Evolution of distributed detection performance over balanced binary relay tree networks with bit errors

Target detection based on wireless sensor networks can be considered as a distributed binary hypothesis testing problem. In this paper, the evolution of detection error probability with the increase in the network scale is studied for the balanced binary relay tree network with channel noise. Firstly, the iterative expressions of false-alarm probability and missed-detection probability depending on the number of tree network layers are given. Then, the iterative process of two types of error probabilities in the network space is described as a discrete nonlinear switched dynamic system, and the dynamic properties of two types of error probabilities are analyzed in a plane rectangular coordinate system. A globally attractive invariant set of the state of the dynamic system, which is not related to the channel noise, is derived. The switching mode of the system and the total error probability in the invariant set are further analyzed, and a necessary and sufficient convergence condition of the total error probability is provided. Based on this condition, the following detection properties of the network are revealed: (1) as long as the channel bit error probability is not zero, the total error probability does not tend to zero as the network size grows to infinity; (2) when the channel bit error probability is greater than 2-32\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\frac{2-\sqrt{3}}{2}$$\end{document}, the total error probability will continue to increase with the increase in the network size.


Introduction
Target detection based on wireless sensor networks (WSNs) has a very wide and important application background and has received constant attention from industry and academia [1][2][3][4][5][6][7]. In particular, the relationship between distributed detection performance and sensor network size has been studied in detail for different network structures. Parallel structure is the first to receive attention. In the network of this structure, each sensor directly submits its detection results to the fusion center (FC). When each sensor has independent observations, the parallel structure can ensure that the total error probability of FC decreases exponentially with the increase in the network size [8]. However, when the physical distance between a sensor and FC is far away, the parallel structure is not conducive to energy saving and is usually difficult to implement in the case of large-scale networks. An alternative structure is tandem network [9][10][11]. In tandem networks, each node fuses its own detection results with the detection results of lower-level nodes and then transmits the fusion results to an upper-level node. Under this structure, the decision-making error probability at the FC decreases sub-exponentially with the increase in the network size [11]. A tree network is something between parallel structure and tandem structure. In this kind of network, in addition to the FC and the leaf nodes which directly detects the target, there are also a number of relay nodes which are only responsible for local information fusion and transmission. Tree networks have attracted more and more attention because they can take into account both the energy consumption and detection performance in practical applications.
For the tree networks with finite tree height (i.e., when the network size tends to infinity, the tree height is finite), paper [12] proves that under Bayesian criterion the error probability of FC exponentially decays with the increase in the network size although the decay exponent is smaller than that of parallel network. Paper [13] studies a kind of tree network with infinite tree height (i.e., when the network size tends to infinity, the tree height also tends to infinity), the so-called balanced binary relay tree (BBRT) network. Under the premise that all leaf nodes' observations are independent and all relay nodes adopt the same likelihood ratio test (LRT), it is proved by the Lyapunov method that the total error probability of FC converges to zero. In [14], the derivation of optimal fusion rules in the BBRT network is investigated by using dynamic programming method. Under the unit threshold LRT (UT-LRT) rule, paper [15] studies the binary hypothesis testing in the BBRT network, analyzes the dynamical properties of false-alarm probability (probability of Type I error) and missed-detection probability (probability of Type II error), and derives a globally attractive invariant set in the state space of error probabilities of two types. Based on the dynamic property analysis in this invariant set, it is proved that the decay exponent of the total error probability of the network is the square root of the number of network layers.
All the above-mentioned researches are aimed at dealing with ideal networks. In practical applications, however, WSNs are often deployed in complex environments. Many environmental factors may interfere with the communication between sensor nodes, which will reduce the stability and accuracy of communication to a certain extent [16,17]. More and more attention has been paid to the distributed detection performance of WSNs with network failure or communication noise. For the problem of network failure, paper [18] intro-duces node and link failure probabilities and studies the asymptotic detection performance of the BBRT network under the UT-LRT criterion. In paper [19], the asymptotic detection performance of the M-ary tree network with node and link failures is studied under the majority dominance rule. The upper and lower bounds of the convergence rate of the FC's error probability with respect to the network size are also given. Both [18] and [19] show that the existence of node and link failures with a certain probability does not overthrow the fact that the error probability at the FC decreases to zero as the network size grows to infinity. Besides, the influence of Gaussian noise in communication channels on the distributed detection performance of WSNs has also been extensively investigated in the literature [20][21][22].
