Convergence Analysis for the SMC-MeMBer and SMC-CBMeMBer Filters

Abstract

The convergence for the sequential Monte Carlo (SMC) implementations of the multitarget multi-Bernoulli (MeMBer) filter and cardinality-balanced MeMBer (CBMeMBer) filters is studied here. This paper proves that the SMC-MeMBer and SMC-CBMeMBer filters, respectively, converge to the true MeMBer and CBMeMBer filters in the mean-square sense and the corresponding bounds for the mean-square errors are given. The significance of this paper is in theory to present the convergence results of the SMC-MeMBer and SMC-CBMeMBer filters and the conditions under which the two filters satisfy mean-square convergence.

1. Introduction

Recently, the random finite-set- (RFS-) based multitarget tracking (MTT) approaches [1] have attracted extensive attention. Although theoretically solid, the RFS-based approaches usually involve intractable computations. By introducing the finite-set statistics (FISSTs) [2], Mahler developed the probability hypothesis density (PHD) [3], and cardinalized PHD (CPHD) [4] filters, which have been shown to be a computationally tractable alternative to full multitarget Bayes filters in the RFS framework. The sequential Monte Carlo (SMC) implementations for the PHD and CPHD filters were devised by Zajic and Mahler [5], Sidenbladh [6], and Vo et al. [7]. Vo et al. [8, 9] devised the Gaussian mixture (GM) implementation for the PHD and CPHD filters under the linear, Gaussian assumption on target dynamics, birth process, and sensor model. However, the SMC-PHD and SMC-CPHD approaches require clustering to extract state estimates from the particle population, which is expensive and unreliable [10, 11].

In 2007, Mahler proposed the multitarget multi-Bernoulli (MeMBer) [2] recursion, which is an approximation to the full multitarget Bayes recursion using multi-Bernoulli RFSs under low clutter density scenarios. In 2009, Vo et al. showed that the MeMBer filter overestimates the number of targets and proposed a cardinality-balanced MeMBer (CBMeMBer) filter [12] to reduce the cardinality bias. Then, the SMC and GM implementations for the MeMBer and CBMeMBer filters were, respectively, proposed in nonlinear and linear-Gaussian dynamic and measurement models. The key advantage of this approach is that the multi-Bernoulli representation allows reliable and inexpensive extraction of state estimates. The Monte Carlo simulations given by Vo et al. showed that the SMC-CBMeMBer filter outperforms the SMC-CPHD (and hence SMC-PHD) filter despite having smaller complexity under certain range of signal settings.

Although the convergence results for the SMC-PHD and GM-PHD filters were established by Clark and Bell [13] in 2006 and by Clark and Vo [14] in 2007, respectively, there have been no results showing the asymptotic convergence for the SMC-MeMBer and SMC-CBMeMBer filters. This paper demonstrates the mean-square convergence of the errors [15–17] for the two filters. In other words, given simple sufficient conditions, the approximation error of the multi-Bernoulli parameter set comprised of a set of weighted samples is proved to converge to zero as the number of the samples tends to infinity at each stage of the two algorithms. In addition, the corresponding bounds for the mean-square errors are obtained.

2. MeMBer and CBMeMBer Filters

A Bernoulli RFS 𝑌^(𝑖) has probability 1−𝑟^(𝑖) of being empty, and probability 𝑟^(𝑖) (0≤𝑟^(𝑖)≤1) of being a singleton whose only element is distributed according to a probability density 𝑝^(𝑖). The probability density of 𝑌^(𝑖) is 𝜋𝑌^(𝑖)=1−𝑟^(𝑖)𝑟𝑌^(𝑖)=∅,_(𝑖)𝑝^(𝑖)𝐲_𝑖𝑌^(𝑖)=𝐲_𝑖.(2.1)

A multi-Bernoulli RFS 𝑌 is a union of a fixed number of independent Bernoulli RFSs 𝑌^(𝑖), 𝑖=1,…,𝑀, that is, 𝑌=⋃^𝑀𝑖=1𝑌^(𝑖). 𝑌 is thus completely described by the multi-Bernoulli parameter set {(𝑟^(𝑖),𝑝^(𝑖))}^𝑀𝑖=1 with the mean cardinality ∑^𝑀𝑖=1𝑟^(𝑖)  and the probability density [2]: 𝜋(𝑌)=^𝑀_𝑗=11−𝑟^(𝑗)_{1≤𝑖1≠⋅⋅⋅≠𝑖𝑛𝑛≤𝑀}_𝑗=1𝑟^(𝑖𝑗)𝑝^(𝑖𝑗)𝐲_𝑗1−𝑟^(𝑖𝑗).(2.2)

Throughout this paper, we abbreviate a probability density of the form (2.2) by 𝜋={(𝑟^(𝑖),𝑝^(𝑖))}^𝑀𝑖=1.

By approximating the multitarget RFS as a multi-Bernoulli RFS at each time step, Mahler proposed the MeMBer recursion, which propagated the multi-Bernoulli parameters of the posterior multitarget density forward in time [2]. The MeMBer filter is summarized as follows.

MeMBer Prediction
If at time 𝑘−1, the posterior multitarget density is a multi-Bernoulli of the form 𝜋_𝑘−1={(𝑟^{(𝑖)𝑘−1},𝑝^{(𝑖)𝑘−1})}^𝑀𝑘−1_𝑖=1, then the predicted multitarget density is also a multi-Bernoulli and is given by 𝜋_{𝑘∣𝑘−1}=𝑟^{(𝑖)𝑃,𝑘∣𝑘−1},𝑝^{(𝑖)𝑃,𝑘∣𝑘−1}^𝑀𝑘−1_𝑖=1𝑟^{(𝑖)Γ,𝑘},𝑝^{(𝑖)Γ,𝑘}^𝑀Γ,𝑘_𝑖=1,(2.3) where {(𝑟^{(𝑖)Γ,𝑘},𝑝^{(𝑖)Γ,𝑘})}^𝑀Γ,𝑘_𝑖=1 are the parameters of the multi-Bernoulli RFS of births at time 𝑘: 𝑝𝑟^{(𝑖)𝑃,𝑘∣𝑘−1}=𝑟^{(𝑖)𝑘−1}𝑝^{(𝑖)𝑘−1},𝑝_𝑆,𝑘,𝑖=1,…,𝑀^𝑘−1,(2.4)_{(𝑖)𝑃,𝑘∣𝑘−1}𝐱_𝑘=𝑓^{𝑘∣𝑘−1}𝐱_𝑘∣⋅,𝑝^{(𝑖)𝑘−1}𝑝_𝑆,𝑘𝑝^{(𝑖)𝑘−1},𝑝_𝑆,𝑘,𝑖=1,…,𝑀^𝑘−1.(2.5)

MeMBer Update
If at time 𝑘, the predicted multitarget density is a multi-Bernoulli of the form 𝜋_{𝑘∣𝑘−1}={(𝑟^{(𝑖)𝑘∣𝑘−1},𝑝^{(𝑖)𝑘∣𝑘−1})}^{𝑀𝑘∣𝑘−1}_𝑖=1; then the posterior multitarget density can be approximated by a multi-Bernoulli as follows: 𝜋_𝑘≈𝑟^{(𝑖)𝐿,𝑘},𝑝^{(𝑖)𝐿,𝑘}^{𝑀𝑘∣𝑘−1}_𝑖=1𝑟^𝑈,𝑘𝐳_𝑘,𝑝_𝑈,𝑘⋅;𝐳_𝑘_{𝐳𝑘∈𝑍𝑘},(2.6) where, 𝑟^{(𝑖)𝐿,𝑘}=𝑟^{(𝑖)𝑘∣𝑘−1}1−𝑝^{(𝑖)𝑘∣𝑘−1},𝑝_𝐷,𝑘1−𝑟^{(𝑖)𝑘∣𝑘−1}𝑝^{(𝑖)𝑘∣𝑘−1},𝑝_𝐷,𝑘,𝑖=1,…,𝑀^{𝑘∣𝑘−1}𝑝,(2.7)_{(𝑖)𝐿,𝑘}𝐱_𝑘=𝑝^{(𝑖)𝑘∣𝑘−1}𝐱_𝑘1−𝑝^𝐷,𝑘𝐱_𝑘1−𝑝^{(𝑖)𝑘∣𝑘−1},𝑝_𝐷,𝑘,𝑖=1,…,𝑀^{𝑘∣𝑘−1}𝑟,(2.8)_𝑈,𝑘𝐳_𝑘=1𝜅_𝑘𝐳_𝑘+∑^{𝑀𝑘∣𝑘−1}_𝑖=1𝑟^{(𝑖)𝛼,𝑘}𝐳_𝑘^{𝑀𝑘∣𝑘−1}_𝑖=1𝑝𝑟^{(𝑖)𝛼,𝑘}𝐳_𝑘,𝐳_𝑘∈𝑍_𝑘,(2.9)_𝑈,𝑘𝐱_𝑘;𝐳_𝑘=1∑^{𝑀𝑘∣𝑘−1}_𝑖=1𝑟^{(𝑖)𝛼,𝑘}𝐳_𝑘^{𝑀𝑘∣𝑘−1}_𝑖=1𝑝𝑝^{(𝑖)𝛼,𝑘}𝐱_𝑘;𝐳_𝑘,𝐳_𝑘∈𝑍_𝑘,(2.10)_{(𝑖)𝛼,𝑘}𝐱_𝑘;𝐳_𝑘=𝑟^{(𝑖)𝑘∣𝑘−1}𝑝^{(𝑖)𝑘∣𝑘−1}𝐱_𝑘𝜓_{𝑘,𝐳𝑘}𝐱_𝑘1−𝑟^{(𝑖)𝑘∣𝑘−1}𝑝^{(𝑖)𝑘∣𝑘−1},𝑝_𝐷,𝑘,𝑖=1,…,𝑀^{𝑘∣𝑘−1}𝑟,(2.11)_{(𝑖)𝛼,𝑘}𝐳_𝑘=𝑝^{(𝑖)𝛼,𝑘}𝐱_𝑘;𝐳_𝑘,1=𝑟^{(𝑖)𝑘∣𝑘−1}𝑝^{(𝑖)𝑘∣𝑘−1},𝜓_{𝑘,𝐳𝑘}1−𝑟^{(𝑖)𝑘∣𝑘−1}𝑝^{(𝑖)𝑘∣𝑘−1},𝑝_𝐷,𝑘,𝑖=1,…,𝑀^{𝑘∣𝑘−1}.(2.12)

By correcting the cardinality bias in the 𝑟_𝑈,𝑘(𝐳_𝑘) of the MeMBer update step, Vo et al. proposed the CBMeMBer filter [12]. The CBMeMBer recursions are the same as the MeMBer recursions except the update of 𝑟_𝑈,𝑘(𝐳_𝑘), which is revised as𝑟^∗𝑈,𝑘𝐳_𝑘=1𝜅_𝑘𝐳_𝑘+∑^{𝑀𝑘∣𝑘−1}_𝑖=1𝑟^{(𝑖)𝛼,𝑘}𝐳_𝑘^{𝑀𝑘∣𝑘−1}_𝑖=11−𝑟^{(𝑖)𝑘∣𝑘−1}𝑟^{(𝑖)𝛼,𝑘}𝐳_𝑘1−𝑟^{(𝑖)𝑘∣𝑘−1}𝑝^{(𝑖)𝑘∣𝑘−1},𝑝_𝐷,𝑘.(2.13)

Note that not (38) in [12] but (2.10) in our paper is used in the CBMeMBer update step here. The reasons are (1) the (38) in [12] and the (2.10) in our paper are both the approximations of (36) in [12] under the same assumption ⟨𝑝^{(𝑖)𝑘∣𝑘−1},𝑝_𝐷,𝑘⟩≈1, but the latter is more precise than former; (2) the (38) in [12] is unbounded at 𝑟^{(𝑖)𝑘∣𝑘−1}=1 while (2.10) in our paper is bounded at 𝑟^{(𝑖)𝑘∣𝑘−1}=1 as long as 𝑝_𝐷,𝑘(𝐱_𝑘)≠1.

For the multi-Bernoulli representation 𝜋_𝑘={(𝑟^𝑘(𝑖),𝑝^𝑘(𝑖))}^𝑀𝑘_𝑖=1, the probability 𝑟^𝑘(𝑖) indicates how likely the 𝑖th hypothesized track is a true track, and the posterior density 𝑝^𝑘(𝑖) describes the distribution of the estimated current state of the track. Hence, ∑^𝑀𝑘_𝑖=1𝑟^𝑘(𝑖) denotes the multitarget number and the multitarget state estimate can be obtained by choosing the means or modes from the posterior densities of the hypothesized tracks with existence probabilities exceeding a given threshold.

3. SMC-MeMBer and SMC-CBMeMBer Filters

The SMC implementations of the MeMBer and CBMeMBer recursions are summarized as follows.

