i = 1}^{n}{{{\left| {{a}_{ni}} \right|}^{\alpha }}} = O\left(n \right) $ for some $ 1 < \alpha \le 2 $. Set $ {{b}_{n}} = {{n}^{1/\alpha }}{{\left(\log n \right)}^{1/\gamma }} $ for $ 0 < \gamma < \alpha $. If $ E{{{\left| X \right|}^{\alpha }}}/{{{\left(\log \left(1+\left| X \right| \right) \right)}^{\alpha /\gamma -1}}}\; < \infty $, then
| $ \begin{equation} \sum\limits_{n = 1}^{\infty }{\frac{1}{n}}P\left( {\mathop {\max }\limits_{1 \le j \le n} }\, \left| \sum\limits_{i = 1}^{j}{{{a}_{ni}}{{X}_{i}}} \right| > \varepsilon {{b}_{n}} \right) < \infty \quad \text{for} \quad\forall \varepsilon > 0. \end{equation} $ | (1.3) |
In addition, Huang et al.[10] proved the following complete $ \alpha $-th moment convergence theorem for weighted sums of $ {{\rho }^{*}} $-mixing random variables under some moment conditions.
Theorem B. Let $ \left\{ {{X}_{n}}; n\ge 1 \right\} $ be a sequence of $ {{\rho }^{*}} $-mixing random variables, which is stochastically dominated by a random variable $ X $, let $ \left\{ {{a}_{ni}}; 1\le i\le n, n\ge 1 \right\} $ be an array of real constants such that $ \sum\limits_{i = 1}^{n}{{{\left| {{a}_{ni}} \right|}^{\alpha }}} = O\left(n \right) $ for some $ 0 < \alpha \le 2 $. Set $ {{b}_{n}} = {{n}^{1/\alpha }}{{\left(\log n \right)}^{1/\gamma }} $ for some $ \gamma > 0 $. Assume further that $ E{{X}_{n}} = 0 $ when $ 1 < \alpha \le 2 $. If
| $ \begin{equation} \begin{array}{ll} E{{|X|}^{\alpha }} < \infty, &\;{\rm{for}}\;\quad\alpha > \gamma, \\ E|X|^{\alpha}\log (1+|X|) < \infty, &\;{\rm{for}}\;\quad \alpha = \gamma, \\ E|X|^{\gamma} < \infty, &\;{\rm{for}}\;\quad \alpha < \gamma, \\ \end{array} \end{equation} $ | (1.4) |
then
| $ \begin{equation} \sum\limits_{n = 1}^{\infty }{\frac{1}{n}}E\left( \frac{1}{{{b}_{n}}}{\mathop {\max }\limits_{1 \le j \le n} }\, \left| \sum\limits_{i = 1}^{j}{{{a}_{ni}}{{X}_{i}}} \right|-\varepsilon \right)_{+}^{\alpha } < \infty \quad \text{ for } \forall \varepsilon > 0. \end{equation} $ | (1.5) |
It is interesting to find the optimal moment conditions for (1.5). Huang et al.[10] also posed a worth pondering problem whether the result (1.5) holds for the case $ \alpha > \gamma $ under the almost optimal moment condition $ E{{{\left| X \right|}^{\alpha }}}/{{{\left(\log \left(1+\left| X \right| \right) \right)}^{\alpha /\gamma -1}}}\; < \infty $?
Mainly inspired by the related results of Li et al.[12], Chen and Sung[6] and Huang et al.[10], the authors will further study the convergence rate for weighted sums of $ {{\rho }^{*}} $-mixing random variables without assumptions of identical distribution. Under the almost optimal moment condition $ E{{{\left| X \right|}^{\alpha }}}/{{{\left(\log \left(1+\left| X \right| \right) \right)}^{\alpha /\gamma -1}}}\; < \infty $ for $ 0 < \gamma < \alpha $ with $ 1 < \alpha \le 2 $, a version of the complete $ \alpha $-th moment convergence theorem for weighted sums of $ {{\rho }^{*}} $-mixing random variables is established. The main result not only improves the corresponding ones of Li et al.[12], Chen and Sung[6], but also partially settles the open problem posed by Huang et al.[10].
Now, we state the main result as follows. Some important auxiliary lemmas and the proof of the theorem will be detailed in the next section.
Theorem 1.1. Let $ \left\{ {{X}_{n}}; n\ge 1 \right\} $ be a sequence of $ {{\rho }^{*}} $-mixing random variables with $ E{{X}_{n}} = 0 $, which is stochastically dominated by a random variable $ X $, let $ \left\{ {{a}_{ni}}; 1\le i\le n, n\ge 1 \right\} $ be an array of real constants such that $ \sum\limits_{i = 1}^{n}{{{\left| {{a}_{ni}} \right|}^{\alpha }}} = O\left(n \right) $ for some $ 0 < \alpha \le 2 $. Set $ {{b}_{n}} = {{n}^{1/\alpha }}{{\left(\log n \right)}^{1/\gamma }} $ for $ \gamma > 0 $. If $ E{{{\left| X \right|}^{\alpha }}}/{{{\left(\log \left(1+\left| X \right| \right) \right)}^{\alpha /\gamma -1}}}\; < \infty $ for $ \alpha > \gamma $ with $ 1 < \alpha \le 2 $, then (1.5) holds.
Throughout this paper, let $ I\left(A \right) $ be the indicator function of the event $ A $ and $ I(A, B) = I(A\bigcap B) $. The symbol $ C $ always presents a positive constant, which may be different in various places, and $ {{a}_{n}} = O\left({{b}_{n}} \right) $ stands for $ {{a}_{n}}\le C{{b}_{n}} $.
To prove our main result of this paper, we need the following important lemmas.
Lemma 2.1. (Utev and Peligrad[18]) Let $ p\ge 2 $, $ \left\{ {{X}_{n}}; n\ge 1 \right\} $ be a sequence of $ {{\rho }^{*}} $-mixing random variables with $ E{{X}_{n}} = 0 $ and $ E{{\left| {{X}_{n}} \right|}^{p}} < \infty $ for all $ n\ge 1 $. Then there exists a positive constant $ C $ depending only on $ p $, $ s $ and $ {{\rho }^{*}}\left(s \right) $ such that
| $ \begin{equation} E\left( {\mathop {\max }\limits_{1 \le j \le n} }\, {{\left| \sum\limits_{i = 1}^{j}{{{X}_{i}}} \right|}^{p}} \right)\le C\left( \sum\limits_{i = 1}^{n}{E{{\left| {{X}_{i}} \right|}^{p}}}+{{\left( \sum\limits_{i = 1}^{n}{EX_{i}^{2}} \right)}^{p/2}} \right). \end{equation} $ | (2.1) |
In particular, if $ p = 2 $,
| $ \begin{equation} E\left( {\mathop {\max }\limits_{1 \le j \le n} }\, {{\left| \sum\limits_{i = 1}^{j}{{{X}_{i}}} \right|}^{2}} \right)\le C\sum\limits_{i = 1}^{n}{EX_{i}^{2}}. \end{equation} $ | (2.2) |
The following one is a basic property for stochastic domination. For the details, one refers to Adler and Rosalsky[1] and Adler et al.[2], or Wu[22]. In fact, we can remove the constant $ C $ in those of Adler and Rosalsky[1] and Adler et al.[2], or Wu[22], since it was proved in Reference [[14], Theorem 2.4] (or [[17], Corollary 2.3]) that this is indeed equivalent to $ C = 1 $.