The bit error caused by the communication noise in binary communication channels has a completely different impact in comparison with the network failure, and it can change the transferred information from "true" to "false." Paper [16] studies the impact of errors on a distributed detection conducted by belief propagation algorithm in WSN and shows that the detection can effectively be modeled as a distributed linear data fusion scheme. Paper [17] discusses fusion rules for distributed detection in clustered WSNs with communication channels experiencing additive white Gaussian noise and derives an optimal log-likelihood ratio fusion rule. Paper [23] studies the design of a parameter estimator with binary quantitative observations under the binary symmetric channels (BSC) model. Paper [24] compares the advantages and disadvantages of various information fusion rules in the BSC model. A distributed detection method based on Rao test is studied for noisy channels in [25]. Paper [26] considers the case that the bit noise exists in the communication between sensor node and FC, and constructs an extended maximum likelihood estimator. However, there are few relevant reports concerning the questions such as: How does the FC's error probability evolve in WSNs with bit errors with respect to the network size? Will it decay to zero as the networks size grows to infinity? Recently, the authors' group has investigated these questions for the M-ary tree network and has given a negative answer to the second question [27].
In this paper, we consider the distributed detection problem in the BBRT network with bit errors and analyze the asymptotic detection performance of the network described by the BSC model. Firstly, we derive FC the iterative expression of the false-alarm probability and missed-detection probability with respect to the number of tree network layers, and model the iterative process of the two kinds of error probabilities as a statedependent nonlinear switched dynamic system. Comparing the dynamic model of the BBRT model derived in this paper and the model of the M-ary tree network given in [27], we find that the former is much more complicated because it is a switched system besides nonlinearity while the latter is just a nonlinear stationary system. Afterward, since it is difficult to make the equilibrium analysis like [27] for this nonlinear switched system, we analyze the dynamic properties of the system by dividing the state space as mode-holding regions and mode-switching regions. Based on the state space partitioning, a globally attractive invariant set of the state of the dynamic system is obtained. Finally, we study the switching mode of the dynamic system in the globally attractive invariant set and give a necessary and sufficient condition for the convergence of the total error probability of the FC. It is pointed out that the critical value of the bit error probability that makes the total error probability unable to converge is 2− √ 3 2 . This critical value is meaningful for the design of large-scale wireless sensor networks.
The research method of this paper is greatly inspired by paper [18]. However, unlike the node and link failure discussed in [18], the BSC channel discussed in this paper will produce a signal with the opposite meaning to the source signal. The introduction of bit error probability in the dynamic model makes the evolution process of detection error probability much more complex, and the derivation and proof of the invariant set become much more difficult. And the result obtained in this paper is also quite different from that of [18]: The conclusion that the total error probability will decay to zero with the infinite growth of network size is no longer true for the BSC model.
The following notations are used in this paper. P(·) denotes probability. α k , β k and d k = 1 − β k denote the false-alarm probability, missed-detection probability and detection probability of a node at level k in a BBRT network, respectively. The channel bit error probability is denoted by b. And the closure of set A is denoted by A.
The rest of this paper is organized as follows. The problem formulation and dynamic model of detection performance over the BBRT network is presented in Sect 2. In Sect 3, we study the evolution of the detection error probabilities of two types by dividing their state space into different regions. In Sect 4, a channel noise-independent and globally attractive invariant set of the detection error probabilities is constructed. The switching mode of the dynamic system in the invariant set is also studied in this section. The asymptotic property of the total error probability is analyzed in Sect. 5. The conclusion is drawn in Sect. 6. We consider the target detection as a binary hypothesis testing problem: Hypothesis H 1 corresponds to the appearance of the target and hypothesis H 0 corresponds to the absence of the target. Let such a detection be conducted in a BBRT network with channel noise as shown in Fig. 1, where circles represent sensors, also called leaf nodes, which make measurements, squares represent relay nodes which only fuse messages, rectangle represents the fusion center which makes an overall decision, k denotes the layer number of the tree and N is the total number of sensors. Each leaf node makes independent detections for the same events, then summarizes it into binary message and sends the message to its parent node. Each parent node receives message sent by two child nodes; then, it will fuse two binary messages into one binary message and forward the new binary message to its parent node. This process is repeating until the message is transmitted to the root node, which is the fusion center. The channel noise modeled by BSC is shown in Fig. 2. The bit error probability of the BSC is denoted by b. When a child node sends X to its parent node, the parent node receives 1 − X with probability b and receives X with probability 1 − b.
The following assumptions are needed for our further study: Assumption 1 All sensors are independent of identical false-alarm probability α 0 and identical misseddetection probability β 0 , which are subject to α 0 +β 0 < 1.

