Below is an open-problem in my field, and I'm wondering if someone has insights I'm missing. (cross-posted on math.se)
Suppose observation $x$ is drawn from some distribution $\mathcal{D}$, $w_0\in \mathbb{R}^\mathrm{d}$, and my update has the following form
$$w_{k+1}=(I-xx')w_k+b$$
When are average trajectories $u_k=E[w_k]$ bounded? Expectation is taken over all sequences of IID observations $x_1,\ldots,x_k$. Motivation for this recurrence is here.
I have worked out the case of $b=0$ and I'm wondering if global asymptotic stability for $b=0$ also implies boundedness for some other value of $b$. An interesting special case is when $x$ comes from Gaussian centered at 0.
Cases of increasing difficulty are
- $b=0$
- $b=c$ is some vector $c$ from $\mathbb{R}^\mathrm{d}$
- $b=Bx$ for some $\mathrm{d}\times \mathrm{d}$ matrix $B$
- $b=d$ where $d$ is drawn from some distribution $\mathcal{D}_2$ independent of $x$
- $b=c+Bx+d$
- $b,x$ are drawn jointly from some distribution $\mathcal{D}_3$
For the case of $b=0$, condition below seems to be a necessary and sufficient condition for all trajectories $u_k$ to converge to 0. The following must hold for all symmetric matrices $A$
$$E[(x'Ax)^2]<2 E[x'A^2 x]$$
For the case of $x$ coming from zero-centered Gaussian with covariance $\Sigma$, this becomes the following equation, which can be checked in practice using generalized eigenvalue solver (using identities 20.18 and 20.25c of Seber's Matrix Handbook book)
$$(\text{Tr}(A\Sigma))^2+2\text{Tr}(A\Sigma)^2<2\text{Tr}A^2\Sigma$$