Lemma 2.2. Let $ \left\{ {{X}_{n}}, n\ge 1 \right\} $ be a sequence of random variables which is stochastically dominated by a random variable $ X $. For all $ \beta > 0 $ and $ b > 0 $, the following statements hold:
| $ \begin{equation} E{{\left| {{X}_{n}} \right|}^{\beta }}I\left( \left| {{X}_{n}} \right|\le b \right)\le \left( E{{\left| X \right|}^{\beta }}I\left( \left| X \right|\le b \right)+{{b}^{\beta }}P\left( \left| X \right| > b \right) \right), \end{equation} $ | (2.3) |
| $ \begin{equation} E{{\left| {{X}_{n}} \right|}^{\beta }}I\left( \left| {{X}_{n}} \right| > b \right)\le E{{\left| X \right|}^{\beta }}I\left( \left| X \right| > b \right). \end{equation} $ | (2.4) |
Consequently, $ E{{\left| {{X}_{n}} \right|}^{\beta }}\le E{{\left| X \right|}^{\beta }} $.
Lemma 2.3. Under the conditions of Theorem 1.1, if $ E{{{\left| X \right|}^{\alpha }}}/{{{\left(\log \left(1+\left| X \right| \right) \right)}^{\alpha /\gamma -1}}}\; < \infty $ for $ 0 < \gamma < \alpha $ with $ 0 < \alpha \le 2 $, then
| $ \begin{equation} \sum\limits_{n = 1}^{\infty }{\frac{1}{n}\int_{1}^{\infty }{\sum\limits_{i = 1}^{n}{P\left( \left| {{a}_{ni}}{{X}_{i}} \right| > {{b}_{n}}{{t}^{1/\alpha }} \right)}dt}} < \infty. \end{equation} $ | (2.5) |
Proof. By Definition 1.2, noting that
| $ \begin{eqnarray} \sum\limits_{n = 1}^{\infty }{\frac{1}{n}\int_{1}^{\infty }{\sum\limits_{i = 1}^{n}{P\left( \left| {{a}_{ni}}{{X}_{i}} \right| > {{b}_{n}}{{t}^{1/\alpha }} \right)}dt}}&\le& \sum\limits_{n = 1}^{\infty }{\frac{1}{n}\int_{1}^{\infty }{\sum\limits_{i = 1}^{n}{P\left( \left| {{a}_{ni}}X \right| > {{b}_{n}}{{t}^{1/\alpha }} \right)}dt}} \\ &\le& \sum\limits_{n = 1}^{\infty }{\frac{1}{n}\int_{0}^{\infty }{\sum\limits_{i = 1}^{n}{P\left( \frac{{{\left| {{a}_{ni}}X \right|}^{\alpha }}}{b_{n}^{\alpha }} > t \right)}dt}} \\ &\le& \sum\limits_{n = 1}^{\infty }{{{n}^{-1}}b_{n}^{-\alpha }\sum\limits_{i = 1}^{n}{E{{\left| {{a}_{ni}}X \right|}^{\alpha }}}}. \end{eqnarray} $ | (2.6) |
It is easy to show that
| $ \begin{eqnarray} \sum\limits_{n = 1}^{\infty }{{{n}^{-1}}b_{n}^{-\alpha }\sum\limits_{i = 1}^{n}{{{\left| {{a}_{ni}} \right|}^{\alpha }}E{{\left| X \right|}^{\alpha }}I\left( \left| X \right|\le {{b}_{n}} \right)}}&\le& C\sum\limits_{n = 1}^{\infty }{b_{n}^{-\alpha }E{{\left| X \right|}^{\alpha }}I\left( \left| X \right|\le {{b}_{n}} \right)} \\ &\le& C\sum\limits_{n = 1}^{\infty }{b_{n}^{-\alpha }\sum\limits_{k = 1}^{n}{E{{\left| X \right|}^{\alpha }}I\left( {{b}_{k}} < \left| X \right|\le {{b}_{k+1}} \right)}} \\ &\le& C\sum\limits_{k = 1}^{\infty }{E{{\left| X \right|}^{\alpha }}I\left( {{b}_{k}} < \left| X \right|\le {{b}_{k+1}} \right){{\left( \log k \right)}^{1-\left( \alpha /\gamma \right)}}} \\ &\le& CE{{{\left| X \right|}^{\alpha }}}/{{{\left( \log \left( 1+\left| X \right| \right) \right)}^{\left( \alpha /\gamma \right)-1}}}\; < \infty, \end{eqnarray} $ | (2.7) |
and
| $ \begin{eqnarray} \sum\limits_{n = 1}^{\infty }{{{n}^{-1}}b_{n}^{-\alpha }\sum\limits_{i = 1}^{n}{{{\left| {{a}_{ni}} \right|}^{\alpha }}E{{\left| X \right|}^{\alpha }}I\left( \left| X \right| > {{b}_{n}} \right)}}&\le& C\sum\limits_{n = 1}^{\infty }{b_{n}^{-\alpha }E{{\left| X \right|}^{\alpha }}I\left( \left| X \right| > {{b}_{n}} \right)} \\ & = &C\sum\limits_{n = 1}^{\infty }{b_{n}^{-\alpha }\sum\limits_{j = n}^{\infty }{E{{\left| X \right|}^{\alpha }}I\left( {{b}_{j}} < \left| X \right|\le {{b}_{j+1}} \right)}} \\ & = &C\sum\limits_{j = 1}^{\infty }{E{{\left| X \right|}^{\alpha }}I\left( {{b}_{j}} < \left| X \right|\le {{b}_{j+1}} \right)\sum\limits_{n = 1}^{j}{{{n}^{-1}}{{\left( \log n \right)}^{-\alpha /\gamma }}}} \\ &\le& C\sum\limits_{j = 1}^{\infty }{{{\left( \log j \right)}^{1-\left( \alpha /\gamma \right)}}E{{\left| X \right|}^{\alpha }}I\left( {{b}_{j}} < \left| X \right|\le {{b}_{j+1}} \right)} \\ &\le& CE{{{\left| X \right|}^{\alpha }}}/{{{\left( \log \left( 1+\left| X \right| \right) \right)}^{\left( \alpha /\gamma \right)-1}}}\; < \infty. \end{eqnarray} $ | (2.8) |
Hence, (2.5) holds by (2.6)–(2.8).