SMC-MeMBer and SMC-CBMeMBer Predictions
Suppose that at time 𝑘−1 the (multi-Bernoulli) posterior multitarget density 𝜋_𝑘−1={(𝑟^{(𝑖),𝐿(𝑖)𝑘−1}_𝑘−1,𝑝^{(𝑖),𝐿(𝑖)𝑘−1}_𝑘−1)}^𝑀𝑘−1_𝑖=1 is given and each 𝑝^{(𝑖),𝐿(𝑖)𝑘−1}_𝑘−1, 𝑖=1,…,𝑀_𝑘−1, is comprised of a set of weighted samples {𝜔^{(𝑖,𝑗)𝑘−1},𝐱^{(𝑖,𝑗)𝑘−1}}^{𝐿(𝑖)𝑘−1}_𝑗=1: 𝑝^{(𝑖),𝐿(𝑖)𝑘−1}_𝑘−1𝐱_𝑘=^{𝐿(𝑖)𝑘−1}_𝑗=1𝜔^{(𝑖,𝑗)𝑘−1}𝛿_𝐱^{(𝑖,𝑗)𝑘−1}𝐱_𝑘,𝑖=1,…,𝑀_𝑘−1.(3.1)
Then, given proposal densities 𝑞^𝑘(𝑖)(⋅∣𝐱^{(𝑖,𝑗)𝑘−1},𝑍_𝑘) and 𝑏^𝑘(𝑖)(⋅∣𝑍_𝑘), the predicted (multi-Bernoulli) multitarget density 𝜋_{𝑘∣𝑘−1}={(𝑟^{(𝑖),𝐿(𝑖)𝑘−1}_{𝑃,𝑘∣𝑘−1},𝑝^{(𝑖),𝐿(𝑖)𝑘−1}_{𝑃,𝑘∣𝑘−1})}^𝑀𝑘−1_𝑖=1∪{(𝑟^{(𝑖),𝐿(𝑖)Γ,𝑘}_Γ,𝑘,𝑝^{(𝑖),𝐿(𝑖)Γ,𝑘}_Γ,𝑘)}^𝑀Γ,𝑘_𝑖=1 can be computed as follows: 𝑟^{(𝑖),𝐿(𝑖)𝑘−1}_{𝑃,𝑘∣𝑘−1}=𝑟^{(𝑖),𝐿(𝑖)𝑘−1}_𝑘−1𝑝^{(𝑖),𝐿(𝑖)𝑘−1𝑘−1},𝑝_𝑆,𝑘,𝑖=1,…,𝑀^𝑘−1𝑝,(3.2)_{(𝑖),𝐿(𝑖)𝑘−1𝑃,𝑘∣𝑘−1}𝑝_{(𝑖),𝐿(𝑖)Γ,𝑘Γ,𝑘}𝐱_𝑘=^{𝐿(𝑖)𝑘−1}_𝑗=1𝜔^{(𝑖,𝑗)𝑃,𝑘∣𝑘−1}𝛿_𝐱^{(𝑖,𝑗)𝑃,𝑘∣𝑘−1}𝐱_𝑘,𝑖=1,…,𝑀_𝑘−1,𝐱_𝑘=^{𝐿(𝑖)Γ,𝑘}_𝑗=1𝜔^{(𝑖,𝑗)Γ,𝑘}𝛿_𝐱^{(𝑖,𝑗)Γ,𝑘}𝐱_𝑘,𝑖=1,…,𝑀_Γ,𝑘,(3.3) where 𝑟^{(𝑖),𝐿(𝑖)Γ,𝑘}_Γ,𝑘 (𝑖=1,…,𝑀_Γ,𝑘) is given by birth model; 𝐱^{(𝑖,𝑗)𝑃,𝑘∣𝑘−1}, 𝜔^{(𝑖,𝑗)𝑃,𝑘∣𝑘−1} (𝑖=1,…,𝑀_𝑘−1) and 𝐱^{(𝑖,𝑗)Γ,𝑘}, 𝜔^{(𝑖,𝑗)Γ,𝑘} (𝑖=1,…,𝑀_Γ,𝑘) are, respectively, given by 𝐱^{(𝑖,𝑗)𝑃,𝑘∣𝑘−1}∼𝑞^𝑘(𝑖)⋅∣𝐱^{(𝑖,𝑗)𝑘−1},𝑍_𝑘𝜔,𝑗=_{(𝑖,𝑗)𝑃,𝑘∣𝑘−1}1,…,𝐿^{(𝑖)𝑘−1},𝐱^{(𝑖,𝑗)Γ,𝑘}∼𝑏^𝑘(𝑖)⋅∣𝑍_𝑘,𝑗=1,…,𝐿^{(𝑖)Γ,𝑘},(3.4)=1∑^{𝐿(𝑖)𝑘−1}_𝑗=1𝜔^{(𝑖,𝑗)𝑃,𝑘∣𝑘−1}𝜔_{(𝑖,𝑗)𝑃,𝑘∣𝑘−1}𝜔^{(𝑖,𝑗)𝑃,𝑘∣𝑘−1},=𝜔^{(𝑖,𝑗)𝑘−1}𝑓_{𝑘∣𝑘−1}𝐱^{(𝑖,𝑗)𝑃,𝑘∣𝑘−1}∣𝐱^{(𝑖,𝑗)𝑘−1}𝑝^𝑆,𝑘𝐱^{(𝑖,𝑗)𝑘−1}𝑞^𝑘(𝑖)𝐱^{(𝑖,𝑗)𝑃,𝑘∣𝑘−1}∣𝐱^{(𝑖,𝑗)𝑘−1},𝑍_𝑘,𝑗=1,…,𝐿^{(𝑖)𝑘−1}𝜔_{(𝑖,𝑗)Γ,𝑘},(3.5)=1∑^{𝐿(𝑖)Γ,𝑘}_𝑗=1𝜔^{(𝑖,𝑗)Γ,𝑘}𝜔^{(𝑖,𝑗)Γ,𝑘},𝜔^{(𝑖,𝑗)Γ,𝑘}=𝑝^Γ,𝑘𝐱^{(𝑖,𝑗)Γ,𝑘}𝑏^𝑘(𝑖)𝐱^{(𝑖,𝑗)Γ,𝑘}∣𝑍_𝑘,𝑗=1,…,𝐿^{(𝑖)Γ,𝑘}.(3.6)

SMC-MeMBer and SMC-CBMeMBer Updates
Suppose that at time 𝑘 the predicted (multi-Bernoulli) multitarget density 𝜋_{𝑘∣𝑘−1}={(𝑟^{(𝑖),𝐿(𝑖)𝑘∣𝑘−1}_{𝑘∣𝑘−1},𝑝^{(𝑖),𝐿(𝑖)𝑘∣𝑘−1}_{𝑘∣𝑘−1})}^{𝑀𝑘∣𝑘−1}_𝑖=1 is given and each 𝑝^{(𝑖),𝐿(𝑖)𝑘∣𝑘−1}_{𝑘∣𝑘−1}, 𝑖=1,…,𝑀_{𝑘∣𝑘−1}, is comprised of a set of weighted samples {𝜔^{(𝑖,𝑗)𝑘∣𝑘−1},𝐱^{(𝑖,𝑗)𝑘∣𝑘−1}}^{𝐿(𝑖)𝑘∣𝑘−1}_𝑗=1: 𝑝^{(𝑖),𝐿(𝑖)𝑘∣𝑘−1}_{𝑘∣𝑘−1}𝐱_𝑘=^{𝐿(𝑖)𝑘∣𝑘−1}_𝑗=1𝜔^{(𝑖,𝑗)𝑘∣𝑘−1}𝛿_𝐱^{(𝑖,𝑗)𝑘∣𝑘−1}𝐱_𝑘,𝑖=1,…,𝑀_{𝑘∣𝑘−1}.(3.7)
Then, the multi-Bernoulli approximation of the SMC-MeMBer-updated multitarget density 𝜋_𝑘≈{(𝑟^{(𝑖),𝐿(𝑖)𝑘∣𝑘−1}_𝐿,𝑘,𝑝^{(𝑖),𝐿(𝑖)𝑘∣𝑘−1}_𝐿,𝑘)}^{𝑀𝑘∣𝑘−1}_𝑖=1∪{(𝑟^{𝐿(𝑖)𝑘∣𝑘−1}_𝑈,𝑘(𝐳_𝑘),𝑝^{𝐿(𝑖)𝑘∣𝑘−1}_𝑈,𝑘(⋅;𝐳_𝑘))}_{𝐳𝑘∈𝑍𝑘} and SMC-CBMeMBer-updated multitarget density 𝜋^∗𝑘≈{(𝑟^{(𝑖),𝐿(𝑖)𝑘∣𝑘−1}_𝐿,𝑘,𝑝^{(𝑖),𝐿(𝑖)𝑘∣𝑘−1}_𝐿,𝑘)}^{𝑀𝑘∣𝑘−1}_𝑖=1∪{(𝑟^{∗,𝐿(𝑖)𝑘∣𝑘−1}_𝑈,𝑘(𝐳_𝑘),𝑝^{𝐿(𝑖)𝑘∣𝑘−1}_𝑈,𝑘(⋅;𝐳_𝑘))}_{𝐳𝑘∈𝑍𝑘} can be computed as follows: 𝑟^{(𝑖),𝐿(𝑖)𝑘∣𝑘−1}_𝐿,𝑘=𝑟^{(𝑖),𝐿(𝑖)𝑘∣𝑘−1}_{𝑘∣𝑘−1}1−𝑝^{(𝑖),𝐿(𝑖)𝑘∣𝑘−1}_{𝑘∣𝑘−1},𝑝_𝐷,𝑘1−𝑟^{(𝑖),𝐿(𝑖)𝑘∣𝑘−1}_{𝑘∣𝑘−1}𝑝^{(𝑖),𝐿(𝑖)𝑘∣𝑘−1𝑘∣𝑘−1},𝑝_𝐷,𝑘,𝑖=1,…,𝑀^{𝑘∣𝑘−1}𝑝,(3.8)_{(𝑖),𝐿(𝑖)𝑘∣𝑘−1𝐿,𝑘}𝐱_𝑘=^{𝐿(𝑖)𝑘∣𝑘−1}_𝑗=1𝜔^{(𝑖,𝑗)𝑘∣𝑘−1}1−𝑝_𝐷,𝑘𝐱^{(𝑖,𝑗)𝑘∣𝑘−1}1−𝑝^{(𝑖),𝐿(𝑖)𝑘∣𝑘−1}_{𝑘∣𝑘−1},𝑝_𝐷,𝑘𝛿^{𝐱(𝑖,𝑗)𝑘∣𝑘−1}𝑟𝐱_𝑘,𝑖=1,…,𝑀_{𝑘∣𝑘−1},(3.9)_{𝐿(𝑖)𝑘∣𝑘−1𝑈,𝑘}𝐳_𝑘=1𝜅_𝑘𝐳_𝑘+∑^{𝑀𝑘∣𝑘−1}_𝑖=1𝑟^{(𝑖),𝐿(𝑖)𝑘∣𝑘−1}_𝛼,𝑘𝐳_𝑘^{𝑀𝑘∣𝑘−1}_𝑖=1𝑟𝑟^{(𝑖),𝐿(𝑖)𝑘∣𝑘−1}_𝛼,𝑘𝐳_𝑘,𝐳_𝑘∈𝑍_𝑘,(3.10)_{∗,𝐿(𝑖)𝑘∣𝑘−1𝑈,𝑘}𝐳_𝑘=1𝜅×_𝑘𝐳_𝑘_{𝑀𝑘∣𝑘−1}+∑^{𝑀𝑘∣𝑘−1}_𝑖=1𝑟^{(𝑖),𝐿(𝑖)𝑘∣𝑘−1}_𝛼,𝑘𝐳_𝑘_𝑖=11−𝑟^{(𝑖),𝐿(𝑖)𝑘∣𝑘−1𝑘∣𝑘−1}𝑟^{(𝑖),𝐿(𝑖)𝑘∣𝑘−1𝛼,𝑘}𝐳_𝑘1−𝑟^{(𝑖),𝐿(𝑖)𝑘∣𝑘−1}_{𝑘∣𝑘−1}𝑝^{(𝑖),𝐿(𝑖)𝑘∣𝑘−1𝑘∣𝑘−1},𝑝_𝐷,𝑘,𝐳^𝑘𝑝∈𝑍_𝑘,(3.11)_{𝐿(𝑖)𝑘∣𝑘−1𝑈,𝑘}𝐱_𝑘;𝐳_𝑘=1∑^{𝑀𝑘∣𝑘−1}_𝑖=1𝑟^{(𝑖),𝐿(𝑖)𝑘∣𝑘−1}_𝛼,𝑘𝐳_𝑘^{𝑀𝑘∣𝑘−1}_𝑖=1𝑝^{(𝑖),𝐿(𝑖)𝑘∣𝑘−1}_𝛼,𝑘𝐱_𝑘;𝐳_𝑘,𝐳_𝑘∈𝑍_𝑘,(3.12) where, 𝑝^{(𝑖),𝐿(𝑖)𝑘∣𝑘−1}_𝛼,𝑘𝐱_𝑘;𝐳_𝑘=^{𝐿(𝑖)𝑘∣𝑘−1}_𝑗=1𝜔^{(𝑖,𝑗)𝑘∣𝑘−1}𝑟^{(𝑖),𝐿(𝑖)𝑘∣𝑘−1}_{𝑘∣𝑘−1}𝜓_{𝑘,𝐳𝑘}𝐱^{(𝑖,𝑗)𝑘∣𝑘−1}1−𝑟^{(𝑖),𝐿(𝑖)𝑘∣𝑘−1}_{𝑘∣𝑘−1}𝑝^{(𝑖),𝐿(𝑖)𝑘∣𝑘−1𝑘∣𝑘−1},𝑝_𝐷,𝑘𝛿^{𝐱(𝑖,𝑗)𝑘∣𝑘−1}𝑟𝐱_𝑘,𝑖=1,…,𝑀_{𝑘∣𝑘−1},(3.13)_{(𝑖),𝐿(𝑖)𝑘∣𝑘−1𝛼,𝑘}𝐳_𝑘==𝑟𝑝^{(𝑖),𝐿(𝑖)𝑘∣𝑘−1}_𝛼,𝑘𝐱_𝑘;𝐳_𝑘,1_{(𝑖),𝐿(𝑖)𝑘∣𝑘−1𝑘∣𝑘−1}𝑝^{(𝑖),𝐿(𝑖)𝑘∣𝑘−1𝑘∣𝑘−1},𝜓_{𝑘,𝐳𝑘}1−𝑟^{(𝑖),𝐿(𝑖)𝑘∣𝑘−1}_{𝑘∣𝑘−1}𝑝^{(𝑖),𝐿(𝑖)𝑘∣𝑘−1𝑘∣𝑘−1},𝑝_𝐷,𝑘,𝑖=1,…,𝑀^{𝑘∣𝑘−1}.(3.14)

Resampling
To reduce the effect of degeneracy, we resample the particles for the multi-Bernoulli parameter set after the update step.

