Why lattice-structured all-pole IIR filter becomes stable, if the k-coefficients satisfies $|k_i| < 1$?
I'm referring to the formulas on page 9 of this document. The denominator polynomial of the transfer function at lattice stage number $m$ is given by

$$\begin{align}A_m(z)&amp;=A_{m-1}(z)+k_mz^{-1}B_{m-1}(z)\\
\text{with}\quad B_m(z)&amp;=z^{-m}A_m(1/z)\\
\text{and}\quad A_0(z)&amp;=B_0(z)=1\end{align}\tag{1}$$

where $k_m$ is the $m^{th}$ reflection coefficient.

You can prove that $A_m(z)$ has all its zeros inside the unit circle if this is the case for $A_{m-1}(z)$ and if $|k_m|&lt;1$. This follows from Rouché's Theorem, which says that if $F(z)$ and $G(z)$ are two complex-valued functions that are analytic in and on some closed contour $C$ (which in our case is the unit circle), then $F(z)$ and $F(z)+G(z)$ have the same number of zeros inside $C$ if $|G(z)|&lt;|F(z)|$ is satisfied on $C$.

The proof goes as follows. Define $F(z)=z^mA_{m-1}(z)$ and $G(z)=k_mz^{m-1}B_{m-1}(z)$, which, according to $(1)$, means that $F(z)+G(z)=z^mA_{m}(z)$. The zeros of $F(z)$ are the $m-1$ zeros of $A_{m-1}(z)$ plus one zero at $z=0$. Consequently, if all zeros of $A_{m-1}(z)$ are inside the unit circle, then also all zeros of $F(z)$ are inside the unit circle. From the definition of $B_m(z)$ in $(1)$ it follows that $B_{m-1}(z)$ has the same magnitude as $A_{m-1}(z)$ on the unit circle (i.e., for $|z|=1$). This means that $|F(z)|&lt;|G(z)|$ is satisfied on the unit circle if $|k_m|&lt;1$. In this case $F(z)+G(z)$ and, consequently, $A_m(z)$ have all their zeros inside the unit circle. This is the case because $F(z)$ has all its $m$ zeros inside the unit circle, and according to Rouché's theorem $F(z)+G(z)$, which has exactly $m$ zeros, has the same number of zeros inside the unit circle as $F(z)$. So $F(z)+G(z)$ must have all its $m$ zeros inside the unit circle.

This proves that all denominator polynomials $A_m(z)$ in the iteration given by $(1)$ correspond to stable transfer functions if the iteration is started with $A_0(z)=B_0(z)=1$ and if $|k_m|&lt;1$ is satisfied for all $m$.

I've learned the above proof based on Rouché's theorem from Jackson's excellent book Digital Filters and Signal Processing.
Another good explanation of lattice structures can be found in this book by Zelniker and Taylor.