pr.probability – When are all average trajectories of $w_{k+1}=Aw_k+b$ bounded?


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 $mathcal{D}$, $w_0in mathbb{R}^d$, 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

  1. $b=0$
  2. $b=c$ is some fixed vector $c$ from $mathbb{R}^d$
  3. $b=Bx$ for some matrix $B$
  4. $b=d$ where $d$ is drawn from some distribution $mathcal{D}_2$ independent of $x$
  5. $b=c+Bx+d$
  6. $b,x$ 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 (using identities 20.18 and 20.25c of Seber’s Matrix Handbook book)

$$(text{Tr}(ASigma))^2+2text{Tr}(ASigma)^2<2text{Tr}ASigma$$