**5. Convergence Analysis**

**Theorem 1.** *For the system in (1)–(2) and the AM-MIFSG algorithm in (31)–(47), assume that the noise sequence* {*vk*} *satisfies*

$$\mathbb{E}(\mathbf{A}1) \to [v\_k | \mathcal{F}\_t] = 0, \text{ a.s., } \operatorname{E}[v\_k^2 | \mathcal{F}\_t] \lessapprox \sigma^2 \prec \infty, \text{ a.s., } \mathcal{F}\_t$$

*and there exist an integer Nk and a positive constant independent of k such that the following persistent excitation condition holds,*

$$(\text{A2})\sum\_{i=0}^{N\_k} \frac{\hat{\Phi}\_{a,p,k+i}^{\text{r}} \hat{\Phi}\_{a,p,k+i}}{s\_{k+i}} \gtrless \varrho I,\text{ a.s.},\tag{48}$$

*where* **Φ**ˆ *<sup>α</sup>*,*p*,*k* = [*ϕ*ˆ *k θ<sup>α</sup>*,*ϕ*<sup>ˆ</sup> *k*−1 *θ<sup>α</sup>*, ··· ,*ϕ*<sup>ˆ</sup> *k*−*p*+<sup>1</sup> *<sup>θ</sup>α*]*, θα* := 1*n* + *θ*ˆ 1−*<sup>α</sup> k*−1 *, denotes an elementby-element multiplication of vectors. Then the parameter estimation error given by the AM-MIFSG algorithm satisfies* lim*k*→∞ E[*θ*<sup>ˆ</sup> *k* − *θ*<sup>2</sup>] → 0*.*

**Proof.** Define the parameter estimation error *θ* ¯ *k* = *θ* ˆ *k* − *θ* ∈ R*<sup>n</sup>*. To simplify the proof, assuming *<sup>s</sup>α*,*<sup>k</sup>* = *sk*/Γ(<sup>2</sup> − *<sup>α</sup>*). Inserting (32) into (31) and rearranging, we have

$$\begin{array}{rcl} \bar{\theta}\_{k} &=& \bar{\theta}\_{k-1} + \frac{\bar{\Phi}\_{p,k}}{s\_{k}} \left[ \mathbf{Y}\_{p,k} - \hat{\Phi}\_{p,k}^{\top} \hat{\theta}\_{k-1} \right] \odot \boldsymbol{\theta}\_{\boldsymbol{a}} \\\\ &=& \bar{\theta}\_{k-1} + \frac{\bar{\Phi}\_{p,k}}{s\_{k}} \left[ \boldsymbol{\Phi}\_{p,k}^{\top} \theta\_{k-1} + \mathbf{V}\_{p,k} - \hat{\Phi}\_{p,k}^{\top} \hat{\theta}\_{k-1} \right] \odot \boldsymbol{\theta}\_{\boldsymbol{a}} \\\\ &=& \boldsymbol{\theta}\_{k-1} + \frac{\bar{\Phi}\_{p,k}}{s\_{k}} \left[ \mu\_{p,k} - \mathfrak{c}\_{p,k} + \mathbf{V}\_{p,k} \right] \odot \boldsymbol{\theta}\_{\boldsymbol{a}} \end{array} \tag{49}$$

where

$$\begin{array}{rcl} \mu\_{q,t} & := & [\boldsymbol{\Phi}\_{p,k} - \boldsymbol{\hat{\Phi}}\_{p,k}]^\top \boldsymbol{\theta} \in \mathbb{R}^p, \quad \boldsymbol{\mathfrak{g}}\_{q,t} := \boldsymbol{\hat{\Phi}}\_{p,k}^\top \boldsymbol{\bar{\theta}}\_{k-1} \in \mathbb{R}^p, \\\ \mathcal{V}\_{p,k} & := & [\boldsymbol{v}\_{k}, \boldsymbol{v}\_{k-1}, \cdot, \cdot, \boldsymbol{v}\_{k-p+1}] \in \mathbb{R}^p. \end{array}$$

Pre-multiplying (49) by *θ* ¯ T *k* gives

$$\begin{aligned} \boldsymbol{\theta}\_{k}^{\mathrm{r}} \boldsymbol{\theta}\_{k} &= \quad \boldsymbol{\theta}\_{k-1}^{\mathrm{r}} \boldsymbol{\theta}\_{k-1} + \frac{2}{s\_{k}} \boldsymbol{\theta}\_{k-1}^{\mathrm{r}} \boldsymbol{\Phi}\_{a,p,k} [\boldsymbol{\mu}\_{p,k} - \boldsymbol{\xi}\_{p,k} + \boldsymbol{V}\_{p,k}] \\ &+ \frac{1}{r\_{k}^{2}} [\boldsymbol{\mu}\_{p,k} - \boldsymbol{\xi}\_{p,k} + \boldsymbol{V}\_{p,k}]^{\mathrm{r}} \boldsymbol{\Phi}\_{a,p,k}^{\mathrm{r}} \boldsymbol{\Phi}\_{a,p,k} [\boldsymbol{\mu}\_{p,k} - \boldsymbol{\xi}\_{p,k} + \boldsymbol{V}\_{p,k}]. \end{aligned}$$

The rest can be proved in a similar to the way in [66].
