Consider a Toeplitz matrix $T$, indexed by $\mathbb{N}\_0 \times \mathbb{N}\_0$. given by the sequence $t\_k,k \in \mathbb{Z}$ where $t\_k \geq 0,\sum\_{k=-\infty}^\infty t\_k=1$. By this I mean that $T\_{i,i+k}=t\_k$ for all $i \in \mathbb{N}\_0$ and $k \in \{ -i,-(i-1),\dots,0,1,\dots \}$.


It is easy to see that $T$ defines a bounded operator on all the $\ell^p(\mathbb{N}\_0)$ spaces. Suppose additionally that there are vectors $b,x^\*$ with $Tx^\*+b=x^\*$. 


I would like to understand under what assumptions the iteration $x^{(k+1)}= Tx^{(k)}+b$ will converge to $x^\*$.


It is not hard to see that this will not happen if $\lim\_{i \to \infty} x^{(0)}\_i$ and $\lim\_{i \to \infty} x^\*\_i$ both exist but are different. Thus $\ell^\infty$ perturbations will in general not decay. But it seems that this should be the only obstacle, i.e. $\ell^p$ perturbations should decay. 


However, Szego's theorem tells us that the spectrum of $T$ accumulates around $1$, since it traces out the curve $\left \{ \sum\_{k=-\infty}^\infty t\_k e^{2 \pi i k \lambda} : \lambda \in [0,1] \right \}$. So it seems that there is no hope of using a simple Banach fixed point theorem argument. Is there some workaround? Perhaps some kind of Riemann-Lebesgue lemma type argument based on the idea that the eigenvectors whose eigenvalues are near $1$ cannot contribute too much to any fixed $x^{(0)}$? Or perhaps some kind of Perron-Frobenius type result?



When does iteration of an infinite Toeplitz matrix converge?
OK. The case when one $t\_k=1$ and the rest are $0$ is easy to figure out (left shifts are good, the rest are bad).
Assume that all $t\_k<1$. The $\ell\_1$ problem is essentially equivalent to the question when the corresponding random walk on $\mathbb Z$ has positive chance to stay above $0$ forever (this is not obvious to me except in non-interesting cases like that of finite step variance, say, and the answer *does* depend on $t\_k$). However, any other $\ell^p$ ($1<p<\infty$) is trivial again because it will suffice to show that the images of every $x$ consisting of a single $1$ and a bunch of zeroes tend to $0$ in the uniform norm since their $\ell^1$ norms are bounded by $1$, which is true even for the full convolution on $\mathbb Z$ (if $f(z)=\sum t\_kz^k$, then all entries of $T^kx$ are bounded by $\int\_{\mathbb T}|f(z)|^k\,dm(z)$ and the dominated convergence theorem finishes the story). 


So, the question is now just this. Given a random walk on $\mathbb Z$, find a reasonable necessary and sufficient condition for its having a positive chance to stay above $0$ forever starting with some sufficiently positive point. I'm inclined to wait for real probabilists to say something about it because that has definitely been considered in the literature and I do not feel like it is an undergraduate level question.