4. Convergence of the Mean-Square Errors for the SMC-MeMBer and SMC-CBMeMBer Filters

To show the convergence results for the SMC-MeMBer and SMC-CBMeMBer filters, certain conditions on the functions need to be met:(1)the transition kernel 𝑓_{𝑘∣𝑘−1}(𝐱_𝑘∣𝐱_𝑘−1) satisfies the Feller property [18], that is, for all 𝜑∈𝐶_𝑏(ℝ^𝑑), ∫𝜑(𝐱_𝑘−1)𝜙_{𝑘∣𝑘−1}(𝐱_𝑘∣𝐱_𝑘−1)𝑑𝐱_𝑘−1∈𝐶_𝑏(ℝ^𝑑);(2)single-sensor/target likelihood density 𝜓_{𝑘,𝐳𝑘}(𝐱_𝑘)∈𝐵(ℝ^𝑑);(3)𝑄^𝑘(𝑖) are rational-valued random variables such that there exists 𝑝>1, some constant 𝐶, and 𝛼<𝑝−1 so that 𝐸⎡⎢⎢⎣|||||^𝑁_𝑖=1𝑄^𝑘(𝑖)−𝑁𝜔^𝑘(𝑖)𝑞^(𝑖)|||||^𝑝⎤⎥⎥⎦≤𝐶𝑁_𝛼‖𝑞‖^𝑝,with^𝑁_𝑖=1𝑄^𝑘(𝑖)=𝑁(4.1) for all vectors 𝑞=(𝑞⁽¹⁾,…,𝑞^(𝑁));(4)the importance sampling ratios are bounded, that is, there exists constants 𝐵₁ and 𝐵₂ such that ‖𝑝^{(𝑖)Γ,𝑘}/𝑏^𝑘(𝑖)‖≤𝐵₁, 𝑖=1,…,𝑀_Γ,𝑘, and ‖𝑓_{𝑘∣𝑘−1}/𝑞^𝑘(𝑖)‖≤𝐵₂, 𝑖=1,…,𝑀_𝑘−1;(5)the resampling strategy is multinomial and hence unbiased [19].

First, the convergence of the mean-square errors for the initialization steps of the two filters can easily be established by Lemma  0 in [13]. Assuming that at time 𝑘=0, we can sample exactly from the initial distribution 𝑝^0(𝑖) (𝑖=1,…,𝑀₀). Then, for all 𝜑∈𝐵(ℝ^𝑑), 𝐸𝑟^{(𝑖),𝐿0(𝑖)}₀−𝑟^0(𝑖)²≤𝐿𝑐^00(𝑖)𝐸𝑝_{(𝑖),𝐿0(𝑖)0},𝑖=1,…,𝑀₀,,𝜑−𝑝^0(𝑖),𝜑²≤‖𝜑‖²𝑑₀𝐿^0(𝑖),𝑖=1,…,𝑀₀(4.2) hold for some real numbers 𝑐₀>0 and 𝑑₀>0 which are independent of the number 𝐿^0(𝑖) of the sampled particles at time 𝑘=0, 𝑖=1,…,𝑀₀.

Also, the convergence of the mean-square errors for the resampling steps of the two filters can easily be established by Assumption 5 and Lemma  5 in [19].

The main difficulty and greatest challenge is to prove the mean-square convergence for the prediction steps and update steps of the two filters. They are, respectively, established by Propositions 4.1 and 4.2.

Proposition 4.1. Suppose that, for all 𝜑∈𝐵(ℝ^𝑑), 𝐸𝑟^{(𝑖),𝐿(𝑖)𝑘−1}_𝑘−1−𝑟^{(𝑖)𝑘−1}²≤𝐿𝑐^{𝑘−1(𝑖)𝑘−1}𝐸𝑝_{(𝑖),𝐿(𝑖)𝑘−1𝑘−1},𝑖=1,…,𝑀_𝑘−1,(4.3),𝜑−𝑝^{(𝑖)𝑘−1},𝜑²≤‖𝜑‖²𝑑_𝑘−1𝐿^{(𝑖)𝑘−1},𝑖=1,…,𝑀_𝑘−1,(4.4) hold for some real numbers 𝑐_𝑘−1>0 and 𝑑_𝑘−1>0 which are independent of the number 𝐿^{(𝑖)𝑘−1} of the resampled particles at time 𝑘−1, 𝑖=1,…,𝑀_𝑘−1.

Then, after the prediction steps of the SMC-MeMBer and SMC-CBMeMBer filters at time 𝑘:𝐸𝑟^{(𝑖),𝐿(𝑖)𝑘−1}_{𝑃,𝑘∣𝑘−1}−𝑟^{(𝑖)𝑃,𝑘∣𝑘−1}²≤𝑐𝐿^{𝑃,𝑘∣𝑘−1(𝑖)𝑘−1}𝐸𝑝_{(𝑖),𝐿(𝑖)𝑘−1𝑃,𝑘∣𝑘−1},𝑖=1,…,𝑀_𝑘−1,(4.5),𝜑−𝑝^{(𝑖)𝑃,𝑘∣𝑘−1},𝜑²≤‖𝜑‖²𝑑_{𝑃,𝑘∣𝑘−1}𝐿^{(𝑖)𝑘−1}𝐸𝑝_{(𝑖),𝐿(𝑖)Γ,𝑘Γ,𝑘∣𝑘−1},𝑖=1,…,𝑀_𝑘−1,(4.6),𝜑−𝑝^{(𝑖)Γ,𝑘∣𝑘−1},𝜑²≤‖𝜑‖²𝑑_Γ,𝑘𝐿^{(𝑖)Γ,𝑘},𝑖=1,…,𝑀_Γ,𝑘,(4.7) hold for a constant 𝑑_Γ,𝑘>0 and some real numbers 𝑐_{𝑃,𝑘∣𝑘−1}>0 and 𝑑_{𝑃,𝑘∣𝑘−1}>0 which are independent of 𝐿^{(𝑖)𝑘−1}, 𝑖=1,…,𝑀_𝑘−1. 𝑐_{𝑃,𝑘∣𝑘−1} and 𝑑_{𝑃,𝑘∣𝑘−1} are defined by (A.8) and (A.18), respectively. The proof of Proposition 4.1 can be found in Appendix A.1.

Proposition 4.2. Suppose that, for all 𝜑∈𝐵(ℝ^𝑑), 𝐸𝑟^{(𝑖),𝐿(𝑖)𝑘∣𝑘−1}_{𝑘∣𝑘−1}−𝑟^{(𝑖)𝑘∣𝑘−1}²≤𝐿𝑐^{𝑘∣𝑘−1(𝑖)𝑘∣𝑘−1}𝐸𝑝_{(𝑖),𝐿(𝑖)𝑘∣𝑘−1𝑘∣𝑘−1},𝑖=1,…,𝑀_{𝑘∣𝑘−1},(4.8),𝜑−𝑝^{(𝑖)𝑘∣𝑘−1},𝜑²≤‖𝜑‖²𝑑_{𝑘∣𝑘−1}𝐿^{(𝑖)𝑘∣𝑘−1},𝑖=1,…,𝑀_{𝑘∣𝑘−1}(4.9) hold for some real numbers 𝑐_{𝑘∣𝑘−1}>0 and 𝑑_{𝑘∣𝑘−1}>0 which are independent of the number 𝐿^{(𝑖)𝑘∣𝑘−1} of the predicted particles, 𝑖=1,…,𝑀_{𝑘∣𝑘−1}. Then, after the update steps of the SMC-MeMBer and SMC-CBMeMBer filters at time 𝑘: 𝐸𝑟^{(𝑖),𝐿(𝑖)𝑘∣𝑘−1}_𝐿,𝑘−𝑟^{(𝑖)𝐿,𝑘}²≤𝐿𝑐^{𝐿,𝑘(𝑖)𝑘∣𝑘−1}𝐸𝑝_{(𝑖),𝐿(𝑖)𝑘∣𝑘−1𝐿,𝑘},𝑖=1,…,𝑀_{𝑘∣𝑘−1},(4.10),𝜑−𝑝^{(𝑖)𝐿,𝑘},𝜑²≤‖𝜑‖²𝑑_𝐿,𝑘𝐿^{(𝑖)𝑘∣𝑘−1}𝐸𝑟,𝑖=1,…,𝑀_{𝑘∣𝑘−1},(4.11)_{𝐿(𝑖)𝑘∣𝑘−1𝑈,𝑘}(𝐳_𝑘)−𝑟_𝑈,𝑘𝐳_𝑘²≤𝐿𝑐^{𝑈,𝑘min𝑘∣𝑘−1}𝐸𝑟,𝐳_𝑘∈𝑍_𝑘,(4.12)_{∗,𝐿(𝑖)𝑘∣𝑘−1𝑈,𝑘}(𝐳_𝑘)−𝑟^∗𝑈,𝑘𝐳_𝑘²≤𝐿𝑐^{∗𝑈,𝑘min𝑘∣𝑘−1}𝐸𝑝_{𝐿(𝑖)𝑘∣𝑘−1𝑈,𝑘},𝐳_𝑘∈𝑍_𝑘,(4.13)⋅;𝐳_𝑘,𝜑−𝑝_𝑈,𝑘⋅;𝐳_𝑘,𝜑²≤‖𝜑‖²𝑑_𝑈,𝑘𝐿^{min𝑘∣𝑘−1},𝐳_𝑘∈𝑍_𝑘(4.14) hold for some real numbers 𝑐_𝐿,𝑘>0, 𝑑_𝐿,𝑘>0, 𝑐_𝑈,𝑘>0, 𝑐^∗𝑈,𝑘>0, and 𝑑_𝑈,𝑘>0, which are independent of 𝐿^{(𝑖)𝑘∣𝑘−1}. 𝑐_𝐿,𝑘, 𝑑_𝐿,𝑘, 𝑐_𝑈,𝑘, 𝑐^∗𝑈,𝑘 and 𝑑_𝑈,𝑘 are defined by (A.29), (A.35), (A.47), (A.55), and (A.61), respectively. 𝐿^{min𝑘∣𝑘−1}=min(𝐿^{(1)𝑘∣𝑘−1},…,L^{(𝑀𝑘∣𝑘−1)𝑘∣𝑘−1}), min(⋅) denotes the minimum. In addition, 𝑐_𝑈,𝑘, 𝑐^∗𝑈,𝑘, and 𝑑_𝑈,𝑘 depend on the number of targets and decrease with the increase of the target number. From (A.47) and (A.55), it can also be seen that 𝑐^∗𝑈,𝑘≥𝑐_𝑈,𝑘. It indicates that 𝑟^{∗,𝐿(𝑖)𝑘∣𝑘−1}_𝑈,𝑘(𝐳_𝑘) may need more particles than 𝑟^{𝐿(𝑖)𝑘∣𝑘−1}_𝑈,𝑘(𝐳_𝑘) to achieve the same mean-square error bound. The proof of the Proposition 4.2 can be found in Appendix A.2.

Propositions 4.1 and 4.2 show that the bounds for the mean-square error of the SMC-MeMBer and SMC-CBMeMBer prediction steps and update steps at each stage depend on the number of particles. The mean-square errors tend to zero as the number of particles tends to infinity. The bounds for the mean-square errors of these quantities are inversely proportional to the corresponding particle number.

Moreover, from the proofs of Propositions 4.1 and 4.2, it can be seen that(1)Assumptions 1, 3, and 4 ensure that (4.6) holds;(2)Assumption 4 ensures that (4.7) holds;(3)Assumption 2 ensures that (4.12), (4.13), and (4.14) hold;(4)Assumption 5 ensures the convergence of the mean-square errors for the resampling steps of the two filters.

Assumptions 3, 4, and 5 are concerned with the SMC method. They can be satisfied as long as the appropriate sampling strategies are chosen. Assumptions 1 and 2 are concerned with the likelihood and target transition kernel. They may be too restrictive or unrealistic for some practical applications. However, these convergence results give justification to the SMC implementations of the MeMBer and CBMeMBer filters and show how the order of the mean-square errors are reduced as the number of particles increases.

5. Simulations

Here, we briefly describe the application of the convergence results for the SMC-CBMeMBer filter to the nonlinear MTT example presented in Example 1 of [12]. The experiment settings are the same as those of Example 1 except that the number of the particles 𝐿^𝑘(𝑖) used for each hypothesized track at time 𝑘. For convenience, we assume 𝐿^𝑘(𝑖)=𝐿. Assumptions 1–5 are satisfied in this example. So, the SMC-CBMeMBer filter converges to the ground truth in the mean-square sense.

For the SMC-CBMeMBer filter, the estimates of the multitarget number and states, which are derived from the particle multi-Bernoulli parameter set, are unbiased. Therefore, via comparing the tracking performance of the algorithm in the various particle number 𝐿, the convergence results for the SMC-CBMeMBer filter can be verified to a great extent.

The standard deviation of the estimated cardinality distribution and the optimal subpattern assignment (OSPA) multitarget miss-distance [20] of order 𝑝=2 with cut-off 𝑐=100m, which jointly captures differences in cardinality and individual elements between two finite sets, are used to evaluate the performance of the method. Table 1 shows the time-averaged standard deviation of the estimated cardinality distribution and the time-averaged OSPA in various 𝐿 via 200 MC simulation experiments.

Table 1 

Time-averaged standard deviation of the estimated cardinality distribution and time-averaged OSPA (m) in various 𝐿.

Table 1 shows that both the standard deviation of the estimated cardinality distribution and OSPA decrease with the increase of the particle number 𝐿. This phenomenon can be reasonably explained by the convergence results derived in this paper: first, the mean-square error of the particle multi-Bernoulli parameter set decreases as the number of the particles increases; then, the more precise estimates of the cardinality distribution and multitarget states can be derived from the more precise particle multi-Bernoulli parameter set, which eventually leads to the results presented in Table 1.

6. Conclusions and Future Work

This paper presents the mathematical proofs of the convergence for the SMC-MeMBer and SMC-CBMeMBer filters and gives the bounds for the mean-square errors. In the linear-Gaussian condition, Vo et al. presented the analytic solutions to the MeMBer and CBMeMBer recursions: GM-MeMBer and GM-CBMeMBer filters [12]. The future work is focused on studying the convergence results and error bounds for the two filters.

Appendix

A.

