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Theorem \[th1\] describes the relationship between the dynamics of a process and those of its sums. When the matrix $B$ is stable, $x_t = \sum_{s=0}^{t-1} A^s x_0$ for all $t$. In this case, the state of $Y_t$ is a stable function of the state of $X_t$, and thus $\rho(Y_t,X_t) = 0$.
For each $t$, $\rho(X_t,Y_t) = \rho(A^t x_0, \sum_{s=0}^{t-1} A^s x_0) \le \max_{0 \le u \le t-1} \rho(A^u x_0, A^{u+1} x_0)$. Because $B$ is a diagonal matrix, this implies $\rho(Y_t,X_t) \le \max_{i,j} \rho(A^{t-i}x_0, A^{t-j}x_0) = \rho(x_0,A^{t}x_0) = (\rho(x_0,Ax_0))^t$. Thus $\rho(Y_t,X_t) \le \rho(Y_0,X_0)^{t}$. Since $\rho(Y_0,X_0) < 1$, the theorem follows. These results are based on the Lipschitz property of $\rho$. Consider $h(y,x) = \rho(y,x) + g(y)$ where $g$ is a function of $y$. If $y' = A^t y$, then $\rho(y',x) \le \max_{0 \le i \le t-1} \rho(
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