Proof of Theorem 1.1. For any given $ \varepsilon > 0 $, observing that
| $ \begin{eqnarray} \sum\limits_{n = 1}^{\infty }{\frac{1}{n}}E\left( \frac{1}{{{b}_{n}}}{\mathop {\max }\limits_{1 \le j \le n} }\, \left| \sum\limits_{i = 1}^{j}{{{a}_{ni}}{{X}_{i}}} \right|-\varepsilon \right)_{+}^{\alpha} & = & \sum\limits_{n = 1}^{\infty }{\frac{1}{n}\int_{0}^{\infty }{P\left( \frac{1}{{{b}_{n}}}{\mathop {\max }\limits_{1 \le j \le n} }\, \left| \sum\limits_{i = 1}^{j}{{{a}_{ni}}{{X}_{i}}} \right|-\varepsilon > {{t}^{1/\alpha}} \right)dt}} \\ & = & \sum\limits_{n = 1}^{\infty }{\frac{1}{n}\int_{0}^{1}{P\left( \frac{1}{{{b}_{n}}}{\mathop {\max }\limits_{1 \le j \le n} }\, \left| \sum\limits_{i = 1}^{j}{{{a}_{ni}}{{X}_{i}}} \right| > \varepsilon +{{t}^{1/\alpha}} \right)dt}} \\ &&+ \sum\limits_{n = 1}^{\infty }{\frac{1}{n}\int_{1}^{\infty }{P\left( \frac{1}{{{b}_{n}}}{\mathop {\max }\limits_{1 \le j \le n} }\, \left| \sum\limits_{i = 1}^{j}{{{a}_{ni}}{{X}_{i}}} \right| > \varepsilon +{{t}^{1/\alpha}} \right)dt}} \\ &\le& \sum\limits_{n = 1}^{\infty }{\frac{1}{n}P\left( {\mathop {\max }\limits_{1 \le j \le n} }\, \left| \sum\limits_{i = 1}^{j}{{{a}_{ni}}{{X}_{i}}} \right| > \varepsilon {{b}_{n}} \right)} \\ &&+ \sum\limits_{n = 1}^{\infty }{\frac{1}{n}\int_{1}^{\infty }{P\left( {\mathop {\max }\limits_{1 \le j \le n} }\, \left| \sum\limits_{i = 1}^{j}{{{a}_{ni}}{{X}_{i}}} \right| > {{b}_{n}}{{t}^{1/\alpha}} \right)dt}} \\ &\triangleq& I+J. \end{eqnarray} $ | (2.9) |
By Theorem A of Li et al.[12] declared in the first section, we get directly $ I < \infty $. In order to prove (1.5), it suffices to show that $ J < \infty $.
Without loss of generality, assume that $ {{a}_{ni}}\ge 0 $. For all $ t\ge 1 $ and $ 1\le i\le n $, $ n\in \text{N} $, define
| $ \begin{equation*} {{Y}_{i}} = {{a}_{ni}}{{X}_{i}}I\left( \left| {{a}_{ni}}{{X}_{i}} \right|\le {{b}_{n}}{{t}^{1/\alpha }} \right). \end{equation*} $ |
It is easy to check that
| $ \begin{equation*} \left( {\mathop {\max }\limits_{1 \le j \le n} }\, \left| \sum\limits_{i = 1}^{j}{{{a}_{ni}}{{X}_{i}}} \right| > {{b}_{n}}{{t}^{1/\alpha }} \right)\subset \left( {\mathop {\max }\limits_{1 \le j \le n} }\, \left| \sum\limits_{i = 1}^{j}{{{Y}_{i}}} \right| > {{b}_{n}}{{t}^{1/\alpha }} \right)\bigcup \left( \bigcup\limits_{i = 1}^{n}{\left( \left| {{a}_{ni}}{{X}_{i}} \right| > {{b}_{n}}{{t}^{1/\alpha }} \right)} \right), \end{equation*} $ |
which implies
| $ \begin{eqnarray} P\left( {\mathop {\max }\limits_{1 \le j \le n} }\, \left| \sum\limits_{i = 1}^{j}{{{a}_{ni}}{{X}_{i}}} \right| > {{b}_{n}}{{t}^{1/\alpha }} \right)&\le& P\left( {\mathop {\max }\limits_{1 \le j \le n} }\, \left| \sum\limits_{i = 1}^{j}{{{Y}_{i}}} \right| > {{b}_{n}}{{t}^{1/\alpha }} \right) \\ && +P\left( \bigcup\limits_{i = 1}^{n}{\left( \left| {{a}_{ni}}{{X}_{i}} \right| > {{b}_{n}}{{t}^{1/\alpha }} \right)} \right). \end{eqnarray} $ | (2.10) |
To prove $ J < \infty $, we need only to show that
| $ \begin{equation*} {{J}_{1}} = \sum\limits_{n = 1}^{\infty }{\frac{1}{n}\int_{1}^{\infty }{P\left( {\mathop {\max }\limits_{1 \le j \le n} }\, \left| \sum\limits_{i = 1}^{j}{{{Y}_{i}}} \right| > {{b}_{n}}{{t}^{1/\alpha }} \right)dt}} < \infty, \end{equation*} $ |
| $ \begin{equation*} {{J}_{2}} = \sum\limits_{n = 1}^{\infty }{\frac{1}{n}\int_{1}^{\infty }{P\left( \bigcup\limits_{i = 1}^{n}{\left( \left| {{a}_{ni}}{{X}_{i}} \right| > {{b}_{n}}{{t}^{1/\alpha }} \right)} \right)dt}} < \infty. \end{equation*} $ |
Since
| $ P\left( \bigcup\limits_{i = 1}^{n}{\left( \left| {{a}_{ni}}{{X}_{i}} \right| > {{b}_{n}}{{t}^{1/\alpha }} \right)} \right)\le \sum\limits_{i = 1}^{n}{P\left( \left| {{a}_{ni}}{{X}_{i}} \right| > {{b}_{n}}{{t}^{1/\alpha }} \right)}, $ |
it follows from Lemma 2.3 that
| $ \begin{equation*} {{J}_{2}}\le \sum\limits_{n = 1}^{\infty }{\frac{1}{n}\int_{1}^{\infty }{\sum\limits_{i = 1}^{n}{P\left( \left| {{a}_{ni}}{{X}_{i}} \right| > {{b}_{n}}{{t}^{1/\alpha }} \right)}dt}} < \infty. \end{equation*} $ |
Next, we prove that