Assumption 2
All channels in the network have the same bit error probability b ∈ [0, 1 2 ).
The part of Assumption 1 that all sensors are independent of identical false-alarm probability and identical missed-detection probability is made for the possibility of iterative analysis of the BBRT network (see, e.g. [15,18]). For the BSC model, the assumption α 0 + β 0 < 1 and Assumption 2 do not reduce the generality of the results of this paper because the case of α 0 +β 0 > 1 or the case b ∈ ( 1 2 , 1] can be considered as a folded case: If we redefine the meaning of the binary decision 0 as 1 and 1 as 0, then we will return to the case α 0 + β 0 < 1 or the case b ∈ [0, 1 2 ). Suppose that the UT-LRT rule is applied at each relay node and the fusion center. It has been shown that this fusion rule is locally optimal if the event detected has equal a priori probability, i.e., P(H 0 ) = P(H 1 ) = 1 2 (see, e.g., [13]).
Considering x as the type I (or type II) error probability of a decision, then F(b, x) is actually the type I (or type II) probability of the decision signal passing through the BSC channel with bit error probability b. Apparently, for any x ∈ [0, 1] and y ∈ [0, 1], the following equivalence holds Theorem 1 Under Assumptions 1, 2 and the UT-LRT rule, the relation of the false-alarm probability and missed-detection probability between two consecutive layers of the BBRT network with bit error can be described by where k ≥ 0 denotes the layer of the tree.
Proof At first, let us analyze a local network which consists of a parent node and two child nodes, both child nodes are leaves. By Assumption 1, the leaf nodes have identical false-alarm probability α 0 and identical detection probability d 0 = 1 − β 0 . We denote by X the message that the parent node receives from one child node, Y the message from the other child node, and let Z = (X, Y ). By Assumption 1 on the independence of the leaf nodes and Assumption 2 on the bit error probability, it is easy to get: Denote by D the decision made by the parent node based on the observation Z and the UT-LRT rule. By the meaning of false-alarm probability and misseddetection probability, we know that Case 1: Z = (0, 0). In this case, the UT-LRT rule implies that which contradicts Assumption 1. This implies that when Z = (0, 0), the decision D = 1 can never be taken under UT-LRT.
Case 2: Z = (1, 0). In this case, the UT-LRT rule implies that Case 3: Z = (0, 1). It is easy to see that this case is equivalent to the case that Z = (1, 0).
Case 4: Z = (1, 1). In this case, the UT-LRT rule implies that This implies that when Z = (1, 1), it always takes the decision D = 1 under UT-LRT. Summarizing the above four cases, we know that for the case α 0 ≤ β 0 , and for the case α 0 ≤ β 0 , and Thus, we have proved Eq. (4) in Theorem 1 holds for k = 0. This equation also shows that all nodes at level 1 have identical false-alarm probability α 1 and identical missed-detection probability β 1 . Moreover, by Eqs. (2), (3) and (4), it follows from α 0 +β 0 < 1 that α 1 +β 1 < 1 also holds. Since the UT-LRT rule is applied to all nodes in the tree, and all channels in the tree have the same bit error probability, one can repeat the above proof procedure from k = 1 to any positive integer and obtain the conclusion that the Eq. (4) holds for all k ≥ 0.
The iterative Eq. (4) can be regarded as a discrete dynamic system, which determines how the false-alarm probability α k and the missed-detection probability β k evolve in the BBRT with respect to the number of the tree level k. Obviously, it is a state-dependent nonlinear switched system. Paper [15] has studied this system for a special case that b = 0, i.e., the case of no bit error. An invariant set for the system state has been constructed, and with the help of the analysis of the system dynamics in this invariant set, it has been shown that both the false-alarm probability and missed-detection probability converge to zero as the number of sensor in the network increases to infinity. However, in this paper we will show that neither the false-alarm probability nor missed-detection probability converge to zero in the case that b = 0, and the system has very rich and complicated nonlinear dynamic properties. In the following sections, we will study the relationship between the convergence property of the system and the channel parameter b via dividing the state space according to the iterative mode of the system.