In deriving the proofs, we use the Minkowski inequality, which states that, for any two random variables 𝑋 and 𝑌 in 𝐿², 𝐸(𝑋+𝑌)²^1/2≤𝐸𝑋²^1/2+𝐸𝑋²^1/2.(A.1)

Using Minkowski’s inequality, we obtain that, for all𝜑∈𝐵(ℝ^𝑑), 𝐸𝑟^{(𝑖),𝐿(𝑖)}𝑝^{(𝑖),𝐿(𝑖)},𝜑−𝑟^(𝑖)𝑝^(𝑖),𝜑²^1/2=𝐸𝑟^{(𝑖),𝐿(𝑖)}𝑝^{(𝑖),𝐿(𝑖)},𝜑−𝑟^(𝑖)𝑝^{(𝑖),𝐿(𝑖)},𝜑+𝑟^(𝑖)𝑝^{(𝑖),𝐿(𝑖)},𝜑−𝑟^(𝑖)𝑝^(𝑖),𝜑²^1/2≤𝐸𝑝_{(𝑖),𝐿(𝑖)}(A.2),𝜑²𝑟^{(𝑖),𝐿(𝑖)}−𝑟^(𝑖)²^1/2+𝑟^(𝑖)𝐸𝑝^{(𝑖),𝐿(𝑖)},𝜑−𝑝^(𝑖),𝜑²^1/2≤‖𝜑‖𝐸𝑟_{(𝑖),𝐿(𝑖)}(A.3)−𝑟^(𝑖)²^1/2+𝑟^(𝑖)𝐸𝑝^{(𝑖),𝐿(𝑖)},𝜑−𝑝^(𝑖),𝜑²^1/2(A.4) holds, 𝑖=1,…,𝑀, for the multi-Bernoulli density 𝜋={(𝑟^(𝑖),𝑝^(𝑖))}^𝑀𝑖=1 and its particle approximation 𝜋^𝐿(𝑖)={(𝑟^{(𝑖),𝐿(𝑖)},𝑝^{(𝑖),𝐿(𝑖)})}^𝑀𝑖=1.

A.1. Proof of Proposition 4.1

We first prove (4.5). From (2.4) and (3.2), we have 𝐸𝑟^{(𝑖),𝐿(𝑖)𝑘−1}_{𝑃,𝑘∣𝑘−1}−𝑟^{(𝑖)𝑃,𝑘∣𝑘−1}²^1/2=𝐸𝑟^{(𝑖),𝐿(𝑖)𝑘−1}_𝑘−1𝑝^{(𝑖),𝐿(i)𝑘−1𝑘−1},𝑝_𝑆,𝑘−𝑟^{(𝑖)𝑘−1}𝑝^{(𝑖)𝑘−1},𝑝_𝑆,𝑘²^1/2(A.5) (by (A.4)) ≤‖‖𝑝_𝑆,𝑘‖‖𝐸𝑟^{(𝑖),𝐿(𝑖)𝑘−1}_𝑘−1−𝑟^{(𝑖)𝑘−1}²^1/2+𝑟^{(𝑖)𝑘−1}𝐸𝑝^{(𝑖),𝐿(𝑖)𝑘−1}_𝑘−1,𝑝_𝑆,𝑘−𝑝^{(𝑖)𝑘−1},𝑝_𝑆,𝑘²^1/2(A.6) (by (4.3), (4.4), and 0≤𝑟^{(𝑖)𝑘−1}≤1) ≤‖‖𝑝_𝑆,𝑘‖‖√𝑐_𝑘−1+√𝑑_𝑘−1𝐿^{(𝑖)𝑘−1}.(A.7)

So that (4.5) is proved with 𝑐_{𝑃,𝑘∣𝑘−1}=‖‖𝑝_𝑆,𝑘‖‖²√𝑐_𝑘−1+√𝑑_𝑘−1².(A.8)

Now turn to (4.6). From (2.5), we have 𝐸𝑝^{(𝑖),𝐿(𝑖)𝑘−1}_{𝑃,𝑘∣𝑘−1},𝜑−𝑝^{(𝑖)𝑃,𝑘∣𝑘−1},𝜑²^1/2=𝐸⎡⎢⎢⎢⎣⎛⎜⎜⎝𝑝^{(𝑖),𝐿(𝑖)𝑘−1}_{𝑃,𝑘∣𝑘−1},𝜑−𝑓^{𝑘∣𝑘−1},𝑝^{(𝑖)𝑘−1}𝑝_𝑆,𝑘𝑝^{(𝑖)𝑘−1},𝑝_𝑆,𝑘,𝜑⎞⎟⎟⎠²⎤⎥⎥⎥⎦^1/2(A.9) (adding and subtracting a new term) =𝐸⎡⎢⎢⎢⎢⎣⎛⎜⎜⎜⎜⎝𝑝^{(𝑖),𝐿(𝑖)𝑘−1}_{𝑃,𝑘∣𝑘−1},𝜑−𝑓^{𝑘∣𝑘−1},𝑝^{(𝑖),𝐿(𝑖)𝑘−1}_𝑘−1𝑝_𝑆,𝑘𝑝^{(𝑖),𝐿(𝑖)𝑘−1𝑘−1},𝑝_𝑆,k+𝑓,𝜑_{𝑘∣𝑘−1},𝑝^{(𝑖),𝐿(𝑖)𝑘−1}_𝑘−1𝑝_𝑆,𝑘𝑝^{(𝑖),𝐿(𝑖)𝑘−1𝑘−1},𝑝_𝑆,𝑘,𝜑−𝑓^{𝑘∣𝑘−1},𝑝^{(𝑖)𝑘−1}𝑝_𝑆,𝑘𝑝^{(𝑖)𝑘−1},𝑝_𝑆,𝑘,𝜑⎞⎟⎟⎟⎟⎠²⎤⎥⎥⎥⎥⎦^1/2(A.10) (using Minkowski’s inequality) ≤𝐸⎡⎢⎢⎢⎢⎣⎛⎜⎜⎜⎜⎝𝑝^{(𝑖),𝐿(𝑖)𝑘−1}_{𝑃,𝑘∣𝑘−1},𝜑−𝑓^{𝑘∣𝑘−1},𝑝^{(𝑖),𝐿(𝑖)𝑘−1}_𝑘−1𝑝_𝑆,𝑘𝑝^{(𝑖),𝐿(𝑖)𝑘−1𝑘−1},𝑝_𝑆,𝑘,𝜑⎞⎟⎟⎟⎟⎠²⎤⎥⎥⎥⎥⎦^1/2+𝐸⎡⎢⎢⎢⎢⎣⎛⎜⎜⎜⎜⎝𝑝^{(𝑖),𝐿(𝑖)𝑘−1𝑘−1},𝑝_𝑆,𝑘𝑓_{𝑘∣𝑘−1},𝜑𝑝^{(𝑖),𝐿(𝑖)𝑘−1𝑘−1},𝑝_𝑆,𝑘−𝑝^{(𝑖)𝑘−1},𝑝_𝑆,𝑘𝑓_{𝑘∣𝑘−1},𝜑𝑝^{(𝑖)𝑘−1},𝑝_𝑆,𝑘⎞⎟⎟⎟⎟⎠²⎤⎥⎥⎥⎥⎦.^1/2(A.11)

By Assumption 3 and Lemma  1 in [13], we easily obtain that the first term in (A.11) becomes 𝐸⎡⎢⎢⎢⎢⎣⎛⎜⎜⎜⎜⎝𝑝^{(𝑖),𝐿(𝑖)𝑘−1}_{𝑃,𝑘∣𝑘−1},𝜑−𝑓^{𝑘∣𝑘−1},𝑝^{(𝑖),𝐿(𝑖)𝑘−1}_𝑘−1𝑝_𝑆,𝑘𝑝^{(𝑖),𝐿(𝑖)𝑘−1𝑘−1},𝑝_𝑆,𝑘,𝜑⎞⎟⎟⎟⎟⎠²⎤⎥⎥⎥⎥⎦^1/2≤‖𝜑‖𝐿^{(𝑖)𝑘−1}‖‖‖‖𝑓^{𝑘∣𝑘−1}𝑝_𝑆,𝑘𝑞^𝑘(𝑖)‖‖‖‖²+‖‖𝑓_{𝑘∣𝑘−1}𝑝_𝑆,𝑘‖‖²^1/2(A.12) (since ‖𝑓_{𝑘∣𝑘−1}/𝑞^𝑘(𝑖)‖≤𝐵₂ by Assumption 4 and 𝑓_{𝑘∣𝑘−1}𝑝_𝑆,𝑘∈𝐶_𝑏(ℝ^𝑑) by Assumption 1) ≤‖𝜑‖𝐿^{(𝑖)𝑘−1}‖‖𝑝^S,𝑘‖‖²𝐵²²+‖‖𝑓_{𝑘∣𝑘−1}𝑝_𝑆,𝑘‖‖²^1/2.(A.13)

Adding and subtracting a new term in the second term of (A.11), we have 𝐸⎡⎢⎢⎢⎢⎣⎛⎜⎜⎜⎜⎝𝑝^{(𝑖),𝐿(𝑖)𝑘−1𝑘−1},𝑝_𝑆,𝑘𝑓_{𝑘∣𝑘−1},𝜑𝑝^{(𝑖),𝐿(𝑖)𝑘−1𝑘−1},𝑝_𝑆,𝑘−𝑝^{(𝑖)𝑘−1},𝑝_𝑆,𝑘𝑓_{𝑘∣𝑘−1},𝜑𝑝^{(𝑖)𝑘−1},𝑝_𝑆,𝑘⎞⎟⎟⎟⎟⎠²⎤⎥⎥⎥⎥⎦^1/2=𝐸⎡⎢⎢⎢⎢⎣⎛⎜⎜⎜⎜⎝𝑝^{(𝑖),𝐿(𝑖)𝑘−1𝑘−1},𝑝_𝑆,𝑘𝑓_{𝑘∣𝑘−1},𝜑𝑝^{(𝑖),𝐿(𝑖)𝑘−1𝑘−1},𝑝_𝑆,𝑘−𝑝^{(𝑖),𝐿(𝑖)𝑘−1𝑘−1},𝑝_𝑆,𝑘𝑓_{𝑘∣𝑘−1},𝜑+𝑝_{(𝑖),𝐿(𝑖)𝑘−1𝑘−1}𝑝^{(𝑖)𝑘−1},𝑝_𝑆,𝑘,𝑝_𝑆,𝑘𝑓_{𝑘∣𝑘−1},𝜑𝑝^{(𝑖)𝑘−1},𝑝_𝑆,𝑘−𝑝^{(𝑖)𝑘−1},𝑝_𝑆,𝑘𝑓_{𝑘∣𝑘−1},𝜑𝑝^{(𝑖)𝑘−1},𝑝_𝑆,𝑘⎞⎟⎟⎟⎟⎠²⎤⎥⎥⎥⎥⎦^1/2(A.14) (using Minkowski’s inequality) ≤1𝑝^{(𝑖)𝑘−1},𝑝_𝑆,𝑘𝐸⎡⎢⎢⎢⎢⎣⎛⎜⎜⎜⎜⎝𝑝^{(𝑖),𝐿(𝑖)𝑘−1𝑘−1},𝑝_𝑆,𝑘𝑓_{𝑘∣𝑘−1},𝜑𝑝^{(𝑖),𝐿(𝑖)𝑘−1𝑘−1},𝑝_𝑆,𝑘𝑝^{(𝑖)𝑘−1},𝑝_𝑆,𝑘−𝑝^{(𝑖),𝐿(𝑖)𝑘−1𝑘−1},𝑝_𝑆,𝑘⎞⎟⎟⎟⎟⎠²⎤⎥⎥⎥⎥⎦^1/2+1𝑝^{(𝑖)𝑘−1},𝑝_𝑆,𝑘𝐸𝑝^{(𝑖),𝐿(𝑖)𝑘−1𝑘−1},𝑝_𝑆,𝑘𝑓_{𝑘∣𝑘−1},𝜑−𝑝^{(𝑖)𝑘−1},𝑝_𝑆,𝑘𝑓_{𝑘∣𝑘−1},𝜑²^1/2≤2‖‖𝑓_{𝑘∣𝑘−1}(A.15),𝜑‖‖𝑝^{(𝑖)𝑘−1},𝑝_𝑆,𝑘𝐸𝑝^{(𝑖)𝑘−1},𝑝_𝑆,𝑘−𝑝^{(𝑖),𝐿(𝑖)𝑘−1𝑘−1},𝑝_𝑆,𝑘²^1/2(A.16) (by (4.4)) ≤2‖𝜑‖⋅‖‖𝑝^𝑆,𝑘‖‖inf𝑝_𝑆,𝑘𝑑_𝑘−1𝐿^{(𝑖)𝑘−1},(A.17) where inf(⋅) denotes the infimum.

Finally, substituting (A.12) and (A.17) into (A.11), (4.6) is proved with 𝑑_{𝑃,𝑘∣𝑘−1}=‖‖𝑝_𝑆,𝑘‖‖²𝐵²²+‖‖𝑓_{𝑘∣𝑘−1}𝑝_𝑆,𝑘‖‖²+2‖‖𝑝^𝑆,𝑘‖‖√𝑑_𝑘−1inf(𝑝_𝑆,𝑘)².(A.18)

Now, turn to (4.7). By Lemma  0 in [13] and the boundedness of ‖𝑝^{(𝑖)Γ,𝑘}/𝑏^𝑘(𝑖)‖≤𝐵₁ (𝑖=1,…,𝑀_𝑘−1) in Assumption 4, we get that (4.7) holds for a constant 𝑑_Γ,𝑘. This completes the proof.

