# 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)

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