| $ \begin{equation} {\mathop {\sup }\limits_{t \ge 1} }\, \frac{1}{{{b}_{n}}{{t}^{1/\alpha }}}{\mathop {\max }\limits_{1 \le j \le n} }\, \left| \sum\limits_{i = 1}^{j}{E{{Y}_{i}}} \right|\to 0. \end{equation} $ | (2.11) |
By $ E{{X}_{n}} = 0 $ and (2.4) of Lemma 2.2, it follows that
| $ \begin{array}{l} {\mathop {\sup }\limits_{t \ge 1} }\, \frac{1}{{{b}_{n}}{{t}^{1/\alpha }}}{\mathop {\max }\limits_{1 \le j \le n} }\, \left| \sum\limits_{i = 1}^{j}{E{{Y}_{i}}} \right| = {\mathop {\sup }\limits_{t \ge 1} }\, \frac{1}{{{b}_{n}}{{t}^{1/\alpha }}}{\mathop {\max }\limits_{1 \le j \le n} }\, \left| \sum\limits_{i = 1}^{j}{E{{a}_{ni}}{{X}_{i}}I\left( \left| {{a}_{ni}}{{X}_{i}} \right|\le {{b}_{n}}{{t}^{1/\alpha }} \right)} \right|\\ = {\mathop {\sup }\limits_{t \ge 1} }\, \frac{1}{{{b}_{n}}{{t}^{1/\alpha }}}{\mathop {\max }\limits_{1 \le j \le n} }\, \left| \sum\limits_{i = 1}^{j}{E{{a}_{ni}}{{X}_{i}}I\left( \left| {{a}_{ni}}{{X}_{i}} \right| > {{b}_{n}}{{t}^{1/\alpha }} \right)} \right|\\ \le C{\mathop {\sup }\limits_{t \ge 1} }\, \frac{1}{{{b}_{n}}{{t}^{1/\alpha }}}\sum\limits_{i = 1}^{n}{E\left| {{a}_{ni}}X \right|I\left( \left| {{a}_{ni}}X \right| > {{b}_{n}}{{t}^{1/\alpha }} \right)}. \end{array} $ |
Observe that,
| $ \begin{eqnarray} E\left| {{a}_{ni}}X \right|I\left( \left| {{a}_{ni}}X \right| > {{b}_{n}}{{t}^{1/\alpha }} \right)& = &E\left| {{a}_{ni}}X \right|I\left( \left| {{a}_{ni}}X \right| > {{b}_{n}}{{t}^{1/\alpha }}, \left| X \right|\le {{b}_{n}} \right) \\ &&+E\left| {{a}_{ni}}X \right|I\left( \left| {{a}_{ni}}X \right| > {{b}_{n}}{{t}^{1/\alpha }}, \left| X \right| > {{b}_{n}} \right). \end{eqnarray} $ | (2.12) |
For $ 0 < \gamma < \alpha $ and $ 1 < \alpha \le 2 $, it is clearly shown that
| $ \begin{align} & E\left| {{a}_{ni}}X \right|I\left( \left| {{a}_{ni}}X \right| > {{b}_{n}}{{t}^{1/\alpha }}, \left| X \right|\le {{b}_{n}} \right) \le {{C}}b_{n}^{1-\alpha }{{t}^{\left( 1/\alpha \right)-1}}{{\left| {{a}_{ni}} \right|}^{\alpha }}E{{\left| X \right|}^{\alpha }}I\left( \left| X \right|\le {{b}_{n}} \right) \\ & \le {{C}}b_{n}^{1-\alpha }{{t}^{\left( 1/\alpha \right)-1}}{{\left| {{a}_{ni}} \right|}^{\alpha }}E\left( \frac{{{\left| X \right|}^{\alpha }}}{{{\left( \log \left( 1+\left| X \right| \right) \right)}^{\alpha /\gamma -1}}}{{\left( \log \left( 1+\left| X \right| \right) \right)}^{\alpha /\gamma -1}} \right)I\left( \left| X \right|\le {{b}_{n}} \right) \\ & \le {{C}}{{t}^{\left( 1/\alpha \right)-1}}{{n}^{-1+\left( 1/\alpha \right)}}{{\left| {{a}_{ni}} \right|}^{\alpha }}{{(\log n)}^{\left( 1/\gamma \right)-1}}, \end{align} $ | (2.13) |
and
| $ \begin{eqnarray} E\left| {{a}_{ni}}X \right|I\left( \left| {{a}_{ni}}X \right| > {{b}_{n}}{{t}^{1/\alpha }}, \left| X \right| > {{b}_{n}} \right)&\le& {{C}}\left| {{a}_{ni}} \right|E\left| X \right|I\left( \left| X \right| > {{b}_{n}} \right) \\ &\le& {{C}}b_{n}^{1-\alpha }{{\left( \log \left( 1+{{b}_{n}} \right) \right)}^{\left( \alpha /\gamma \right)-1}}\left| {{a}_{ni}} \right| \\ &\le& {{C}}{{n}^{-1+\left( 1/\alpha \right)}}{{(\log n)}^{-1+\left( 1/\gamma \right)}}\left| {{a}_{ni}} \right|. \end{eqnarray} $ | (2.14) |
Thus,
| $ \begin{eqnarray} {\mathop {\sup }\limits_{t \ge 1} }\, \frac{1}{{{b}_{n}}{{t}^{1/\alpha }}}\sum\limits_{i = 1}^{n}{E\left| {{a}_{ni}}X \right|I\left( \left| {{a}_{ni}}X \right| > {{b}_{n}}{{t}^{1/\alpha }}, \left| X \right|\le {{b}_{n}} \right)}&\le& Cb_{n}^{-1}{{n}^{-1+\left( 1/\alpha \right)}}{{(\log n)}^{\left( 1/\gamma \right)-1}}\sum\limits_{i = 1}^{n}{{{\left| {{a}_{ni}} \right|}^{\alpha }}} \\ &\le& C{{(\log n)}^{-1}}\to 0, \end{eqnarray} $ | (2.15) |
and
| $ \begin{eqnarray} {\mathop {\sup }\limits_{t \ge 1} }\, \frac{1}{{{b}_{n}}{{t}^{1/\alpha }}}\sum\limits_{i = 1}^{n}{E\left| {{a}_{ni}}X \right|I\left( \left| {{a}_{ni}}X \right| > {{b}_{n}}{{t}^{1/\alpha }}, \left| X \right| > {{b}_{n}} \right)}&\le& Cb_{n}^{-1}{{n}^{-1+\left( 1/\alpha \right)}}{{(\log n)}^{-1+\left( 1/\gamma \right)}}\sum\limits_{i = 1}^{n}{\left| {{a}_{ni}} \right|} \\ &\le& C{{(\log n)}^{-1}}\to 0. \end{eqnarray} $ | (2.16) |
Then, (2.11) holds by the argumentation of (2.12)–(2.16).
