## Appendix A  
We here investigate the evaluation of \(q_{w}(n+1)\). \[ \begin{array}{ll}q_{w}(n+1)&=q_{w}(n)-2\mu_{w}\mathrm{E}(\Delta\mathbf{w}^{ \mathsf{T}}(n)(\mathbf{A}(n)+\mathbf{B}(n))^{\mathsf{T}}\Delta\mathbf{w}(n)) \\ &+\mu_{w}^{2}\mathrm{E}(\Delta\mathbf{w}^{\mathsf{T}}(n)(\mathbf{A}(n)+ \mathbf{B}(n))^{\mathsf{T}}\\ &\times(\mathbf{A}(n)+\mathbf{B}(n))\Delta\mathbf{w}(n))\\ &+\mu_{w}^{2}\mathrm{E}(\mathbf{b}^{\mathsf{T}}(n)\mathbf{b}(n))\end{array} . \tag{10}\]  
For the third term:  
\[ \begin{array}{ll}\Delta\mathbf{w}^{\mathsf{T}}(n)(\mathbf{A}(n)+\mathbf{B}( n))^{\mathsf{T}}(\mathbf{A}(n)+\mathbf{B}(n))\Delta\mathbf{w}(n)\\ =\big{|}\big{|}\Delta\mathbf{w}^{\mathsf{T}}(n)(\mathbf{A}(n)+\mathbf{B}(n))^{ \mathsf{T}}(\mathbf{A}(n)+\mathbf{B}(n))\Delta\mathbf{w}(n)\big{|}\big{|}_{ \infty}\\ \leq\big{|}\big{|}\Delta\mathbf{w}^{\mathsf{T}}(n)\big{|}\big{|}_{\infty} \big{|}\big{|}(\mathbf{A}(n)+\mathbf{B}(n))^{\mathsf{T}}(\mathbf{A}(n)+ \mathbf{B}(n))\big{|}\big{|}_{\infty}\big{|}\big{|}\Delta\mathbf{w}(n)\big{|} \big{|}_{\infty}\\ =\big{|}\big{|}(\mathbf{A}(n)+\mathbf{B}(n))^{\mathsf{T}}(\mathbf{A}(n)+ \mathbf{B}(n))\big{|}\big{|}_{\infty}\big{|}\big{|}\Delta\mathbf{w}^{\mathsf{ T}}(n)\big{|}_{\infty}\big{|}\big{|}\Delta\mathbf{w}(n)\big{|}\big{|}_{\infty} \end{array} . \tag{11}\]  
Using the definition of \(\left\|\cdot\right\|_{\infty}\):  
\[ \begin{array}{ll}\left\|\Delta\mathbf{w}(n)\right\|_{\infty}\big{\|}\Delta \mathbf{w}^{\mathsf{T}}(n)\big{\|}_{\infty}&=\max_{(}|\Delta w_{i}(n)|,0\leq i \leq N-1)\\ &\times(\sum\limits_{i=0}^{N-1}\big{|}\Delta w_{i}(n)|)\\ &\leq N(\sum\limits_{i=0}^{N-1}\Delta w_{i}^{2}(n))=Nq_{w}(n)\end{array} . \tag{12}\]  
Note, \(\mathbf{A}(n)\) is a symmetric matrix:  
\[ \begin{array}{ll}\big{|}\big{|}(\mathbf{A}(n)+\mathbf{B}(n))^{\mathsf{T}}( \mathbf{A}(n)+\mathbf{B}(n))\big{|}\big{|}_{\infty}\\ \leq\big{|}\big{|}\big{|}(\mathbf{A}^{2}(n)\big{|}\big{|}_{\infty}+\big{|} \big{|}\mathbf{B}^{\mathsf{T}}(n)\mathbf{B}(n)\big{|}\big{|}_{\infty}\\ +\big{|}\big{|}\mathbf{A}(n)\mathbf{B}(n)\big{|}\big{|}_{\infty}+\big{|}\big{|} \mathbf{B}^{\mathsf{T}}(n)\mathbf{A}(n)\big{|}\big{|}_{\infty}\end{array} . \tag{13}\]  
The \((k,m)\)-th (\(0\leq k,m\leq N-1\)) entry of \(\mathbf{A}(n)\), \(\mathbf{B}(n)\) is:  
\[a_{km}(n)=\sum\limits_{j=0}^{L-1}\sum\limits_{i=0}^{L-1}(s_{L-1-i}s_{L-1-j}x(n -i-k)x(n-j-m)), \tag{14}\]  
\[b_{km}(n)=\sum\limits_{j=0}^{L-1}\sum\limits_{i=0}^{L-1}(\Delta s_{L-1-i}(n)s_{ L-1-j}x(n-i-k)x(n-j-m)). \tag{15}\]The \((k1,m1)\)-th \((0\leq k1,m1\leq N-1)\) entry of \(\mathbf{A}^{2}(n)\) is:  
\[ \begin{array}{ll}a^{\prime}{}_{k1m1}(n)&=\sum\limits_{q=0}^{N-1}\sum\limits_{p= 0}^{N-1}a_{k1q}a_{pm1}\\ &=\sum\limits_{q=0}^{N-1}\sum\limits_{p=0}^{N-1}\sum\limits_{i=0}^{L-1}\sum \limits_{j=1}^{L-1}\sum\limits_{i=0}^{L-1}\sum\limits_{j=0}^{L-1}(s_{L-1-i}s_{ L-1-j}s_{L-1-i1}s_{L-1-j1}\\ &\times x(n-i-k1)x(n-j-q)x(n-i1-p)x(n-j1-m1))\\ &\leq N^{2}\sigma_{x}^{4}(\sum\limits_{j=0}^{L-1}\sum\limits_{i=0}^{L-1}s_{L-1- j}^{2}s_{L-1-i}^{2}+2\sum\limits_{i=0}^{L-1}s_{L-1-i}^{4})\\ &\leq 3R^{2}N^{2}\sigma_{x}^{4}\end{array} . \tag{10}\]  
By (10), it can derive that:  
\[ \begin{array}{ll}E(a^{\prime}{}_{k1m1}(n))&\leq N^{2}\sigma_{x}^{4}(\sum \limits_{j=0}^{L-1}\sum\limits_{i=0}^{L-1}s_{L-1-j}^{2}s_{L-1-i}^{2}+2\sum \limits_{i=0}^{L-1}s_{L-1-i}^{4})\\ &\leq 3R^{2}N^{2}\sigma_{x}^{4}\end{array} , \tag{11}\]  
\[ \begin{array}{ll}E(\left\|\mathbf{A}^{2}\right\|_{\infty})\leq 3R^{2}N^{3} \sigma_{x}^{4}.\end{array}  \tag{12}\]  
The \((k1,m1)\)-th \((0\leq k1,m1\leq N-1)\) entry of \(\mathbf{A}(n)\mathbf{B}(n)\) is:  
\[ \begin{array}{ll}c_{k1m1}(n)=\sum\limits_{q=0}^{N-1}\sum\limits_{p=0}^{N-1}a _{k1q}b_{pm1}\\ =\sum\limits_{q=0}^{N-1}\sum\limits_{p=0}^{N-1}\sum\limits_{j=0}^{L-1}\sum \limits_{j=1}^{L-1}\sum\limits_{i=0}^{L-1}\sum\limits_{i=0}^{L-1}(s_{L-1-i}s_ {L-1-i})s_{L-1-i}\\ \times x(n-i-k1)x(n-j-q)x(n-i1-p)x(n-j1-m1)))\end{array} . \tag{13}\]  
By (10), it can be derived that:  
\[