A.2. Proof of Proposition 4.2

Now turn to (4.10). From (2.7) and (3.8), we have𝐸𝑟^{(𝑖)𝐿,𝑘}−𝑟^{(𝑖),𝐿(𝑖)𝑘∣𝑘−1}_𝐿,𝑘²^1/2=𝐸⎡⎢⎢⎢⎢⎣⎛⎜⎜⎜⎜⎝r^{(𝑖)𝑘∣𝑘−1}1−𝑝^{(𝑖)𝑘∣𝑘−1},𝑝_𝐷,𝑘1−𝑟^{(𝑖)𝑘∣𝑘−1}𝑝^{(𝑖)𝑘∣𝑘−1},𝑝_𝐷,𝑘−𝑟^{(𝑖),𝐿(𝑖)𝑘∣𝑘−1𝑘∣𝑘−1}1−𝑝^{(𝑖),𝐿(𝑖)𝑘∣𝑘−1}_{𝑘∣𝑘−1},𝑝_𝐷,𝑘1−𝑟^{(𝑖),𝐿(𝑖)𝑘∣𝑘−1}_{𝑘∣𝑘−1}𝑝^{(𝑖),𝐿(𝑖)𝑘∣𝑘−1𝑘∣𝑘−1},𝑝_𝐷,𝑘⎞⎟⎟⎟⎟⎠²⎤⎥⎥⎥⎥⎦^1/2(A.19) (adding and subtracting a new term) =𝐸⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝𝑟^{(𝑖)𝑘∣𝑘−1}1−𝑝^{(𝑖)𝑘∣𝑘−1},𝑝_𝐷,𝑘1−𝑟^{(𝑖)𝑘∣𝑘−1}𝑝^{(𝑖)𝑘∣𝑘−1},𝑝_𝐷,𝑘−𝑟^{(𝑖),𝐿(𝑖)𝑘∣𝑘−1𝑘∣𝑘−1}1−𝑝^{(𝑖),𝐿(𝑖)𝑘∣𝑘−1}_{𝑘∣𝑘−1},𝑝_𝐷,𝑘+𝑟_{(𝑖),𝐿(𝑖)𝑘∣𝑘−1𝑘∣𝑘−1}1−𝑟^{(𝑖)𝑘∣𝑘−1}𝑝^{(𝑖)𝑘∣𝑘−1},𝑝_𝐷,𝑘1−𝑝^{(𝑖),𝐿(𝑖)𝑘∣𝑘−1}_{𝑘∣𝑘−1},𝑝_𝐷,𝑘1−𝑟^{(𝑖)𝑘∣𝑘−1}𝑝^{(𝑖)𝑘∣𝑘−1},𝑝_𝐷,𝑘−𝑟^{(𝑖),𝐿(𝑖)𝑘∣𝑘−1𝑘∣𝑘−1}1−𝑝^{(𝑖),𝐿(𝑖)𝑘∣𝑘−1}_{𝑘∣𝑘−1},𝑝_𝐷,𝑘1−𝑟^{(𝑖),𝐿(𝑖)𝑘∣𝑘−1}_{𝑘∣𝑘−1}𝑝^{(𝑖),𝐿(𝑖)𝑘∣𝑘−1𝑘∣𝑘−1},𝑝_𝐷,𝑘⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠²⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦^1/2(A.20) (using Minkowski’s inequality) ≤𝐸𝑟^{(𝑖)𝑘∣𝑘−1}1−𝑝^{(𝑖)𝑘∣𝑘−1},𝑝_𝐷,𝑘−𝑟^{(𝑖),𝐿(𝑖)𝑘∣𝑘−1𝑘∣𝑘−1}1−𝑝^{(𝑖),𝐿(𝑖)𝑘∣𝑘−1𝑘∣𝑘−1},𝑝_𝐷,𝑘²^1/2+𝐸⎡⎢⎢⎢⎢⎣⎛⎜⎜⎜⎜⎝𝑟_{(𝑖),𝐿(𝑖)𝑘∣𝑘−1𝑘∣𝑘−1}1−𝑟^{(𝑖)𝑘∣𝑘−1}𝑝^{(𝑖)𝑘∣𝑘−1},𝑝_𝐷,𝑘1−𝑝^{(𝑖),𝐿(𝑖)𝑘∣𝑘−1}_{𝑘∣𝑘−1},𝑝_𝐷,𝑘1−𝑟^{(𝑖)𝑘∣𝑘−1}𝑝^{(𝑖)𝑘∣𝑘−1},𝑝_𝐷,𝑘−𝑟^{(𝑖),𝐿(𝑖)𝑘∣𝑘−1𝑘∣𝑘−1}1−𝑝^{(𝑖),𝐿(𝑖)𝑘∣𝑘−1}_{𝑘∣𝑘−1},𝑝_𝐷,𝑘1−𝑟^{(𝑖),𝐿(𝑖)𝑘∣𝑘−1}_{𝑘∣𝑘−1}𝑝^{(𝑖),𝐿(𝑖)𝑘∣𝑘−1𝑘∣𝑘−1},𝑝_𝐷,𝑘⎞⎟⎟⎟⎟⎠²⎤⎥⎥⎥⎥⎦^1/2.(A.21)

The numerator of the first term in (A.21) is 𝐸𝑟^{(𝑖)𝑘∣𝑘−1}1−𝑝^{(𝑖)𝑘∣𝑘−1},𝑝_𝐷,𝑘−𝑟^{(𝑖),𝐿(𝑖)𝑘∣𝑘−1𝑘∣𝑘−1}1−𝑝^{(𝑖),𝐿(𝑖)𝑘∣𝑘−1𝑘∣𝑘−1},𝑝_𝐷,𝑘²^1/2=𝐸𝑟^{(𝑖)𝑘∣𝑘−1}−𝑟^{(𝑖),𝐿(𝑖)𝑘∣𝑘−1}_{𝑘∣𝑘−1}+𝑟^{(𝑖),𝐿(𝑖)𝑘∣𝑘−1𝑘∣𝑘−1}𝑝^{(𝑖),𝐿(𝑖)𝑘∣𝑘−1𝑘∣𝑘−1},𝑝_𝐷,𝑘−𝑟^{(𝑖)𝑘∣𝑘−1}𝑝^{(𝑖)𝑘∣𝑘−1},𝑝_𝐷,𝑘²^1/2(A.22) (using Minkowski’s inequality and then (A.4)) ≤𝐸𝑟^{(𝑖)𝑘∣𝑘−1}−𝑟^{(𝑖),𝐿(𝑖)𝑘∣𝑘−1}_{𝑘∣𝑘−1}²^1/2+‖‖𝑝_𝐷,𝑘‖‖𝐸𝑟^{(𝑖)𝑘∣𝑘−1}−𝑟^{(𝑖),𝐿(𝑖)𝑘∣𝑘−1}_{𝑘∣𝑘−1}²^1/2+𝑟^{(𝑖)𝑘∣𝑘−1}𝐸𝑝^{(𝑖),𝐿(𝑖)𝑘∣𝑘−1}_{𝑘∣𝑘−1},𝑝_𝐷,𝑘−𝑝^{(𝑖)𝑘∣𝑘−1},𝑝_𝐷,𝑘²^1/2(A.23) (by (4.8) and (4.9)) ≤1+‖‖𝑝^𝐷,𝑘‖‖√𝑐_{𝑘∣𝑘−1}+𝑟^{(𝑖)𝑘∣𝑘−1}‖‖𝑝_𝐷,𝑘‖‖√𝑑_{𝑘∣𝑘−1}𝐿^{(𝑖)𝑘∣𝑘−1}.(A.24)

The second term in (A.21) is 𝐸⎡⎢⎢⎢⎢⎣⎛⎜⎜⎜⎜⎝𝑟^{(𝑖),𝐿(𝑖)𝑘∣𝑘−1}_{𝑘∣𝑘−1}1−𝑝^{(𝑖),𝐿(𝑖)𝑘∣𝑘−1}_{𝑘∣𝑘−1},𝑝_𝐷,𝑘1−𝑟^{(𝑖)𝑘∣𝑘−1}𝑝^{(𝑖)𝑘∣𝑘−1},𝑝_𝐷,𝑘−𝑟^{(𝑖),𝐿(𝑖)𝑘∣𝑘−1𝑘∣𝑘−1}1−𝑝^{(𝑖),𝐿(𝑖)𝑘∣𝑘−1}_{𝑘∣𝑘−1},𝑝_𝐷,𝑘1−𝑟^{(𝑖),𝐿(𝑖)𝑘∣𝑘−1}_{𝑘∣𝑘−1}𝑝^{(𝑖),𝐿(𝑖)𝑘∣𝑘−1𝑘∣𝑘−1},𝑝_𝐷,𝑘⎞⎟⎟⎟⎟⎠²⎤⎥⎥⎥⎥⎦^1/2=𝐸⎡⎢⎢⎢⎢⎣⎛⎜⎜⎜⎜⎝𝑟^{(𝑖),𝐿(𝑖)𝑘∣𝑘−1𝑘∣𝑘−1}−𝑟^{(𝑖),𝐿(𝑖)𝑘∣𝑘−1}_{𝑘∣𝑘−1}𝑝^{(𝑖),𝐿(𝑖)𝑘∣𝑘−1𝑘∣𝑘−1},𝑝_𝐷,𝑘1−𝑟^{(𝑖),𝐿(𝑖)𝑘∣𝑘−1}_{𝑘∣𝑘−1}𝑝^{(𝑖),𝐿(𝑖)𝑘∣𝑘−1𝑘∣𝑘−1},𝑝_𝐷,𝑘⎞⎟⎟⎟⎟⎠²×⎛⎜⎜⎜⎝𝑟^{(𝑖)𝑘∣𝑘−1}𝑝^{(𝑖)𝑘∣𝑘−1},𝑝_𝐷,𝑘−𝑟^{(𝑖),𝐿(𝑖)𝑘∣𝑘−1𝑘∣𝑘−1}𝑝^{(𝑖),𝐿(𝑖)𝑘∣𝑘−1𝑘∣𝑘−1},𝑝_𝐷,𝑘1−𝑟^{(𝑖)𝑘∣𝑘−1}𝑝^{(𝑖)𝑘∣𝑘−1},𝑝_𝐷,𝑘⎞⎟⎟⎟⎠²⎤⎥⎥⎥⎦^1/2(A.25) (by 0≤𝑟^{(𝑖),𝐿(𝑖)𝑘∣𝑘−1}_{𝑘∣𝑘−1}≤1) ≤𝐸𝑟^{(𝑖)𝑘∣𝑘−1}𝑝^{(𝑖)𝑘∣𝑘−1},𝑝_𝐷,𝑘−𝑟^{(𝑖),𝐿(𝑖)𝑘∣𝑘−1𝑘∣𝑘−1}𝑝^{(𝑖),𝐿(𝑖)𝑘∣𝑘−1𝑘∣𝑘−1},𝑝_𝐷,𝑘²^1/21−𝑟^{(𝑖)𝑘∣𝑘−1}𝑝^{(𝑖)𝑘∣𝑘−1},𝑝_𝐷,𝑘(A.26) (by (A.4), (4.8) and (4.9)) ≤‖‖𝑝^𝐷,𝑘‖‖√𝑐_{𝑘∣𝑘−1}+𝑟^{(𝑖)𝑘∣𝑘−1}‖‖𝑝_𝐷,𝑘‖‖√𝑑_{𝑘∣𝑘−1}1−𝑟^{(𝑖)𝑘∣𝑘−1}𝑝^{(𝑖)𝑘∣𝑘−1},𝑝_𝐷,𝑘𝐿^{(𝑖)𝑘∣𝑘−1}.(A.27)

Substituting (A.24) and (A.27) into (A.21), and then using 0≤𝑟^{(𝑖)𝑘∣𝑘−1}≤1, we get 𝐸𝑟^{(𝑖)𝐿,𝑘}−𝑟^{(𝑖),𝐿(𝑖)𝑘∣𝑘−1}_𝐿,𝑘²^1/2≤1+2‖‖𝑝^𝐷,𝑘‖‖√𝑐_{𝑘∣𝑘−1}+2𝑟^{(𝑖)𝑘∣𝑘−1}‖‖𝑝_𝐷,𝑘‖‖√𝑑_{𝑘∣𝑘−1}1−𝑟^{(𝑖)𝑘∣𝑘−1}𝑝^{(𝑖)𝑘∣𝑘−1},𝑝_𝐷,𝑘𝐿^{(𝑖)𝑘∣𝑘−1}≤1+2‖‖𝑝^𝐷,𝑘‖‖√𝑐_{𝑘∣𝑘−1}+2‖‖𝑝_𝐷,𝑘‖‖√𝑑_{𝑘∣𝑘−1}1−‖‖𝑝_𝐷,𝑘‖‖𝐿^{(𝑖)𝑘∣𝑘−1}.(A.28)

Finally, (4.10) is proved with 𝑐_𝐿,𝑘=1+2‖‖𝑝^𝐷,𝑘‖‖√𝑐_{𝑘∣𝑘−1}+2‖‖𝑝_𝐷,𝑘‖‖√𝑑_{𝑘∣𝑘−1}1−‖‖𝑝_𝐷,𝑘‖‖².(A.29)

Now turn to (4.11). From (2.8) and (3.9), we have 𝐸𝑝^{(𝑖)𝐿,𝑘},𝜑−𝑝^{(𝑖),𝐿(𝑖)𝑘∣𝑘−1}_𝐿,𝑘,𝜑²^1/2=𝐸⎡⎢⎢⎢⎢⎣⎛⎜⎜⎜⎜⎝𝑝^{(𝑖)𝑘∣𝑘−1},𝜑1−𝑝_𝐷,𝑘1−𝑝^{(𝑖)𝑘∣𝑘−1},𝑝_𝐷,𝑘−𝑝^{(𝑖),𝐿(𝑖)𝑘∣𝑘−1𝑘∣𝑘−1},𝜑1−𝑝_𝐷,𝑘1−𝑝^{(𝑖),𝐿(𝑖)𝑘∣𝑘−1}_{𝑘∣𝑘−1},𝑝_𝐷,𝑘⎞⎟⎟⎟⎟⎠²⎤⎥⎥⎥⎥⎦^1/2(A.30) (adding and subtracting a new term) =𝐸⎡⎢⎢⎢⎢⎣⎛⎜⎜⎜⎜⎝𝑝^{(𝑖)𝑘∣𝑘−1},𝜑1−𝑝_𝐷,𝑘1−𝑝^{(𝑖)𝑘∣𝑘−1},𝑝_𝐷,𝑘−𝑝^{(𝑖),𝐿(𝑖)𝑘∣𝑘−1𝑘∣𝑘−1},𝜑1−𝑝_𝐷,𝑘+𝑝_{(𝑖),𝐿(𝑖)𝑘∣𝑘−1𝑘∣𝑘−1}1−𝑝^{(𝑖)𝑘∣𝑘−1},𝑝_𝐷,𝑘,𝜑1−𝑝_𝐷,𝑘1−𝑝^{(𝑖)𝑘∣𝑘−1},𝑝_𝐷,𝑘−𝑝^{(𝑖),𝐿(𝑖)𝑘∣𝑘−1𝑘∣𝑘−1},𝜑1−𝑝_𝐷,𝑘1−𝑝^{(𝑖),𝐿(𝑖)𝑘∣𝑘−1}_{𝑘∣𝑘−1},𝑝_𝐷,𝑘⎞⎟⎟⎟⎟⎠²⎤⎥⎥⎥⎥⎦^1/2(A.31) (using Minkowski’s inequality) ≤𝐸𝑝^{(𝑖)𝑘∣𝑘−1},𝜑1−𝑝_𝐷,𝑘−𝑝^{(𝑖),𝐿(𝑖)𝑘∣𝑘−1}_{𝑘∣𝑘−1},𝜑1−𝑝_𝐷,𝑘²^1/2+𝐸⎡⎢⎢⎢⎢⎣⎛⎜⎜⎜⎜⎝𝑝_{(𝑖),𝐿(𝑖)𝑘∣𝑘−1𝑘∣𝑘−1}1−𝑝^{(𝑖)𝑘∣𝑘−1},𝑝_𝐷,𝑘,𝜑1−𝑝_𝐷,𝑘1−𝑝^{(𝑖),𝐿(𝑖)𝑘∣𝑘−1}_{𝑘∣𝑘−1},𝑝_𝐷,𝑘⎞⎟⎟⎟⎟⎠²⋅⎛⎜⎜⎜⎜⎝𝑝^{(𝑖)𝑘∣𝑘−1},𝑝_𝐷,𝑘−𝑝^{(𝑖),𝐿(𝑖)𝑘∣𝑘−1𝑘∣𝑘−1},𝑝_𝐷,𝑘1−𝑝^{(𝑖)𝑘∣𝑘−1},𝑝_𝐷,𝑘⎞⎟⎟⎟⎟⎠²⎤⎥⎥⎥⎥⎦^1/2≤𝐸𝑝(A.32)_{(𝑖)𝑘∣𝑘−1},𝜑1−𝑝_𝐷,𝑘−𝑝^{(𝑖),𝐿(𝑖)𝑘∣𝑘−1}_{𝑘∣𝑘−1},𝜑1−𝑝_𝐷,𝑘²^1/2+‖𝜑‖𝐸𝑝_{(𝑖)𝑘∣𝑘−1}1−𝑝^{(𝑖)𝑘∣𝑘−1},𝑝_𝐷,𝑘,𝑝_𝐷,𝑘−𝑝^{(𝑖),𝐿(𝑖)𝑘∣𝑘−1𝑘∣𝑘−1},𝑝_𝐷,𝑘²^1/21−𝑝^{(𝑖)𝑘∣𝑘−1},𝑝_𝐷,𝑘(A.33) (by (4.9)) ≤‖𝜑‖√𝑑_{𝑘∣𝑘−1}1−‖‖𝑝_𝐷,𝑘‖‖𝐿^{(𝑖)𝑘∣𝑘−1}.(A.34)