Mode-switching region in state space
As mentioned in the Proof of Theorem 1, α 0 + β 0 < 1 implies that α k + β k < 1 for all k ≥ 0. The state space of the dynamic system (4) is a right triangle in the rectangular coordinate system denoted by (α, β), which consists of upper part U and lower part L given below We note that Eq. (4) is symmetric about the line α = β. Hence, if a dynamic property of system (4) holds in region U , it must also hold in a symmetric way in region L. We refer to this as dynamic symmetry of system (4) and use it to shorten the length of the proof of most conclusions in this paper. The iteration of system (4) switches between two modes depending on whether the state of the system falls in L or U at each step. To study the switching mode of the system, we define the mode-switching region of the dynamic system (4) as follows.

Definition 1 A set A s in
(α, β) is referred to as a mode-switching region of dynamic system (4) if for Apparently, B 1U and B 1L are symmetric with respect to the line β = α.
According to the dynamic symmetry of the system, it is also true that Proposition 1 says that B 1 := B 1U ∪ B 1L is a modeswitching region of system (4).
The following proposition reveals the relationship of the mode-switching regions of the system in the noisy channel case and noise-free case.
Proof of Proposition 2 is given in Appendix A. Proposition 2 is illustrated by Fig. 3,  Corresponding to mode-switching region, mode -holding region is defined as follows. (α, β) is referred to as a mode-holding region of dynamic system (4) if for all

Definition 2 A set A h in
For convenience of discussion, let us define a func- and denote the mth composition of this function as And then, we further divide the state space of system (4) as follows. F(b, β)) < 1}, where F(b, * ) is defined by (1). Apparently, B mU and B mL are symmetric about line β = α.
Proof of Proposition 3 is given in Appendix B. According to Proposition 3, if the initial state (α 0 , β 0 ) ∈ B mU , where m ≥ 2, the iterative trajectory of (α k , β k ) in the plane of (α, β) is subject to the order: B mU → B (m−1)U → · · · B 2U → B 1U ; similarly, if (α 0 , β 0 ) ∈ B mL , where m ≥ 2, the iterative trajectory of (α k , β k ) is subject to the order: B mL → B (m−1)L → · · · B 2L → B 1L . Therefore, the regions B m , m ≥ 2, are mode-holding regions because the iterations from B mU to B 1U or the iterations from B mL to B 1L share a same iteration mode f 1 or f 2 .

Evolution in the whole state space
We have shown that B 1 is a mode-switching region, B m := B mU ∪ B mL are mode-holding regions for all m ≥ 2, in other words, · · · B m ∪ B (m−1) ∪ · · · ∪ B 2 can be considered as a mode-holding region composed of the upper and lower parts. In the state space of the system, is there any other area outside ∪ ∞ m=1 B m ? The following proposition gives a negative answer to this question.
Proof of Proposition 4 is given in Appendix C. Proposition 4 says ∪ ∞ m=1 B m covers the whole state space U ∪ L. Therefore, by Proposition 3, all the areas outside B 1 in the state space is the mode-holding region, and the following corollary is obvious.

Corollary 1
For any initial state (α 0 , β 0 ) ∈ U ∪ L, there must exists a positive integer k * such that the trajectory of system (4) starting from(α 0 , β 0 ) must enter the region B 1 after k * -step iteration , i.e., (α k * , β k * ) ∈ B 1 . Figure 4 presents a simulation example of the iterative trajectory when (α 0 , β 0 ) ∈ U \(B 1U ∪ B 2U ). From the trajectory, we see that before it enters B 1 there is no area switching from U to L or from L to U , which implies that there is no iteration mode switching between f 1 and f 2 . Once it enters B 1 , at the next state it must jump from U to L or from L to U , which implies that the iteration mode switches from f 1 to f 2 or from f 2 to f 1 . However, the results we have obtained up to now do not tell us into which region it will falls after such a jump, mode-switching region or mode-holding region. This problem is left for the next section.

Globally attractive invariant set
In this section, we investigate the dynamic properties of system (4) after its trajectory enters the region B 1 .
Firstly, let us define and denote R = R U ∪ R L . (4).

Proposition 5 Consider dynamic system
The case of (α k , β k ) ∈ B 1U can be proved similarly.
Proof of Proposition 6 is given Appendix D. and respectively. Observing these equations, we find that the upper boundary of R U goes through the origin of the coordinate system, but the upper boundary of B 1U does not when b = 0. Combining Eqs. (7) and (8) yields It can be considered as a quartic equation of t = √ α.
Note the admissible value range of α is [0, 1 4 ) because the abscissa of the intersection of the upper boundary of R U and the line α + β = 1 is 1 4 . Considering the value range of α and using the discriminant of quartic equation, we know that when b ∈ 3−2 √ 2 2 , 1 2 , Eq. (9) has no solution; and when b ∈ 0, 3−2 √ 2 2 , Eq. (9) has only one solution given by Substituting this solution into (8) gives the corresponding ordinate of the intersection Symmetrically, the intersection coordinate of the lower boundary curves of B 1L and R L is given by Now, we are ready to prove the following theorem.