Hence, for $ n $ sufficiently large, we have that $ {\mathop {\max }\limits_{1 \le j \le n} }\, \left| \sum\limits_{i = 1}^{j}{E{{Y}_{i}}} \right|\le \frac{{{b}_{n}}{{t}^{1/\alpha }}}{2} $ holds uniformly for all $ t\ge 1 $. Therefore,
| $ \begin{equation} {{J}_{1}} = \sum\limits_{n = 1}^{\infty }{\frac{1}{n}\int_{1}^{\infty }{P\left( {\mathop {\max }\limits_{1 \le j \le n} }\, \left| \sum\limits_{i = 1}^{j}{\left( {{Y}_{i}}-E{{Y}_{i}} \right)} \right| > \frac{{{b}_{n}}{{t}^{1/\alpha }}}{2} \right)dt}}. \end{equation} $ | (2.17) |
By the Markov's inequality, (2.2) of Lemma 2.1 and (2.3) of Lemma 2.2, we get that
| $ \begin{eqnarray} {{J}_{1}}&\le& C\sum\limits_{n = 1}^{\infty }{\frac{1}{n}\int_{1}^{\infty }{\frac{1}{b_{n}^{2}{{t}^{2/\alpha }}}E\left( {\mathop {\max }\limits_{1 \le j \le n} }\, {{\left| \sum\limits_{i = 1}^{j}{\left( {{Y}_{i}}-E{{Y}_{i}} \right)} \right|}^{2}} \right)dt}} \\ &\le& C\sum\limits_{n = 1}^{\infty }{\frac{1}{n}\int_{1}^{\infty }{\frac{1}{b_{n}^{2}{{t}^{2/\alpha }}}\left( \sum\limits_{i = 1}^{n}{E{{\left| {{Y}_{i}}-E{{Y}_{i}} \right|}^{2}}} \right)dt}} \\ &\le& C\sum\limits_{n = 1}^{\infty }{\frac{1}{n}\int_{1}^{\infty }{\frac{1}{b_{n}^{2}{{t}^{2/\alpha }}}\left( \sum\limits_{i = 1}^{n}{E{{\left| {{a}_{ni}}{{X}_{i}} \right|}^{2}}I\left( \left| {{a}_{ni}}{{X}_{i}} \right|\le {{b}_{n}}{{t}^{1/\alpha }} \right)} \right)dt}} \\ &\le& C\sum\limits_{n = 1}^{\infty }{\frac{1}{n}\int_{1}^{\infty }{\frac{1}{b_{n}^{2}{{t}^{2/\alpha }}}\left( \sum\limits_{i = 1}^{n}{E{{\left| {{a}_{ni}}X \right|}^{2}}I\left( \left| {{a}_{ni}}X \right|\le {{b}_{n}}{{t}^{1/\alpha }} \right)} \right)dt}} \\ &&+C\sum\limits_{n = 1}^{\infty }{\frac{1}{n}\int_{1}^{\infty }{\sum\limits_{i = 1}^{n}{P\left( \left| {{a}_{ni}}X \right| > {{b}_{n}}{{t}^{1/\alpha }} \right)}dt}} \\ &\le& C\sum\limits_{n = 1}^{\infty }{\frac{1}{n}\int_{1}^{\infty }{\frac{1}{b_{n}^{2}{{t}^{2/\alpha }}}\left( \sum\limits_{i = 1}^{n}{E{{\left| {{a}_{ni}}X \right|}^{2}}I\left( \left| {{a}_{ni}}X \right|\le {{b}_{n}} \right)} \right)dt}} \\ &&+C\sum\limits_{n = 1}^{\infty }{\frac{1}{n}\int_{1}^{\infty }{\frac{1}{b_{n}^{2}{{t}^{2/\alpha }}}\left( \sum\limits_{i = 1}^{n}{E{{\left| {{a}_{ni}}X \right|}^{2}}I\left( {{b}_{n}} < \left| {{a}_{ni}}X \right|\le {{b}_{n}}{{t}^{1/\alpha }} \right)} \right)dt}} \\ &&+C\sum\limits_{n = 1}^{\infty }{\frac{1}{n}\int_{1}^{\infty }{\sum\limits_{i = 1}^{n}{P\left( \left| {{a}_{ni}}X \right| > {{b}_{n}}{{t}^{1/\alpha }} \right)}dt}} \\ & = &{{J}_{11}}+{{J}_{12}}+{{J}_{13}}. \end{eqnarray} $ | (2.18) |
Based on the formula (2.2) of Lemma 2.2 in Li et al.[10], we get that
| $ \begin{eqnarray} {{J}_{11}}& = &\sum\limits_{n = 1}^{\infty }{\frac{1}{n}\int_{1}^{\infty }{\frac{1}{b_{n}^{2}{{t}^{2/\alpha }}}\left( \sum\limits_{i = 1}^{n}{E{{\left| {{a}_{ni}}X \right|}^{2}}I\left( \left| {{a}_{ni}}X \right|\le {{b}_{n}} \right)} \right)dt}} \\ &\le& \sum\limits_{n = 1}^{\infty }{\frac{1}{n}\frac{1}{b_{n}^{\alpha }}\left( \sum\limits_{i = 1}^{n}{E{{\left| {{a}_{ni}}X \right|}^{\alpha }}I\left( \left| {{a}_{ni}}X \right|\le {{b}_{n}} \right)} \right)} < \infty. \end{eqnarray} $ | (2.19) |
Denoting $ t = {{x}^{\alpha }} $, by (2.3) of Lemma 2.2, the Markov's inequality and Lemma 2.3, we also get that
| $ \begin{eqnarray} {{J}_{12}}& = &\sum\limits_{n = 1}^{\infty }{\frac{1}{n}\int_{1}^{\infty }{\frac{1}{b_{n}^{2}{{t}^{2/\alpha }}}\left( \sum\limits_{i = 1}^{n}{E{{\left| {{a}_{ni}}X \right|}^{2}}I\left( {{b}_{n}} < \left| {{a}_{ni}}X \right|\le {{b}_{n}}{{t}^{1/\alpha }} \right)} \right)dt}} \\ &\le& C\sum\limits_{n = 1}^{\infty }{\frac{1}{nb_{n}^{2}}\int_{1}^{\infty }{{{x}^{\alpha -3}}\sum\limits_{i = 1}^{n}{E{{\left| {{a}_{ni}}X \right|}^{2}}I\left( {{b}_{n}} < \left| {{a}_{ni}}X \right|\le {{b}_{n}}x \right)}dx}} \\ &\le& C\sum\limits_{n = 1}^{\infty }{\frac{1}{nb_{n}^{2}}\sum\limits_{m = 1}^{\infty }{\int_{m}^{m+1}{{{x}^{\alpha -3}}\sum\limits_{i = 1}^{n}{E{{\left| {{a}_{ni}}X \right|}^{2}}I\left( {{b}_{n}} < \left| {{a}_{ni}}X \right|\le {{b}_{n}}x \right)}dx}}} \\ &\le& C\sum\limits_{n = 1}^{\infty }{\frac{1}{nb_{n}^{2}}\sum\limits_{m = 1}^{\infty }{{{m}^{\alpha -3}}\sum\limits_{i = 1}^{n}{E{{\left| {{a}_{ni}}X \right|}^{2}}I\left( {{b}_{n}} < \left| {{a}_{ni}}X \right|\le {{b}_{n}}\left( m+1 \right) \right)}}} \\ & = &C\sum\limits_{n = 1}^{\infty }{\frac{1}{nb_{n}^{2}}\sum\limits_{i = 1}^{n}{\sum\limits_{m = 1}^{\infty }{\sum\limits_{s = 1}^{m}{{{m}^{\alpha -3}}E{{\left| {{a}_{ni}}X \right|}^{2}}I\left( {{b}_{n}}s < \left| {{a}_{ni}}X \right|\le {{b}_{n}}\left( s+1 \right) \right)}}}} \\ & = &C\sum\limits_{n = 1}^{\infty }{\frac{1}{nb_{n}^{2}}\sum\limits_{i = 1}^{n}{\sum\limits_{s = 1}^{\infty }{E{{\left| {{a}_{ni}}X \right|}^{2}}I\left( {{b}_{n}}s < \left| {{a}_{ni}}X \right|\le {{b}_{n}}\left( s+1 \right) \right)\sum\limits_{m = s}^{\infty }{{{m}^{\alpha -3}}}}}} \\ &\le& C\sum\limits_{n = 1}^{\infty }{\frac{1}{nb_{n}^{2}}\sum\limits_{i = 1}^{n}{\sum\limits_{s = 1}^{\infty }{E{{\left| {{a}_{ni}}X \right|}^{2}}I\left( {{b}_{n}}s < \left| {{a}_{ni}}X \right|\le {{b}_{n}}\left( s+1 \right) \right){{s}^{\alpha -2}}}}} \\ &\le& C\sum\limits_{n = 1}^{\infty }{\frac{1}{nb_{n}^{\alpha }}\sum\limits_{i = 1}^{n}{E{{\left| {{a}_{ni}}X \right|}^{\alpha }}I\left( \left| {{a}_{ni}}X \right| > {{b}_{n}} \right)}} \\ &\le& CE{{{\left| X \right|}^{\alpha }}}/{{{\left( \log \left( 1+\left| X \right| \right) \right)}^{\alpha /\gamma -1}}}\; < \infty. \end{eqnarray} $ | (2.20) |
Analogous to the argumentation of Lemma 2.3, it is easy to show that
| $ \begin{equation} {{J}_{13}} = \sum\limits_{n = 1}^{\infty }{\frac{1}{n}\int_{1}^{\infty }{\sum\limits_{i = 1}^{n}{P\left( \left| {{a}_{ni}}X \right| > {{b}_{n}}{{t}^{1/\alpha }} \right)}dt}}\le CE{{{\left| X \right|}^{\alpha }}}/{{{\left( \log \left( 1+\left| X \right| \right) \right)}^{\alpha /\gamma -1}}}\; < \infty. \end{equation} $ | (2.21) |
Hence, the desired result $ {{J}_{1}} < \infty $ holds by the above statements. The proof of Theorem 1.1 is completed.
Remark 2.1. Under the conditions of Theorem 1.1, noting that
| $ \begin{eqnarray} \infty & > & \sum\limits_{n = 1}^{\infty }{\frac{1}{n}}E\left( \frac{1}{{{b}_{n}}}{\mathop {\max }\limits_{1 \le j \le n} }\, \left| \sum\limits_{i = 1}^{j}{{{a}_{ni}}{{X}_{i}}} \right|-\varepsilon \right)_{+}^{\alpha} \\ & = & \sum\limits_{n = 1}^{\infty }{\frac{1}{n}}\int_{0}^{\infty }{P\left( \frac{1}{{{b}_{n}}}{\mathop {\max }\limits_{1 \le j \le n} }\, \left| \sum\limits_{i = 1}^{j}{{{a}_{ni}}{{X}_{i}}} \right|-\varepsilon > {{t}^{1/\alpha}} \right)d}t \\ &\ge& C\sum\limits_{n = 1}^{\infty }{\frac{1}{n}\int_{0}^{{{\varepsilon }^{\alpha }}}{P\left( \frac{1}{{{b}_{n}}}{\mathop {\max }\limits_{1 \le j \le n} }\, \left| \sum\limits_{i = 1}^{j}{{{a}_{ni}}{{X}_{i}}} \right| > \varepsilon +{{t}^{1/\alpha }} \right)}dt} \\ &\ge& C\sum\limits_{n = 1}^{\infty }{\frac{1}{n}}P\left( {\mathop {\max }\limits_{1 \le j \le n} }\, \left| \sum\limits_{i = 1}^{j}{{{a}_{ni}}{{X}_{i}}} \right| > 2\varepsilon{{b}_{n}} \right)\quad \text{for} \quad\forall \varepsilon > 0. \end{eqnarray} $ | (2.22) |
Since $ \varepsilon > 0 $ is arbitrary, it follows from (2.22) that the complete moment convergence is much stronger than the complete convergence. Compared with the corresponding results of Li et al.[12], Chen and Sung[6], it is worth pointing out that Theorem 1.1 of this paper is an extension and improvement of those of Li et al.[12], Chen and Sung[6] under the same moment condition. In addition, the main result partially settles the open problem posed by Huang et al.[10] for the case $ 0 < \gamma < \alpha $ with $ 1 < \alpha \le 2 $.
In this work, we consider the problem of complete moment convergence for weighted sums of weakly dependent (or $ {{\rho }^{*}} $-mixing) random variables. The main results of this paper are presented in the form of the main theorem and a remark as well as Lemma 2.3, which plays a vital role to prove the main theorem. The presented main theorem improves and generalizes the corresponding complete convergence results of Li et al.[12] and Chen and Sung[6].
The authors are most grateful to the Editor as well as the anonymous referees for carefully reading the manuscript and for offering some valuable suggestions and comments, which greatly enabled them to improve this paper. This paper is supported by the Doctor and Professor Natural Science Foundation of Guilin University of Aerospace Technology.
All authors declare no conflicts of interest in this paper.