Finally, (4.11) is proved with 𝑑_𝐿,𝑘=𝑑^{𝑘∣𝑘−1}1−‖‖𝑝_𝐷,𝑘‖‖².(A.35)

Now, turn to (4.12). From (2.9) and (3.10), we have 𝐸𝑟^{𝐿(𝑖)𝑘∣𝑘−1}_𝑈,𝑘(𝐳_𝑘)−𝑟_𝑈,𝑘(𝐳_𝑘)²^1/2=𝐸⎡⎢⎢⎢⎣⎛⎜⎜⎝∑^{𝑀𝑘∣𝑘−1𝑖=1}𝑟^{(𝑖),𝐿(𝑖)𝑘∣𝑘−1}_𝛼,𝑘(𝐳_𝑘)𝜅_𝑘(𝐳_𝑘)+∑^{𝑀𝑘∣𝑘−1}_𝑖=1𝑟^{(𝑖),𝐿(𝑖)𝑘∣𝑘−1}_𝛼,𝑘(𝐳_𝑘)−∑^{𝑀𝑘∣𝑘−1𝑖=1}𝑟^{(𝑖)𝛼,𝑘}(𝐳_𝑘)𝜅_𝑘(𝐳_𝑘)+∑^{𝑀𝑘∣𝑘−1}_𝑖=1𝑟^{(𝑖)𝛼,𝑘}(𝐳_𝑘)⎞⎟⎟⎠²⎤⎥⎥⎥⎦^1/2(A.36) (adding and subtracting a new term) =𝐸⎡⎢⎢⎣⎛⎜⎜⎝∑^{𝑀𝑘∣𝑘−1𝑖=1}𝑟^{(𝑖),𝐿(𝑖)𝑘∣𝑘−1}_𝛼,𝑘𝐳_𝑘𝜅_𝑘𝐳_𝑘+∑^{𝑀𝑘∣𝑘−1}_𝑖=1𝑟^{(𝑖),𝐿(𝑖)𝑘∣𝑘−1}_𝛼,𝑘𝐳_𝑘−∑^{𝑀𝑘∣𝑘−1𝑖=1}𝑟^{(𝑖),𝐿(𝑖)𝑘∣𝑘−1}_𝛼,𝑘𝐳_𝑘+∑_{𝑀𝑘∣𝑘−1𝑖=1}𝜅_𝑘𝐳_𝑘+∑^{𝑀𝑘∣𝑘−1}_𝑖=1𝑟^{(𝑖)𝛼,𝑘}𝐳_𝑘𝑟^{(𝑖),𝐿(𝑖)𝑘∣𝑘−1}_𝛼,𝑘(𝐳_𝑘)𝜅_𝑘(𝐳_𝑘)+∑^{𝑀𝑘∣𝑘−1}_𝑖=1𝑟^{(𝑖)𝛼,𝑘}(𝐳_𝑘)−∑^{𝑀𝑘∣𝑘−1𝑖=1}𝑟^{(𝑖)𝛼,𝑘}(𝐳_𝑘)𝜅_𝑘(𝐳_𝑘)+∑^{𝑀𝑘∣𝑘−1}_𝑖=1𝑟^{(𝑖)𝛼,𝑘}(𝐳_𝑘)⎞⎟⎟⎠²⎤⎥⎥⎥⎦^1/2(A.37) (using Minkowski’s inequality) ≤𝐸⎡⎢⎢⎢⎣⎛⎜⎜⎝𝑟^{𝐿(𝑖)𝑘∣𝑘−1}_𝑈,𝑘(𝐳_𝑘)∑^{𝑀𝑘∣𝑘−1𝑖=1}𝑟^{(𝑖)𝛼,𝑘}(𝐳_𝑘)−∑^{𝑀𝑘∣𝑘−1}_𝑖=1𝑟^{(𝑖),𝐿(𝑖)𝑘∣𝑘−1}_𝛼,𝑘(𝐳_𝑘)𝜅_𝑘(𝐳_𝑘)+∑^{𝑀𝑘∣𝑘−1}_𝑖=1𝑟^{(𝑖)𝛼,𝑘}(𝐳_𝑘)⎞⎟⎟⎠²⎤⎥⎥⎥⎦^1/2+𝐸⎡⎢⎢⎢⎣⎛⎜⎜⎝∑^{𝑀𝑘∣𝑘−1𝑖=1}𝑟^{(𝑖),𝐿(𝑖)𝑘∣𝑘−1}_𝛼,𝑘(𝐳_𝑘)−∑^{𝑀𝑘∣𝑘−1}_𝑖=1𝑟^{(𝑖)𝛼,𝑘}(𝐳_𝑘)𝜅_𝑘(𝐳_𝑘)+∑^{𝑀𝑘∣𝑘−1}_𝑖=1𝑟^{(𝑖)𝛼,𝑘}(𝐳_𝑘)⎞⎟⎟⎠²⎤⎥⎥⎥⎦^1/2(A.38) (using 𝜅_𝑘(𝐳_𝑘)≥0, 0≤𝑟^{𝐿(𝑖)𝑘∣𝑘−1}_𝑈,𝑘(𝐳_𝑘)≤1) ≤2∑^{𝑀𝑘∣𝑘−1𝑖=1}𝐸𝑟^{(𝑖),𝐿(𝑖)𝑘∣𝑘−1}_𝛼,𝑘𝐳_𝑘−𝑟^{(𝑖)𝛼,𝑘}𝐳_𝑘²^1/2∑^{𝑀𝑘∣𝑘−1}_𝑖=1𝑟^{(𝑖)𝛼,𝑘}𝐳_𝑘.(A.39)

From (2.12) and (3.14), the expectation in the summation of (A.39) is 𝐸𝑟^{(𝑖),𝐿(𝑖)𝑘∣𝑘−1}_𝛼,𝑘(𝐳_𝑘)−𝑟^{(𝑖)𝛼,𝑘}(𝐳_𝑘)²^1/2=𝐸⎡⎢⎢⎢⎢⎣⎛⎜⎜⎜⎜⎝𝑟^{(𝑖)𝑘∣𝑘−1}𝑝^{(𝑖)𝑘∣𝑘−1},𝜓_{𝑘,𝐳𝑘}1−𝑟^{(𝑖)𝑘∣𝑘−1}𝑝^{(𝑖)𝑘∣𝑘−1},𝑝_𝐷,𝑘−𝑟^{(𝑖),𝐿(𝑖)𝑘∣𝑘−1𝑘∣𝑘−1}𝑝^{(𝑖),𝐿(𝑖)𝑘∣𝑘−1𝑘∣𝑘−1},𝜓_{𝑘,𝐳𝑘}1−𝑟^{(𝑖),𝐿(𝑖)𝑘∣𝑘−1}_{𝑘∣𝑘−1}𝑝^{(𝑖),𝐿(𝑖)𝑘∣𝑘−1𝑘∣𝑘−1},𝑝_𝐷,𝑘⎞⎟⎟⎟⎟⎠²⎤⎥⎥⎥⎥⎦^1/2(A.40) (adding and subtracting a new term) =𝐸⎡⎢⎢⎢⎢⎣⎛⎜⎜⎜⎜⎝𝑟^{(𝑖),𝐿(𝑖)𝑘∣𝑘−1𝑘∣𝑘−1}𝑝^{(𝑖),𝐿(𝑖)𝑘∣𝑘−1𝑘∣𝑘−1},𝜓_{𝑘,𝐳𝑘}1−𝑟^{(𝑖),𝐿(𝑖)𝑘∣𝑘−1}_{𝑘∣𝑘−1}𝑝^{(𝑖),𝐿(𝑖)𝑘∣𝑘−1𝑘∣𝑘−1},𝑝_𝐷,𝑘−𝑟^{(𝑖)𝑘∣𝑘−1}𝑝^{(𝑖)𝑘∣𝑘−1},𝜓_{𝑘,𝐳𝑘}1−𝑟^{(𝑖),𝐿(𝑖)𝑘∣𝑘−1}_{𝑘∣𝑘−1}𝑝^{(𝑖),𝐿(𝑖)𝑘∣𝑘−1𝑘∣𝑘−1},𝑝_𝐷,𝑘+𝑟_{(𝑖)𝑘∣𝑘−1}𝑝^{(𝑖)𝑘∣𝑘−1},𝜓_{𝑘,𝐳𝑘}1−𝑟^{(𝑖),𝐿(𝑖)𝑘∣𝑘−1}_{𝑘∣𝑘−1}𝑝^{(𝑖),𝐿(𝑖)𝑘∣𝑘−1𝑘∣𝑘−1},𝑝_𝐷,𝑘−𝑟^{(𝑖)𝑘∣𝑘−1}𝑝^{(𝑖)𝑘∣𝑘−1},𝜓_{𝑘,𝐳𝑘}1−𝑟^{(𝑖)𝑘∣𝑘−1}𝑝^{(𝑖)𝑘∣𝑘−1},𝑝_𝐷,𝑘⎞⎟⎟⎟⎟⎠²⎤⎥⎥⎥⎥⎦^1/2(A.41) (using Minkowski’s inequality) ≤𝐸⎡⎢⎢⎢⎢⎣⎛⎜⎜⎜⎜⎝𝑟^{(𝑖),𝐿(𝑖)𝑘∣𝑘−1𝑘∣𝑘−1}𝑝^{(𝑖),𝐿(𝑖)𝑘∣𝑘−1𝑘∣𝑘−1},𝜓_{𝑘,𝐳𝑘}−𝑟^{(𝑖)𝑘∣𝑘−1}𝑝^{(𝑖)𝑘∣𝑘−1},𝜓_{𝑘,𝐳𝑘}1−𝑟^{(𝑖),𝐿(𝑖)𝑘∣𝑘−1}_{𝑘∣𝑘−1}𝑝^{(𝑖),𝐿(𝑖)𝑘∣𝑘−1𝑘∣𝑘−1},𝑝_𝐷,𝑘⎞⎟⎟⎟⎟⎠²⎤⎥⎥⎥⎥⎦^1/2+𝑟^{(𝑖)𝑘∣𝑘−1}𝑝^{(𝑖)𝑘∣𝑘−1},𝜓_{𝑘,𝐳𝑘}1−𝑟^(𝑖)_k∣𝑘−1𝑝^{(𝑖)𝑘∣𝑘−1},𝑝_𝐷,𝑘𝐸⎡⎢⎢⎢⎢⎣⎛⎜⎜⎜⎜⎝𝑟^{(𝑖),𝐿(𝑖)𝑘∣𝑘−1𝑘∣𝑘−1}𝑝^{(𝑖),𝐿(𝑖)𝑘∣𝑘−1𝑘∣𝑘−1},𝑝_𝐷,𝑘−𝑟^{(𝑖)𝑘∣𝑘−1}𝑝^{(𝑖)𝑘∣𝑘−1},𝑝_𝐷,𝑘1−𝑟^{(𝑖),𝐿(𝑖)𝑘∣𝑘−1}_{𝑘∣𝑘−1}𝑝^{(𝑖),𝐿(𝑖)𝑘∣𝑘−1𝑘∣𝑘−1},𝑝_𝐷,𝑘⎞⎟⎟⎟⎟⎠²⎤⎥⎥⎥⎥⎦^1/2(A.42) (by 0≤𝑟^{(𝑖),𝐿(𝑖)𝑘∣𝑘−1}_{𝑘∣𝑘−1}≤1, Assumption 2 and (A.4)) ≤‖‖𝜓^{𝑘,𝐳𝑘}‖‖𝐸𝑟^{(𝑖),𝐿(𝑖)𝑘∣𝑘−1}_{𝑘∣𝑘−1}−𝑟^{(𝑖)𝑘∣𝑘−1}²^1/2+𝑟^{(𝑖)𝑘∣𝑘−1}𝐸𝑝^{(𝑖),𝐿(𝑖)𝑘∣𝑘−1}_{𝑘∣𝑘−1},𝜓_{𝑘,𝐳𝑘}−𝑝^{(𝑖)𝑘∣𝑘−1},𝜓_{𝑘,𝐳𝑘}²^1/2+𝑟_{(𝑖)𝑘∣𝑘−1}1−‖‖𝑝_𝐷,𝑘‖‖𝑝^{(𝑖)𝑘∣𝑘−1},𝜓_{𝑘,𝐳𝑘}×‖‖1𝑝−𝑟^{(𝑖)𝑘∣𝑘−1}_𝐷,𝑘𝑝^{(𝑖)𝑘∣𝑘−1},𝑝_𝐷,𝑘‖‖𝐸𝑟^{(𝑖),𝐿(𝑖)𝑘∣𝑘−1}_{𝑘∣𝑘−1}−𝑟^{(𝑖)𝑘∣𝑘−1}²^1/2+𝑟^{(𝑖)𝑘∣𝑘−1}𝐸𝑝^{(𝑖),𝐿(𝑖)𝑘∣𝑘−1}_{𝑘∣𝑘−1},𝑝_𝐷,𝑘−𝑝^{(𝑖)𝑘∣𝑘−1},𝑝_𝐷,𝑘²^1/21−‖‖𝑝_𝐷,𝑘‖‖(A.43) (by (4.8), (4.9), and 0≤𝑟^{(𝑖)𝑘∣𝑘−1}≤1) ≤‖‖𝜓^{𝑘,𝐳𝑘}‖‖√𝑐_{𝑘∣𝑘−1}+√𝑑_{𝑘∣𝑘−1}1−‖‖𝑝_𝐷,𝑘‖‖²𝐿^{(𝑖)𝑘∣𝑘−1}.(A.44)