Theorem 2
The region R is a globally attractive invariant set in the state space of system (4). Proof By Corollary 1, we know that for any initial state (α 0 , β 0 ) ∈ U ∪ L, there must exists a positive integer k * such that the trajectory of system (4) starting from(α 0 , β 0 ) enters the region B 1 after k * -step iteration , i.e., (α k * , β k * ) ∈ B 1 . Then, by Proposition 5, we have (α k * +1 , β k * +1 ) ∈ R. The rest of this proof is to show that R is an invariant set, i.e., the system trajectory will remain in R for all k > k * .
By Proposition 6 we know R ⊂ B 1 ∪ B 2 . If R B 2 = ∅ i.e., R ⊂ B 1 , using Proposition 5 again we have (α k * +2 , β k * +2 ) ∈ R and so on. The theorem is thus true in this case (see Fig. 5 for illustration). Now, we just need to consider the rest case: R B 2 = ∅. For this case we just need to prove that once (α k * , β k * ) ∈ R U B 2U , its next step (α k * +1 , β k * +1 ) will not fall in the region B 1U \R U . By Proposition 7, R B 2 = ∅ implies that there is only one intersection point between two boundary curves, a part of the region R is inside B 2 , and the other part of R is outside B 1 (see Fig. 6 for illustration). Since R ⊂ B 1 ∪ B 2 by Proposition 6, to ensure that R is an invariant set in this case it suffices to prove the following fact: Once the system state falls in the region R U B 2U , its next step will not enter the region B 1U \R U . This fact is indeed true because the region B 1U \R U is on the left of the region B 2U R U (i.e., the abscissa of any point in B 1U \R U is less than the abscissa of any point in B 2U R U ), and α k+1 = 1 − (1− F(b, α k ) The following theorem characterizes the dynamic property of the system trajectory in the invariant set R.

and system (4) in R is in a switching iteration mode with the maximum beat number of 2.
Proof The proof is obvious based on Propositions 6, 7 and Theorem 2.

Asymptotic property of total error probability
The total error probability L k at layer k of the tree network is the sum of false-alarm probability and misseddetection probability at the same layer, i.e., L k = α k + β k . The total error probability is a very important performance index of detection network. Since the false-alarm probability and the missed-detection probability usually do not increase or decrease simultaneously, the asymptotic property of the total error probability with the increase in the detection network size is an important topic worthy of attention. (4). L k+1 (< , >, =)L k is equivalent to

Proposition 8 Consider dynamic system
Proof We just give the proof for the ">" part because the proof for "<" and "=" is the same. Moreover, because of the dynamic symmetry of the system, we just need to consider the case of α k ≤ β k . Since L k = α k +β k and L k+1 = 1−(1 − F(b, α k )) 2 + (F(b, β k )) 2 , we have By Eq. (1), the above inequality is also equivalent to The proposition is thus proved.

Now, let us define
Based on these definitions and Proposition 8, the proof of the following theorem is obvious. (4).

Theorem 4 Consider dynamic system
Theorem 4 provides a necessary and sufficient condition of the convergence of the total error probability. From Theorems 2 and 4, it is natural to draw the following corollary.

Corollary 2
If b = 0, then L k does not approach to zero as k → ∞.
Proof We prove this corollary by contradiction. Suppose that L k → 0 as k → ∞. By Theorem 2, we know that (α k , β k ) will remain in the invariant set R when k is large enough. If L k → 0, then α k → 0 and β k → 0. This implies that (α k , β k ) will enter a sufficiently small neighborhood of (0, 0) as k becomes large enough. Noticing that R is not related to b we know that the intersection of R with this neighborhood must belong to M as long as b = 0. By Theorem 4, once (α k , β k ) enters M, L k+1 > L k always holds. This contradicts the hypothesis L k → 0.
Although (0, 0) is an equilibrium point of dynamic system (4), Corollary 2 tells us that when there are bit errors in communication channels of the tree detection network, it is impossible to make the total error probability of the fusion center tend to zero by increasing the scale of the network.
Since M is related to b but R is not, it can be expected that when b is large enough, the entire invariant set R may fall into M, which means that when k is large enough, the total error probability L k is continuously increasing. Obviously, this is a situation that should be avoided in network design. In addition, since B 1 is the only mode-switching region in the state space, clarifying the relationship between region M and region B 1 is helpful to analyze the iterative mode of the dynamic system when (α k , β k ) falls into the non-decreasing region M for L k .