| [1] | Hudson WE (1991) Landscape Linkages and Biodiversity, Island Press USA, 195. |
| [2] | Noss RF (1991) Landscape Connectivity, Different functions at different scales, Landscape linkages and biodiversity, Island Press, 27-40. |
| [3] | IUCN (1993) Parks for life, Workshop Ⅲ.9, Corridors, transition zones and buffers: tools for enhancing the effectiveness of protected areas, IUCN, Gland Switzerland. |
| [4] | Ahern J (1994) Greenways as ecological networks in rural areas, Landscape planning and ecological networks, Elsevier, Amsterdam, 159-177. |
| [5] | Bennet G (1994) Conserving Europes Natural Heritage: Towards a European Ecological Network, Graham Bennett, Graham & Trotman, Great Britain, 254. |
| [6] | Forman RTT (1995) Some general principles of landscape and regional ecology. Landscape Ecol 10: 133-142. |
| [7] | Jongman RHG (1995) Nature conservation planning in Europe, developing ecological networks. Landscape Urban Plan 32: 169-183. |
| [8] | Collinge SK (1996) Ecological consequences of habitat fragmentation, implications for landscape architecture and planning. Landscape Urban Plan 36: 59-77. |
| [9] | Romano B (1999) Planning and Environmental Continuity. Urbanistica 112: 156-160. |
| [10] | Jongman RHG, Bouwma, I.M, Griffioen, et al. (2011) The Pan European Ecological Network: PEEN. Landscape Ecol 26: 311-326. |
| [11] | Battisti C (2011) Ecological network planning - from paradigms to design and back: a cautionary note. J Land Use Sci 8: 1-9. |
| [12] | EEA (2011) Landscape fragmentation in Europe. EEA-FOEN, 92. |
| [13] | Ragni B (2009) RERU, Rete Ecologica Regionale dell'Umbria. Petruzzi, 241. |
| [14] | Montanari I, Carati M, Costantino, R, et al. (2010) Qualità ecologica, l'approccio emiliano-romagnolo. Ecoscienza 3: 56-59. |
| [15] | Lombardi M, Giunti M, Castelli C (2014) La rete ecologica toscana: aspetti metodologici e applicativi. Ri-Vista XXII(1), 90-101. |
| [16] | Malcevschi S, Lazzarini M (2013) Tecniche e metodi per la realizzazione della Rete Ecologica Regionale. Regione Lombardia, 240. |
| [17] | Frontoni E, Mancino A, Zingaretti P, et al. (2014) SIT-REM: an interoperable and interactive web geographic information system for fauna, flora and plant landscape data management. Int J Geo-Inf 3: 853-867. |
| [18] | Olson DM, Dinerstein E (2002) The Global 200: Priority Ecoregions for Global Conservation. Annals of the Missouri Botanical Garden 89: 199-224. |
| [19] | Jongman R HG, Külvik M, Kristiansen I (2004) European ecological networks and greenways. Landscape Urban Plan 68: 305-319. |
| [20] | Scolozzi R, Morri E, Santolini R (2012) Delphi-based change assessment in ecosystem service values to support strategic spatial planning in Italian landscapes. Ecol Indic 21: 134-144. |
| [21] | Sargolini M (2013) Urban Landscapes, Environmental Networks and Quality of Life. Springer, Milano, 177. |
| [22] | Larson LR, Keith SJ, Fernandez M, et al. (2016) Ecosystem services and urban greenways: What's the public's perspective? Ecosys Ser 22: 111-116. |
| [23] | Leone F, Zoppi C (2016) Conservation Measures and Loss of Ecosystem Services: A Study Concerning the Sardinian Natura 2000 Network. Sustainability 8. |
| [24] | Pungetti G, Romano B (2004) Planning the future landscapes between nature and culture, Ecological Networks and Greenways, Cambridge University Press, UK, 107-127. |
| [25] | Hennig EI. Schwick C, Soukup T, et al. (2015) Multi-scale analysis of urban sprawl in Europe: Towards a European de-sprawling strategy. Land Use Policy 49: 483-498. |
| [26] | Epifani R, Amato F, Murgante B, et al. (2017) Quantitative Measure of Habitat Quality to Support the Implementation of Sustainable Urban Planning Measures. Computational Science and Its Applications - ICCSA 2017. ICCSA 2017. Lecture Notes in Computer Science, Springer, Cham, 10409: 585-600. |
| [27] | Zanfi F (2013) The città abusiva in contemporary southern Italy: illegal building and prospects for change. Urban Stud 50: 3428-3445. |
| [28] | Romano B, Zullo F, Marucci A, et al. (2018) Vintage Urban Planning in Italy: Land Management with the Tools of the Mid-Twentieth Century. Sustainability 10: 4125. |
| [29] | Bossard, M, Feranec J, Otahel J (2000) Corine Land Cover Technical Guide, EEA Technical report 40: 105. |
| [30] | Büttner G (2014) CORINE Land Cover and Land Cover Change Products., Land Use and Land Cover Mapping in Europe, Springer, Dordrecht, 55-74. |
| [31] | Battisti C (2003) Habitat fragmentation, fauna and ecological network planning: Toward a theoretical conceptual framework. Ital J Zool 70: 241-247. |
| [32] | Boitani L, Falcucci A, Maiorano L, et al. (2007) Ecological Networks as Conceptual Frameworks or Operational Tools in Conservation. Conserv Biol 21: 1414-1422. |
| [33] | Romano B, Zullo F (2012) Landscape fragmentation in Italy. Indices implementation to support territorial policies. Planning Support Tools: Policy analysis, Implementation and Evaluation, Franco Angeli, 399-414. |
| [34] | Malcevschi S (2011) Reti ecologiche polivalenti ed alcune considerazioni sui sistemi eco-territoriali. Territorio 58: 54-60. |
| [35] | European Commission (2006) Urban Sprawl in Europe: the Ignored Challenge, EEA Report 10, 2006, 60. |
| [36] | The Worldwatch Institute (2007) State of the World, Our Urban Future. Norton, NY, 250. |
| [37] | Ewing RH (2008) Characteristics, Causes, and Effects of Sprawl: A Literature Review. Urban Ecology Springer, Boston, MA. 519-535. |
| [38] | Pileri P, Maggi M (2010) Sustainable planning? First results in land uptakes in rural, natural and protected areas: the Lombardia case study (Italy). J Land Use Sci 5: 105-122. |
| [39] | Munafò M, Salvati L, Zitti M (2013) Estimating soil sealing rate at national level—Italy as a case study. Ecol Indic 26: 137-140. |
| [40] | Romano B, Zullo F (2013) Models of urban land use in Europe: assessment tools and criticalities. Int J Agri Environ Infor S 4: 80-97. |
| [41] | ISPRA (2017) Consumo di suolo, dinamiche territoriali e servizi ecosistemici. Rapporto 2017. ISPRA, Roma, 186. |
| [42] | ISTAT (2017) Forme, livelli e dinamiche dell'urbanizzazione in Italia, ISTAT, 350. |
| [43] | Romano B, Zullo F, Fiorini L, et al. (2017) Land transformation of Italy due to half a century of urbanisation. Land Use Policy 67: 387-400. |
| [44] | Agostini A (1930) Il problema dei rimboschimenti in Italia, Libreria del Littorio. Roma. |
| [45] | Romano D (1986) I rimboschimenti nella politica forestale italiana. Monti e Boschi 37: 7-12. |
| [46] | Lazzarini A (2002) Disboscamento montano e politiche territoriali. Alpi e Appennini dal Settecento al Duemila. Franco Angeli, Milano. 608. |
| [47] | Tasser E, Walde J, Tappeiner U, et al. (2007) Land-use changes and natural reforestation in the Eastern Central Alps. Agriculture. Ecosyst Environ 118: 115-129. |
