Suppose that $(X_n)_{n=0}^∞$, $(Y_n)_{n=0}^∞$ satisfy the following discrete renewal-style equations for all $n \in \mathbb N$:
$$ \begin{align} X_n &= \sum_{i=1}^n \, (α_i\, X_{n-i} + β_i\,Y_{n-i}) \tag{1}\\ Y_n &= \sum_{i=1}^n \, (γ_i\, X_{n-i} + δ_i\,Y_{n-i}) \tag{2} \end{align}$$
for non-negative constants $(α_i, β_i, γ_i, δ_i)_{i \geq 1}$, summable.
By expanding the $Y_{n-i}$ in $(1)$ using $(2)$, and iteratively expanding any $Y_\bullet$ using $(2)$, we can derive some renewal-style equation for $X_n$:
$$X_n = \sum_{i=1}^∞ \,λ_i\, X_{n-i} $$
for some coefficients, $λ_i \geq 0$, and where we are now treating $X_i = 0\ $ for $i<0$.
In order for this to define a bona fide renewal equation for $(X_n)$, the $λ_i$ need to sum to one.
This seems easy enough:
$$\begin{align} \sum_{i=1}^∞ λ_i &= \sum_{i_1} α_{i_1} + β_{i_1} \left( \sum_{i_2} γ _{i_2} + δ_{i_2}\left(\sum_{i_3} γ _{i_2} + δ_{i_2} \left(\sum_{i_3} γ _{i_3} + δ_{i_3}\left(\sum_{i_4} γ_{i_4} + \cdots \right.\right.\right.\right. \\&\\ &= \sum_i α_i + \sum_i β_i \sum_i γ_i + \sum_i β_i \left(\sum_i γ_i\right)^k \sum_i δ_i + \cdots + \sum_i β_i \left(\sum_i γ_i \right)^{k+1} \left(\sum_i δ_i\right)^k + \cdots \\&\\ &= \sum_i α_i + \frac {\sum_i β_i \sum_i γ_i}{1-\sum_i γ_i \sum_i δ_i}. \end{align}$$
In particular, assuming this convergence is legit, this leads to the following condition for renewal:
$$A + BC + CD = 1 + ACD,$$
where $$A = \sum_{i=1}^∞ α_i,\qquad B = \sum_{i=1}^∞ β_i,\qquad C = \sum_{i=1}^∞ γ_i,\qquad D = \sum_{i=1}^∞ δ_i.$$
But this doesn't seem right. For one thing, I would have expected this to be symmetric under interchanging $X$ and $Y$, as they relate to one another. But this last equation is not the same as got by interchanging $A$ with $D$ and $B$ with $C$.
Has something gone wrong, or is this a legitimate deduction?
Bounty-wise, references to similar situations, applying renewal theory to multiple dimensional settings, would be most welcome.