Theorem 5
The following statements are true for dynamic system (4).

Corollary 3 Consider dynamic system
2 , 1 2 , then for any initial state (α 0 , β 0 ), there must exist a positive integer k * such that the total error probability L k will continuously increasing when k ≥ k * .
In order to avoid the situation mentioned in Corollary 3, in the design of wireless sensor networks, the bit error probability must be kept in the range b < 2−

Conclusion
In this paper, the evolution of detection performance of balanced binary tree networks with channel noise is studied. Firstly, the iterative rules of false-alarm probability and missed-detection probability are modeled as a state-dependent nonlinear switched system. Then, according to the channel bit error probability b, the system state space is divided into two parts: the modeswitching region and the mode-holding region. The iterative trajectory of the system state is described, and a globally attractive invariant set which is independent of the channel bit error probability is found. The necessary and sufficient condition of the convergence of the total error probability of the detection network is obtained. Through our analysis, the following important conclusions are obtained: (1) as long as the channel bit error probability is not zero, the total error probability of the detection network will not tend to zero with the increase in the network size, which is essentially different from the conclusion obtained when the network has nodes and link failures [18]. (2) When the channel bit error probability is greater than 2− , since R\M = ∅ and M\R = ∅ (see Theorem 5), it is currently not clear whether the total error probability converges to a positive constant. Secondly, for the researchers of nonlinear dynamics, the unknown (possibly aperiodic) switching law of the system in the case of b ∈ 0, 3−2 √ 2 2 (see Theorem 3 and Fig. 8) may seem attractive. Finally, all our results are obtained under the assumption that the sensors make independent detection observations and have identical detection property, which is of course restrictive. The asymptotic detection performance of the network with correlated sensors is worth of further study.
It implies that the abscissa of the intersection point of the upper boundary of B 1U with the line α + β = 1 is less than 1 2 . Considering the fact that 1 − 2b > 0 by Assumption 2 and 1 − F(b, α) − F(b, β) ≥ 0 by (3), we get β b > 0. The proposition is thus proved.

Proof of Proposition 3
Suppose that (α k , β k ) ∈ B mU , m ≥ 2. By definition of B mU , we get that and Note that here we let D 0 (b, y) = y 2 for simplicity of notation.
By (4), it is easy to get Hence, from (10) to (11) it follows that Note that here we assume D −1 (b, F(b, x)) = x and D −1 (b, (1 − F(b, x))) = 1 − x for simplicity of notation. The above two inequalities imply that (α k+1 , β k+1 ) ∈ B (m−1)U . Due to the dynamic symmetry of the system, the case of (α k , β k ) ∈ B mL can be proved similarly.

Proof of Proposition 4
Due to the dynamic symmetry of the system, we just need to prove U ⊆ ∪ ∞ m=1 B mU . Let us check the intersection point of the upper boundary of B mU with the line α + β = 1. The upper boundary equation of B mU is given by 1 (b, (1 − F(b, α)) + D m−1 (b, F(b, β)) = 1.
Combining this equation and the line equation α + β = 1, we get In Proof of Proposition 2, we have shown that for m = 1, this equation implies β = Using mathematical induction, we can show that Eq. (12) implies which indicates that the ordinate of the intersection point of the upper boundary of B mU with line α+β = 1 is not less than 1 as m → ∞. This also implies that the ordinate of the intersection point of the upper boundary of B mU with the axis α = 0 is not less than 1.
Since the convexity of the upper boundary curve of B mU is obvious, the upper triangular of the state space of the system is below this boundary as m → ∞, i.e., U ⊆ ∞ m=1 B mU . Proposition 4 is thus proved.

Proof of Proposition 6
Due to the dynamic symmetry of the system, we just need to prove R U ⊂ B 1U ∪ B 2U , i.e, to prove the upper boundary of B 2U is always higher than the upper boundary of R U . Our proof will use the following lemma which is proved in [15].

Proof of Theorem 5
By the dynamic symmetry of the system, the proof is just provided for the region U .