In a kind of signal processing problem I faced the following recursive (boot-strap) algorithm:
$$R_{k} = R_{k-1} + (y_k-H s_{k-1})*(y_k-H s_{k-1})^T$$ $$s_k = (H^T R_k^{-1} H)^{-1} H^T R_k^{-1} y_k$$
where $R_k$ is a $r\times r$ matrix and $H_k$ is a $r\times t$ matrix. $s_k$ is a $t\times 1$ vector and $y_k$ is a $r\times 1$ matrix. the subscript $k$ denotes the iteration time index. $R_0 = I$
I know it seems that such a question is not proper to be asked here. But I wanted to know, is there any method or well-known mathematical scheme to prove the convergence of such recursive algorithms in math? Considering that it is bootstrap, i.e. The variables $s_k$ and $R_k$ are jointly being updated.
I'll be realy thankful if someone help me on this issue.
Regards
This is not an actual answer. I would just have few observations.
$R_k$ can be written as $R_k=R_{k-1}+w_kw_k^T$, where $w_k=y_k-Hs_{k-1}$ is a certain orthogonal projection of $y_k$. This can be observed from the fact that $$ w_k = (I-Q_k)y_k, \quad Q_k = H(H^TR_{k-1}^{-1}H)^{-1}H^TR_{k-1}^{-1}. $$ The matrix $Q_k$ is the projector orthogonal to the range of $H$ with respect to the inner product induced by $R_{k-1}^{-1}$ (and hence $I-Q_k$ projects on its complement orthogonal w.r.t. the same inner product): $$ Q_{k-1}^2 = Q_k, \quad R_{k-1}^{-1}Q_{k-1} = (R_{k-1}^{-1}Q_{k-1})^T. $$
Of course talking about the inner product induced by the inverses of $R_k$'s makes sense, since each of them is SPD.
In any case, I'm not sure to what the algorithm should converge in the first place since the vectors $y_k$ are not given.