Question on the proof of Subadditive Ergodic Theorem in Durrett's textbook. This came up in Durrett's proof of the subadditive ergodic theorem.
Let $X_{m,n}$ satisfy the assumptions of subadditive ergodic theorem, which reads


*

*$X_{0,m}+X_{m,n} \geq X_{0,n}$




*$(X_{nk, (n+1)k})_{n\geq 1}$ is a stationary sequence for eack $k$.





*The distribution of $(X_{m,m+k})_{k\geq 1}$ does not depend on $m$.





*$ E X_{0,1}^+ <\infty$ for each $n$, $EX_{0,n} \geq \gamma_0 n$ where $\gamma_0 >-\infty$.


In STEP 3 of his proof, he defines $Z:= \epsilon + (\underline{X} \vee -M)$ where $\underline{X}:= \liminf_{n \to \infty} X_{0,n} /n$. He first shows that $\underline{X} = \underline{X}_{m} := \liminf_{n \to \infty} X_{m,m+n} /n$ almost surely for any $m$.
Then he says $$Y_{m,n} := X_{m,n}-(n-m)Z$$ also satisfies the assumptions of the subadditive ergodic theorem since $Z_{m,n}=-(n-m)Z$ trivially does satisfy the assumptions.
My question is : why does $\mathbf {Y_{m,n}}$ satisfy the assumtions?
I see that conditions 1 and 4 are trivial to check, but I don't get how the conditions 2 and 3 are satisfied. Sum of two stationary sequence are not in general stationary.
Any help is appreciated.



*Note that $Y_{nk, (n+1)k}=X_{nk, (n+1)k}-kZ.$ Since $(X_{nk, (n+1)k})_{n\ge 1}$ is staionary by assumption, it follows that $Y_{nk, (n+1)k}$ is also stationary. One can write out the details using the fact that $Z$ is independent of any finite collection of random variables $X_{m, n}.$


*is  actually very easy. Just write out the definition and observe that $(Y_{m,m+k})_{k\ge 1}=(X_{m, m+k}-kZ)_{k\ge 1}$ which clearly does not depend on $m.$