From (2.12), 0≤𝑟^{(𝑖)𝑘∣𝑘−1}≤1 and Assumption 2, the denominator of (A.39) is ^𝑀^{𝑘∣𝑘−1}_𝑖=1𝑟^{(𝑖)𝛼,𝑘}𝐳_𝑘=^{𝑀𝑘∣𝑘−1}_𝑖=1𝑟^{(𝑖)𝑘∣𝑘−1}𝑝^{(𝑖)𝑘∣𝑘−1},𝜓_{𝑘,𝐳𝑘}1−𝑟^{(𝑖)𝑘∣𝑘−1}𝑝^{(𝑖)𝑘∣𝑘−1},𝑝_𝐷,𝑘≥^{𝑀𝑘∣𝑘−1}_𝑖=1≥inf𝜓_{𝑘,𝐳𝑘}𝑟^{(𝑖)𝑘∣𝑘−1}𝑝^{(𝑖)𝑘∣𝑘−1},𝜓_{𝑘,𝐳𝑘}^{𝑀𝑘∣𝑘−1}_𝑖=1𝑟^{(𝑖)𝑘∣𝑘−1}=inf𝜓_{𝑘,𝐳𝑘}𝑛_{𝑘∣𝑘−1},(A.45) where 𝑛_{𝑘∣𝑘−1}=∑^{𝑀𝑘∣𝑘−1}_𝑖=1𝑟^{(𝑖)𝑘∣𝑘−1} is the number of the predicted targets at time 𝑘.

Substituting (A.44) and (A.43) into (A.39), we get 𝐸𝑟^{𝐿(𝑖)𝑘∣𝑘−1}_𝑈,𝑘(𝐳_𝑘)−𝑟_𝑈,𝑘(𝐳_𝑘)²^1/2≤2‖‖𝜓^{𝑘,𝐳𝑘}‖‖√𝑐_{𝑘∣𝑘−1}+√𝑑_{𝑘∣𝑘−1}1−‖‖𝑝_𝐷,𝑘‖‖²inf𝜓_{𝑘,𝐳𝑘}𝑛_{𝑀𝑘∣𝑘−1}^{𝑘∣𝑘−1}_𝑖=1⎛⎜⎜⎜⎝1𝐿^{(𝑖)𝑘∣𝑘−1}≤2‖‖𝜓_{𝑘,𝐳𝑘}⎞⎟⎟⎟⎠‖‖√𝑐_{𝑘∣𝑘−1}+√𝑑_{𝑘∣𝑘−1}𝑀^{𝑘∣𝑘−1}inf𝜓_{𝑘,𝐳𝑘}𝑛_{𝑘∣𝑘−1}1−‖‖𝑝_𝐷,𝑘‖‖²𝐿^{min𝑘∣𝑘−1},(A.46) where 𝐿^{min𝑘∣𝑘−1}=min(𝐿^{(1)𝑘∣𝑘−1},…,𝐿^{(𝑀𝑘∣𝑘−1)𝑘∣𝑘−1}), min(⋅) denotes the minimum.

Finally, (4.12) is proved with 𝑐_𝑈,𝑘=⎛⎜⎜⎝2‖‖𝜓^{𝑘,𝐳𝑘}‖‖√𝑐_{𝑘∣𝑘−1}+√𝑑_{𝑘∣𝑘−1}𝑀^{𝑘∣𝑘−1}inf𝜓_{𝑘,𝐳𝑘}𝑛_{𝑘∣𝑘−1}1−‖‖𝑝_𝐷,𝑘‖‖²⎞⎟⎟⎠².(A.47)

Now, turn to (4.13). First, from (2.12), 0≤𝑟^{(𝑖)𝑘∣𝑘−1}≤1 and Assumption 2, we have 𝑟^{(𝑖)𝛼,𝑘}𝐳_𝑘=𝑟^{(𝑖)𝑘∣𝑘−1}𝑝^{(𝑖)𝑘∣𝑘−1},𝜓_{𝑘,𝐳𝑘}1−𝑟^{(𝑖)𝑘∣𝑘−1}𝑝^{(𝑖)𝑘∣𝑘−1},𝑝_𝐷,𝑘≤‖‖𝜓^{𝑘,𝐳𝑘}‖‖1−‖‖𝑝_𝐷,𝑘‖‖.(A.48)

Then, from (2.13), (3.11), and (A.39), we get𝐸𝑟^{∗,𝐿(𝑖)𝑘∣𝑘−1}_𝑈,𝑘(𝐳_𝑘)−𝑟^∗𝑈,𝑘(𝐳_𝑘)²^1/2≤∑^{𝑀𝑘∣𝑘−1𝑖=1}⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝𝐸⎡⎢⎢⎣⎛⎜⎜⎝𝑟_{(𝑖),𝐿(𝑖)𝑘∣𝑘−1𝛼,𝑘}−𝑟_{(𝑖)𝛼,𝑘}(𝐳_𝑘)(𝐳_𝑘)⎞⎟⎟⎠²⎤⎥⎥⎦^1/2+𝐸⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎛⎜⎜⎜⎜⎜⎜⎜⎜⎝𝑟^{(𝑖),𝐿(𝑖)𝑘∣𝑘−1𝛼,𝑘}(𝐳_𝑘)1−𝑟^{(𝑖),𝐿(𝑖)𝑘∣𝑘−1}_{𝑘∣𝑘−1}1−𝑟^{(𝑖),𝐿(𝑖)𝑘∣𝑘−1}_{𝑘∣𝑘−1}𝑝^{(𝑖),𝐿(𝑖)𝑘∣𝑘−1𝑘∣𝑘−1},𝑝_𝐷,𝑘-𝑟^{(𝑖)𝛼,𝑘}(𝐳_𝑘)1−𝑟^{(𝑖)𝑘∣𝑘−1}1−𝑟^{(𝑖)𝑘∣𝑘−1}𝑝^{(𝑖)𝑘∣𝑘−1},𝑝_𝐷,𝑘⎞⎟⎟⎟⎟⎟⎟⎟⎟⎠²⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦^1/2⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠∑^{𝑀𝑘∣𝑘−1}_𝑖=1𝑟^{(𝑖)𝛼,𝑘}𝐳_𝑘(A.49) (adding and subtracting a new term in the second expectation in the summation) ≤∑^{𝑀𝑘∣𝑘−1𝑖=1}⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝𝐸𝑟_(𝑖),𝐿^{(𝑖)𝑘∣𝑘−1}_𝛼,𝑘𝐳_𝑘−𝑟^{(𝑖)𝛼,𝑘}𝐳_𝑘²^1/2+𝐸⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝𝑟^{(𝑖),𝐿(𝑖)𝑘∣𝑘−1𝛼,𝑘}𝐳_𝑘1−𝑟^{(𝑖),𝐿(𝑖)𝑘∣𝑘−1}_{𝑘∣𝑘−1}1−𝑟^{(𝑖),𝐿(𝑖)𝑘∣𝑘−1}_{𝑘∣𝑘−1}𝑝^{(𝑖),𝐿(𝑖)𝑘∣𝑘−1𝑘∣𝑘−1},𝑝_𝐷,𝑘⁻𝑟^{(𝑖)𝛼,𝑘}𝐳_𝑘1−𝑟^{(𝑖)𝑘∣𝑘−1}1−𝑟^{(𝑖),𝐿(𝑖)𝑘∣𝑘−1}_{𝑘∣𝑘−1}𝑝^{(𝑖),𝐿(𝑖)𝑘∣𝑘−1𝑘∣𝑘−1},𝑝_𝐷,𝑘+𝑟^{(𝑖)𝛼,𝑘}𝐳_𝑘1−𝑟^{(𝑖)𝑘∣𝑘−1}1−𝑟^{(𝑖),𝐿(𝑖)𝑘∣𝑘−1}_{𝑘∣𝑘−1}𝑝^{(𝑖),𝐿(𝑖)𝑘∣𝑘−1𝑘∣𝑘−1},𝑝_𝐷,𝑘⁻𝑟^{(𝑖)𝛼,𝑘}𝐳_𝑘1−𝑟^{(𝑖)𝑘∣𝑘−1}1−𝑟^{(𝑖)𝑘∣𝑘−1}𝑝^{(𝑖)𝑘∣𝑘−1},𝑝_𝐷,𝑘⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠²⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦^1/2⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠∑^{𝑀𝑘∣𝑘−1}_𝑖=1𝑟^{(𝑖)𝛼,𝑘}𝐳_𝑘(A.50) It holds that (using Minkowski’s inequality for the second term in the summation) ≤∑^{𝑀𝑘∣𝑘−1𝑖=1}⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝𝐸𝑟_(𝑖),𝐿^{(𝑖)𝑘∣𝑘−1}_𝛼,𝑘(𝐳_𝑘)−𝑟^{(𝑖)𝛼,𝑘}(𝐳_𝑘)²^1/2+𝐸⎡⎢⎢⎢⎢⎣⎛⎜⎜⎜⎜⎝𝑟^{(𝑖),𝐿(𝑖)𝑘∣𝑘−1𝛼,𝑘}(𝐳_𝑘)−𝑟^{(𝑖)𝛼,𝑘}(𝐳_𝑘)+𝑟^{(𝑖)𝛼,𝑘}(𝐳_𝑘)𝑟^{(𝑖)𝑘∣𝑘−1}−𝑟^{(𝑖),𝐿(𝑖)𝑘∣𝑘−1}_𝛼,𝑘(𝐳_𝑘)𝑟^{(𝑖),𝐿(𝑖)𝑘∣𝑘−1}_{𝑘∣𝑘−1}1−𝑟^{(𝑖),𝐿(𝑖)𝑘∣𝑘−1}_{𝑘∣𝑘−1}𝑝^{(𝑖),𝐿(𝑖)𝑘∣𝑘−1𝑘∣𝑘−1},𝑝_𝐷,𝑘⎞⎟⎟⎟⎟⎠²⎤⎥⎥⎥⎥⎦^1/2+𝑟^{(𝑖)𝛼,𝑘}𝐳_𝑘1−𝑟^{(𝑖)𝑘∣𝑘−1}1−𝑟^{(𝑖)𝑘∣𝑘−1}𝑝^{(𝑖)𝑘∣𝑘−1},𝑝_𝐷,𝑘𝐸⎡⎢⎢⎢⎢⎣⎛⎜⎜⎜⎜⎝𝑟^{(𝑖),𝐿(𝑖)𝑘∣𝑘−1𝑘∣𝑘−1}𝑝^{(𝑖),𝐿(𝑖)𝑘∣𝑘−1𝑘∣𝑘−1},𝑝_𝐷,𝑘−𝑟^{(𝑖)𝑘∣𝑘−1}𝑝^{(𝑖)𝑘∣𝑘−1},𝑝_D,𝑘1−𝑟^{(𝑖),𝐿(𝑖)𝑘∣𝑘−1}_{𝑘∣𝑘−1}𝑝^{(𝑖),𝐿(𝑖)𝑘∣𝑘−1𝑘∣𝑘−1},𝑝_𝐷,𝑘⎞⎟⎟⎟⎟⎠²⎤⎥⎥⎥⎥⎦^1/2⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠∑^{𝑀𝑘∣𝑘−1}_𝑖=1𝑟^{(𝑖)𝛼,𝑘}𝐳_𝑘(A.51) (by 0≤𝑟^{(𝑖),𝐿(𝑖)𝑘∣𝑘−1}_{𝑘∣𝑘−1}≤1, (A.4), and Minkowski’s inequality) ≤∑^{𝑀𝑘∣𝑘−1𝑖=1}⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝2−‖‖𝑝_𝐷,𝑘‖‖𝐸𝑟^{(𝑖),𝐿(𝑖)𝑘∣𝑘−1}_𝛼,𝑘(𝐳_𝑘)−𝑟^{(𝑖)𝛼,𝑘}(𝐳_𝑘)²^1/2+𝐸⎡⎢⎢⎢⎣⎛⎜⎜⎝𝑟^{(𝑖)𝛼,𝑘}(𝐳_𝑘)𝑟^{(𝑖)𝑘∣𝑘−1}−𝑟^{(𝑖)𝛼,𝑘}(𝐳_𝑘)𝑟^{(𝑖),𝐿(𝑖)𝑘∣𝑘−1}_{𝑘∣𝑘−1}+𝑟^{(𝑖)𝛼,𝑘}(𝐳_𝑘)𝑟^{(𝑖),𝐿(𝑖)𝑘∣𝑘−1}_{𝑘∣𝑘−1}−𝑟^{(𝑖),𝐿(𝑖)𝑘∣𝑘−1}_𝛼,𝑘(𝐳_𝑘)𝑟^{(𝑖),𝐿(𝑖)𝑘∣𝑘−1}_{𝑘∣𝑘−1}⎞⎟⎟⎠²⎤⎥⎥⎥⎦^1/2+𝑟^{(𝑖)𝛼,𝑘}𝐳_𝑘1−𝑟^{(𝑖)𝑘∣𝑘−1}1−𝑟^{(𝑖)𝑘∣𝑘−1}𝑝^{(𝑖)𝑘∣𝑘−1},𝑝_𝐷,𝑘⎛⎜⎜⎜⎜⎜⎝‖‖𝑝^𝐷,𝑘‖‖𝐸𝑟^{(𝑖),𝐿(𝑖)𝑘∣𝑘−1}_{𝑘∣𝑘−1}−𝑟^{(𝑖)𝑘∣𝑘−1}²^1/2+𝑟^{(𝑖)𝑘∣𝑘−1}𝐸𝑝^{(𝑖),𝐿(𝑖)𝑘∣𝑘−1}_{𝑘∣𝑘−1},𝑝_𝐷,𝑘−𝑝^{(𝑖)𝑘∣𝑘−1},𝑝_𝐷,𝑘²^1/2⎞⎟⎟⎟⎟⎟⎠⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠1−‖‖𝑝_𝐷,𝑘‖‖∑^{𝑀𝑘∣𝑘−1}_𝑖=1𝑟^{(𝑖)𝛼,𝑘}𝐳_𝑘(A.52) (using 0≤𝑟^{(𝑖),𝐿(𝑖)𝑘∣𝑘−1}_{𝑘∣𝑘−1}≤1 and Minkowski’s inequality again for the second term in the summation) ≤∑^{𝑀𝑘∣𝑘−1𝑖=1}⎛⎜⎜⎜⎜⎜⎜⎜⎜⎝3−‖‖𝑝_𝐷,𝑘‖‖𝐸𝑟^{(𝑖),𝐿(𝑖)𝑘∣𝑘−1}_𝛼,𝑘(𝐳_𝑘)−𝑟^{(𝑖)𝛼,𝑘}(𝐳_𝑘)²^1/2+𝑟^(i)𝛼,𝑘𝐳_𝑘𝐸𝑟^{(𝑖)𝑘∣𝑘−1}−𝑟^{(𝑖),𝐿(𝑖)𝑘∣𝑘−1}_{𝑘∣𝑘−1}²^+1/2𝑟^{(𝑖)𝛼,𝑘}𝐳_𝑘1−𝑟^{(𝑖)𝑘∣𝑘−1}1−𝑟^{(𝑖)𝑘∣𝑘−1}𝑝^{(𝑖)𝑘∣𝑘−1},𝑝_𝐷,𝑘^{⎛⎜⎜⎜⎜⎜⎜⎜⎜⎝}‖‖𝑝_𝐷,𝑘‖‖𝐸𝑟^{(𝑖),𝐿(𝑖)𝑘∣𝑘−1}_{𝑘∣𝑘−1}−𝑟^{(𝑖)𝑘∣𝑘−1}²^1/2+𝑟^{(𝑖)𝑘∣𝑘−1}𝐸𝑝^{(𝑖),𝐿(𝑖)𝑘∣𝑘−1}_{𝑘∣𝑘−1},𝑝_𝐷,𝑘−𝑝^{(𝑖)𝑘∣𝑘−1},𝑝_𝐷,𝑘²^{⎞⎟⎟⎟⎟⎟⎟⎟⎟⎠1/2}⎞⎟⎟⎟⎟⎟⎟⎟⎟⎠1−‖‖𝑝_𝐷,𝑘‖‖∑^{𝑀𝑘∣𝑘−1}_𝑖=1𝑟^{(𝑖)𝛼,𝑘}𝐳_𝑘(A.53) (by 0≤𝑟^{(𝑖)𝑘∣𝑘−1}≤1, (4.8), (4.9) (A.44), (A.45), and Assumption 2) ≤‖‖𝜓^{𝑘,𝐳𝑘}‖‖√𝑐_{𝑘∣𝑘−1}+√𝑑_{𝑘∣𝑘−1}1−‖‖𝑝_𝐷,𝑘‖‖³inf𝜓_{𝑘,𝐳𝑘}𝑛_{𝑀𝑘∣𝑘−1}^{𝑘∣𝑘−1}_𝑖=1⎛⎜⎜⎜⎝4−𝑟^{(𝑖)𝑘∣𝑘−1}+1‖‖𝑝_𝐷,𝑘‖‖𝐿^{(𝑖)𝑘∣𝑘−1}≤‖‖𝜓_{𝑘,𝐳𝑘}⎞⎟⎟⎟⎠‖‖√𝑐_{𝑘∣𝑘−1}+√𝑑_{𝑘∣𝑘−1}4𝑀^{𝑘∣𝑘−1}−𝑛_{𝑘∣𝑘−1}+𝑀_{𝑘∣𝑘−1}‖‖𝑝_𝐷,𝑘‖‖1−‖‖𝑝_𝐷,𝑘‖‖³inf𝜓_{𝑘,𝐳𝑘}𝑛_{𝑘∣𝑘−1}𝐿^{min𝑘∣𝑘−1}.(A.54)