| [48] | D'Ippolito A, Ferrari E, Iovino F, et al. (2013) Reforestation and land use change in a drainage basin of southern Italy. iForest - Biogeo For 6: 175-182. |
| [49] | Paolinelli G (2015) Crosscutting Issues in Treating the Fragmentation of Ecosystems and Landscapes. Nature Policies and Landscape Policies. Urban and Landscape Perspectives, Springer, Cham. 18: 283-290. |
| [50] | Marucci A, Zullo F, Morri E, et al. (2017) Spatial Methods to Measure Natura 2000 Sites Insularization in Italy. ICCSA 2017, Part Ⅳ, LNCS 10407: 437-450. |
| [51] | Salvati L, Biasi R, Carlucci M, et al. (2015) Forest transition and urban growth: exploring latent dynamics (1936-2006) in Rome, Italy, using a geographically weighted regression and implications for coastal forest conservation. Rendiconti Lincei 26: 577-585 |
| [52] | Vizzarri M, Tognetti R, Marchetti M (2015) Forest Ecosystem Services: Issues and Challenges for Biodiversity, Conservation, and Management in Italy. Forests 6: 1810-1838. |
| [53] | Corona P, Chirici G, Travaglini D (2004) Forest ecotone survey by line intersect sampling. Can J Forest Res 34: 1776-1783. |
| [54] | Falcucci A, Maiorano L, Boitani L (2007) Changes in land-use/land-cover patterns in Italy and their implications for biodiversity conservation. Landscape Ecol 22: 617-631. |
| [55] | Orsi F, Geneletti D, Borsdorf A (2013) Mapping wildness for protected area management: A methodological approach and application to the Dolomites UNESCO World Heritage Site (Italy). Landscape Urban Plan 120: 1-15. |
| [56] | Petrillo PL, Di Bella O, Di Palo N (2015) The UNESCO World Heritage Convention and the Enhancement of Rural Vine-Growing Landscapes. Cultural Heritage and Value Creation. Springer, Cham. 127-169. |
| [57] | Meier K, Kuusemets V, Luig J, et al. (2005) Riparian buffer zones as elements of ecological networks: Case study on Parnassius mnemosyne distribution in Estonia. Ecol Eng 24: 531-537. |
| [58] | Theobald DM, Reed SE, Fields K, et al. (2012) Connecting natural landscapes using a landscape permeability model to prioritize conservation activities in the United States. Conservation Letters 5: 123-133. |
| [59] | Hepcan S, Hepcan CC, Bouwma IM, et al. (2009) Ecological networks as a new approach for nature conservation in Turkey: A case study of İzmir Province. Landscape Urban Plan 90: 143-154. |
| [60] | Deodatus F, Kruhlov I, Protsenko L, et al. (2013) Creation of ecological corridors in the Ukrainian Carpathians. The Carpathians: Integrating Nature and Society Towards Sustainability, Springer, Berlin, Heidelberg, 701-717. |
| [61] | Izakovičová Z, Świąder M (2017) Building Ecological Networks in Slovakia And Poland. Ekológia (Bratislava) 36: 303-322. |
| [62] | Kozieł M, Michalczuk W, Jędrzejewski W, et al. (2010) Protection of ecological corridors in spatial planning documents in Poland implementation problems. Europa, XXI: 77-89. |
| [63] | Gambino R (2003) APE, Appennino Parco d'Europa. Ministero dell'Ambiente e della Tutela del Territorio, Alinea Firenze. 240. |
| [64] | Gambino R, Romano B (2004) Territorial strategies and environmental continuity in mountain systems, the case of the Apennines (Italy), Managing Mountain Protected Areas: challenges and responses for the 21st Century, IUCN, Andromeda, 66-77. |
| [65] | Saura S, Santini L, Rondinini C (2015) Connectivity of the global network of protected areas. Biod Res 22: 199-211. |
| [66] | Saura S, Bertzky B, Bastin L, et al. (2018) Protected area connectivity: Shortfalls in global targets and country-level priorities. Biol Conserv 219: 53-67. |
| [67] | Saura S, Bertzky B, Bastin L, et al. (2019) Global trends in protected area connectivity from 2010 to 2018. Biol Conserv 238: 108183. |
| [68] | Thrush, SF, Hewitt, JE, Lohrer, et al. (2013) When small changes matter: the role of cross‐scale interactions between habitat and ecological connectivity in recovery. Ecol Appl 23: 226-238. |
| [69] | La Point S, Balkenhol N, Hale J, et al. (2015) Ecological connectivity research in urban areas. Funct Ecol 29: 868-878. |
| [70] | Rosengarten F (2009) The contemporary relevance of Gramsci's views on Italy's "Southern question". Perspectives on Gramsci, Politics, culture and social theory, Routledge, UK, 134-144. |
| [71] | Romano B, Fiorini L, Di Dato C, et al. (2020) Latitudinal Gradient in Urban Pressure and Socio Environmental Quality: The "Peninsula Effect" in Italy. Land 9: 126. |
| [72] | Forman RTT, Sperling D, Bissonette JA, et al. (2002) Road Ecology: Science and Solutions. Island Press, 479. |
| [73] | Jaeger AG, Schwarz-von Raumer HG, Esswein H, et al. (2007) Time series of landscape fragmentation caused by transportation infrastructure and urban development: a case study from Baden-Württemberg, Germany. Ecol Soc 12: 22. |
| [74] | Karlson M, Mörtberg U, Balfors B (2014) Road ecology in environmental impact assessment. Environ Impact Assess 48: 10-19. |
| [75] | Zurlini G, Amadio V, Rossi O (1999) A Landscape Approach to Biodiversity and Biological Health Planning: The Map of Italian. Ecosyst Health 5: 294-311 |
| [76] | Maiorano L, Falcucci A, Boitani L (2006) Gap analysis of terrestrial vertebrates in Italy: Priorities for conservation planning in a human dominated landscape. Biol Conserv 133: 455-473. |
| [77] | Rondinini C, Boitani L (2007) Systematic Conservation Planning and the Cost of Tackling Conservation Conflicts with Large Carnivores in Italy. Conserv Biol 21: 1455-1462. |
| [78] | Peruzzi L, Conti F, Bartolucci F (2014) An inventory of vascular plants endemic to Italy. Phytotaxa 168: 1-75. |
| [79] | Baldwin RF, Reed SE, McRae BH, et al. (2012) Connectivity Restoration in Large Landscapes: Modeling Landscape Condition and Ecological Flows. Ecol Restor 30: 274-279. |
| [80] | Foster ML, Humphrey SR (1995) Use of Highway Underpasses by Florida Panthers and Other Wildlife. Wildlife Soc B 23: 95-100. |
| [81] | Rosillon F (2004) Valley landscape management: the context of the "river contract" in the Semois Valley (Belgium). Landscape Res 29: 413-422. |
| [82] | Brun A (2010) Les contrats de rivière en France: un outil de gestion concertée de la ressource en local. La Découverte 498. |
| [83] | Peng J, Zhao H, Liu Y (2017) Urban ecological corridors construction: A review. Acta Ecologica Sinica 37: 23-30. |
| [84] | Marucci A, Zullo F, Fiorini L, et al. (2019) The role of infrastructural barriers and gaps on Natura 2000 functionality in Italy A case study on Umbria region. Rendiconti Lincei 30: 223-235. |
| 1. | Yukun Xiao, Jianzhi Han, Cocommutative connected vertex (operator) bialgebras, 2025, 212, 03930440, 105461, 10.1016/j.geomphys.2025.105461 |
Figures(8) / Tables(3)
Bernardino Romano, Andrea Agapito Ludovici, Francesco Zullo, Alessandro Marucci, Lorena Fiorini. Italian landscape macrosystem (ILM) from urban pressure to a National Wildway[J]. AIMS Environmental Science, 2020, 7(6): 505-525. doi: 10.3934/environsci.2020032
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