Finally, (4.13) is proved with 𝑐^∗𝑈,𝑘=⎛⎜⎜⎝‖‖𝜓^{𝑘,𝐳𝑘}‖‖√𝑐_{𝑘∣𝑘−1}+√𝑑_{𝑘∣𝑘−1}1−‖‖𝑝_𝐷,𝑘‖‖³inf(𝜓_{𝑘,𝐳𝑘})4−‖‖𝑝^𝐷,𝑘‖‖𝑀_{𝑘∣𝑘−1}𝑛_{𝑘∣𝑘−1}−‖‖𝑝_𝐷,𝑘‖‖⎞⎟⎟⎠².(A.55)

Now turn to (4.14). From (2.10) and (3.12), we get 𝐸𝑝_𝑈,𝑘(⋅;𝐳_𝑘),𝜑−𝑝^{𝐿(𝑖)𝑘∣𝑘−1}_𝑈,𝑘(⋅;𝐳_𝑘),𝜑²^1/2=𝐸⎡⎢⎢⎢⎢⎣⎛⎜⎜⎜⎜⎝∑^{𝑀𝑘∣𝑘−1𝑖=1}𝑝^{(𝑖),𝐿(𝑖)𝑘∣𝑘−1}_𝛼,𝑘𝐱_𝑘;𝐳_𝑘,𝜑∑^{𝑀𝑘∣𝑘−1}_𝑖=1𝑟^{(𝑖),𝐿(𝑖)𝑘∣𝑘−1}_𝛼,𝑘𝐳_𝑘−∑^{𝑀𝑘∣𝑘−1𝑖=1}𝑝^{(𝑖)𝛼,𝑘}𝐱_𝑘;𝐳_𝑘,𝜑∑^{𝑀𝑘∣𝑘−1}_𝑖=1𝑟^{(𝑖)𝛼,𝑘}𝐳_𝑘⎞⎟⎟⎟⎟⎠²⎤⎥⎥⎥⎥⎦^1/2(A.56) (adding and subtracting a new term) =𝐸⎡⎢⎢⎢⎢⎣⎛⎜⎜⎜⎜⎝∑^{𝑀𝑘∣𝑘−1𝑖=1}𝑝^{(𝑖),𝐿(𝑖)𝑘∣𝑘−1}_𝛼,𝑘⋅;𝐳_𝑘,𝜑∑^{𝑀𝑘∣𝑘−1}_𝑖=1𝑟^{(𝑖),𝐿(𝑖)𝑘∣𝑘−1}_𝛼,𝑘𝐳_𝑘−∑^{𝑀𝑘∣𝑘−1𝑖=1}𝑝^{(𝑖),𝐿(𝑖)𝑘∣𝑘−1}_𝛼,𝑘⋅;𝐳_𝑘,𝜑+∑_{𝑀𝑘∣𝑘−1𝑖=1}∑^{𝑀𝑘∣𝑘−1}_𝑖=1𝑟^{(𝑖)𝛼,𝑘}𝐳_𝑘𝑝^{(𝑖),𝐿(𝑖)𝑘∣𝑘−1}_𝛼,𝑘⋅;𝐳_𝑘,𝜑∑^{𝑀𝑘∣𝑘−1}_𝑖=1𝑟^{(𝑖)𝛼,𝑘}(𝐳_𝑘)−∑^{𝑀𝑘∣𝑘−1𝑖=1}𝑝^{(𝑖)𝛼,𝑘}(⋅;𝐳_𝑘),𝜑∑^{𝑀𝑘∣𝑘−1}_𝑖=1𝑟^{(𝑖)𝛼,𝑘}(𝐳_𝑘)⎞⎟⎟⎟⎟⎠²⎤⎥⎥⎥⎥⎦^1/2(A.57) (using Minkowski’s inequality) ≤𝐸⎡⎢⎢⎢⎣⎛⎜⎜⎝^{𝑀𝑘∣𝑘−1}_𝑖=1𝑝^{(𝑖),𝐿(𝑖)𝑘∣𝑘−1}_𝛼,𝑘(⋅;𝐳_𝑘),𝜑⎛⎜⎜⎝∑^{𝑀𝑘∣𝑘−1𝑖=1}𝑟^{(𝑖),𝐿(𝑖)𝑘∣𝑘−1}_𝛼,𝑘(𝐳_𝑘)−∑^{𝑀𝑘∣𝑘−1}_𝑖=1𝑟^{(𝑖)𝛼,𝑘}(𝐳_𝑘)∑^{𝑀𝑘∣𝑘−1}_𝑖=1𝑟^{(𝑖)𝛼,𝑘}(𝐳_𝑘)∑^{𝑀𝑘∣𝑘−1}_𝑖=1𝑟^{(𝑖),𝐿(𝑖)𝑘∣𝑘−1}_𝛼,𝑘(𝐳_𝑘)⎞⎟⎟⎠⎞⎟⎟⎠²⎤⎥⎥⎥⎦^1/2+𝐸∑^{𝑀𝑘∣𝑘−1𝑖=1}𝑝^{(𝑖),𝐿(𝑖)𝑘∣𝑘−1}_𝛼,𝑘(⋅;𝐳_𝑘)−∑^{𝑀𝑘∣𝑘−1}_𝑖=1𝑝^{(𝑖)𝛼,𝑘}(⋅;𝐳_𝑘),𝜑²^1/2∑^{𝑀𝑘∣𝑘−1}_𝑖=1𝑟^{(𝑖)𝛼,𝑘}𝐳_𝑘(A.58) (by (2.12) and (3.14)) ≤2‖𝜑‖∑^{𝑀𝑘∣𝑘−1}_𝑖=1𝑟^{(𝑖)𝛼,𝑘}𝐳_𝑘𝐸⎡⎢⎢⎣⎛⎜⎜⎝^{𝑀𝑘∣𝑘−1}_𝑖=1𝑟^{(𝑖),𝐿(𝑖)𝑘∣𝑘−1}_𝛼,𝑘(𝐳_𝑘)−^{𝑀𝑘∣𝑘−1}_𝑖=1𝑟^{(𝑖)𝛼,𝑘}(𝐳_𝑘)⎞⎟⎟⎠²⎤⎥⎥⎦^1/2(A.59) (by (A.39) and (A.47)) ≤‖𝜑‖𝐿^{min𝑘∣𝑘−1}𝑐_𝑈,𝑘.(A.60)

Finally, (4.14) is proved with 𝑑_𝑈,𝑘=𝑐_𝑈,𝑘=⎛⎜⎜⎝2‖‖𝜓^{𝑘,𝐳𝑘}‖‖√𝑐_{𝑘∣𝑘−1}+√𝑑_{𝑘∣𝑘−1}𝑀^{𝑘∣𝑘−1.}inf(𝜓_{𝑘,𝐳𝑘})𝑛_{𝑘∣𝑘−1}1−‖‖𝑝_𝐷,𝑘‖‖²⎞⎟⎟⎠².(A.61)

This completes the proof.

Nomenclature

𝐱𝑘:	State vector of a single target at time 𝑘
𝐳𝑘:	Single measurement vector at time 𝑘
𝑛𝑘:	Number of existing targets at time 𝑘
𝑚𝑘:	Number of measurements collected at time 𝑘
𝑋𝑘={𝐱𝑖,𝑘}𝑛𝑘𝑖=1:	Finite set of multitarget state-vectors at time 𝑘
𝑍𝑘={𝐳𝑖,𝑘}𝑚𝑘𝑖=1:	Finite set of measurements collected at time 𝑘
𝑓𝑘∣𝑘−1(𝐱𝑘∣𝐱𝑘−1):	Single-target Markov transition density at time 𝑘
𝑝𝑆,𝑘(𝐱𝑘−1):	Probability of target survival at time 𝑘
𝑝𝐷,𝑘(𝐱𝑘):	Probability of detection at time 𝑘
𝜅𝑘(𝐳𝑘):	Intensity of Poisson clutter process at time 𝑘
𝜓𝑘,𝐳𝑘(𝐱𝑘)=𝑓𝑘(𝐳𝑘∣𝐱𝑘):	Single-sensor/target likelihood density at time 𝑘
𝛿𝐱(⋅):	Dirac delta function centered at 𝐱
ℝ𝑑:	𝑑-dimensional real space
𝐶𝑏(ℝ𝑑):	Set of continuous bounded functions on ℝ𝑑
𝐵(ℝ𝑑):	Set of bounded Borel measurable functions on ℝ𝑑
𝜋(𝑌(𝑖)):	Probability density of a Bernoulli random finite set (RFS) 𝑌(𝐢)
𝜋(𝑌):	Probability density of multi-Bernoulli RFS 𝑌=⋃𝑀𝑖=1𝑌(𝑖)
𝜋={(𝑟(𝑖),𝑝(𝑖))}𝑀𝑖=1:	Abbreviation of 𝜋(𝑌). {(𝑟(𝑖),𝑝(𝑖))}𝑀𝑖=1 is the multi-Bernoulli parameter set
𝜋={(𝑟(𝑖),𝐿(𝑖),𝑝(𝑖),𝐿(𝑖))}𝑀𝑖=1:	Particle approximation of 𝜋={(𝑟(𝑖),𝑝(𝑖))}𝑀𝑖=1. (𝑟(𝑖),𝐿(𝑖),𝑝(𝑖),𝐿(𝑖)) denotes that (𝑟(𝑖),𝑝(𝑖)) is comprised of the number 𝐿(𝑖) of the particles
‖⋅‖:	Supremum norm. ‖𝜑‖≜sup(∣𝜑∣), sup(⋅)  denotes the supremum
⟨⋅,⋅⟩:	Inner product. If the measure in ⟨⋅,⋅⟩ is continuous, it defines the integral inner product; if the measure in ⟨⋅,⋅⟩ is discrete, it defines the summation inner product.

Acknowledgments

This research work was supported by the Natural Science Foundation of China (61004087, 61104051, 61005026), China Postdoctoral Science Foundation (20100481338), and Fundamental Research Funds for the Central University.