[proofplan] The proof constructs an absolutely summable tail-sum coefficient sequence $(\phi_j)_{j \geq 0}$ whose first differences reproduce the coefficients of the transitory part of the linear process. The filtering property assumed in the statement then makes the associated causal filter $(V_t)_{t \in \mathbb{Z}}$ a well-defined stationary causal linear process. This converts $Y_t$ into the sum of a drift term $\mu$, a permanent innovation term $\Psi(1)Z_t$, and a stationary first difference $V_t - V_{t-1}$. Summing the identity from time $1$ to time $t$ telescopes the stationary first-difference term and gives the Beveridge–Nelson representation with correction $V_t - V_0$. [/proofplan] [step:Construct the coefficient sequence that factors out the unit root] Define the coefficient sequence $(\phi_j)_{j \geq 0} \subset \mathbb{R}$ by \begin{align*} \phi_j := -\sum_{m=j+1}^{\infty} \psi_m. \end{align*} The tail sum is finite for each $j$ because $\sum_{m=0}^{\infty} |\psi_m| < \infty$. Moreover, \begin{align*} \sum_{j=0}^{\infty} |\phi_j| &\leq \sum_{j=0}^{\infty} \sum_{m=j+1}^{\infty} |\psi_m| \\ &= \sum_{m=1}^{\infty} \sum_{j=0}^{m-1} |\psi_m| \\ &= \sum_{m=1}^{\infty} m |\psi_m| \\ &< \infty. \end{align*} Thus $(\phi_j)_{j \geq 0}$ is absolutely summable. [guided] We need a stationary process whose first difference accounts for the transitory part of $Y_t$. Algebraically, this means that the lag polynomial $\Psi(L)-\Psi(1)$ should contain the factor $1-L$. The coefficients of the quotient are the tail sums of the original coefficients. Define $(\phi_j)_{j \geq 0}$ by \begin{align*} \phi_j := -\sum_{m=j+1}^{\infty} \psi_m. \end{align*} This is well-defined because the coefficient sequence $(\psi_m)$ is absolutely summable. We must also check that the new coefficients define a stationary linear process. For each $N \in \mathbb{N}$, finite rearrangement of a non-negative double sum gives \begin{align*} \sum_{j=0}^{N} |\phi_j| &\leq \sum_{j=0}^{N} \sum_{m=j+1}^{\infty} |\psi_m| \\ &= \sum_{m=1}^{\infty} \sum_{j=0}^{\min\{N,m-1\}} |\psi_m| \\ &\leq \sum_{m=1}^{\infty} m |\psi_m|. \end{align*} Taking the supremum over $N$ gives \begin{align*} \sum_{j=0}^{\infty} |\phi_j| &\leq \sum_{j=0}^{\infty} \sum_{m=j+1}^{\infty} |\psi_m| \\ &= \sum_{m=1}^{\infty} \sum_{j=0}^{m-1} |\psi_m| \\ &= \sum_{m=1}^{\infty} m |\psi_m| \\ &< \infty. \end{align*} The moment condition $\sum_{m=0}^{\infty} m|\psi_m| < \infty$ is precisely what ensures that these tail coefficients are summable. [/guided] [/step] [step:Rewrite the linear process as a permanent innovation plus a stationary difference] Let $L$ denote the lag operator on two-sided sequences, so that $LZ_t = Z_{t-1}$. Define the causal linear process $(V_t)_{t \in \mathbb{Z}}$ by \begin{align*} V_t := \sum_{j=0}^{\infty} \phi_j Z_{t-j}. \end{align*} The coefficient sequence $(\phi_j)_{j \geq 0}$ is absolutely summable by the previous step. Applying the filtering property from the theorem statement with $a_j := \phi_j$ for $j \geq 0$, the causal filter defining $V_t$ is well-defined for every $t \in \mathbb{Z}$, and $(V_t)_{t \in \mathbb{Z}}$ is a stationary causal linear process. For $j=0$, \begin{align*} \phi_0 = -\sum_{m=1}^{\infty}\psi_m = \psi_0 - \Psi(1). \end{align*} For every $j \geq 1$, \begin{align*} \phi_j - \phi_{j-1} &= -\sum_{m=j+1}^{\infty}\psi_m + \sum_{m=j}^{\infty}\psi_m \\ &= \psi_j. \end{align*} Define the coefficient sequence $(\theta_j)_{j \geq 0} \subset \mathbb{R}$ by $\theta_0 := \phi_0$ and $\theta_j := \phi_j - \phi_{j-1}$ for $j \geq 1$. It is absolutely summable because \begin{align*} \sum_{j=0}^{\infty}|\theta_j| &\leq |\phi_0| + \sum_{j=1}^{\infty}|\phi_j| + \sum_{j=1}^{\infty}|\phi_{j-1}| \\ &\leq 2\sum_{j=0}^{\infty}|\phi_j| < \infty. \end{align*} Thus the difference of the two causal filters is the well-defined causal filter with coefficients $(\theta_j)_{j \geq 0}$. Therefore, \begin{align*} V_t - V_{t-1} &= \sum_{j=0}^{\infty}\phi_j Z_{t-j} - \sum_{j=0}^{\infty}\phi_j Z_{t-1-j} \\ &= \theta_0 Z_t + \sum_{j=1}^{\infty}\theta_j Z_{t-j} \\ &= \phi_0 Z_t + \sum_{j=1}^{\infty}(\phi_j-\phi_{j-1})Z_{t-j} \\ &= \bigl(\psi_0-\Psi(1)\bigr)Z_t + \sum_{j=1}^{\infty}\psi_j Z_{t-j}. \end{align*} Adding $\Psi(1)Z_t$ gives \begin{align*} \Psi(1)Z_t + V_t - V_{t-1} = \sum_{j=0}^{\infty}\psi_j Z_{t-j}. \end{align*} Hence \begin{align*} Y_t = \mu + \Psi(1)Z_t + V_t - V_{t-1}. \end{align*} [guided] Now we turn the coefficient identity into a process identity. Let $L$ be the lag operator, defined on the innovation sequence by $LZ_t = Z_{t-1}$. Define \begin{align*} V_t := \sum_{j=0}^{\infty} \phi_j Z_{t-j}. \end{align*} Because $\sum_{j=0}^{\infty}|\phi_j| < \infty$, the filtering property in the theorem statement applies to the coefficient sequence $(\phi_j)_{j \geq 0}$. Taking $a_j := \phi_j$ for every $j \geq 0$, that property says exactly that the series defining $V_t$ converges for each $t \in \mathbb{Z}$ and that $(V_t)_{t \in \mathbb{Z}}$ is a stationary causal linear process. The coefficient computation is the core of the decomposition. First, \begin{align*} \phi_0 = -\sum_{m=1}^{\infty}\psi_m = \psi_0 - \Psi(1). \end{align*} Next, for $j \geq 1$, \begin{align*} \phi_j - \phi_{j-1} &= -\sum_{m=j+1}^{\infty}\psi_m + \sum_{m=j}^{\infty}\psi_m \\ &= \psi_j. \end{align*} Thus the first difference of $(V_t)$ has exactly the coefficients of the original filter after removing the contemporaneous permanent coefficient $\Psi(1)$. To make the regrouping precise, define $(\theta_j)_{j \geq 0}$ by $\theta_0 := \phi_0$ and $\theta_j := \phi_j - \phi_{j-1}$ for $j \geq 1$. This new coefficient sequence is absolutely summable, since \begin{align*} \sum_{j=0}^{\infty}|\theta_j| &\leq |\phi_0| + \sum_{j=1}^{\infty}|\phi_j| + \sum_{j=1}^{\infty}|\phi_{j-1}| \\ &\leq 2\sum_{j=0}^{\infty}|\phi_j| < \infty. \end{align*} Therefore the difference of the two causal filters defining $V_t$ and $V_{t-1}$ is itself a well-defined absolutely summable causal filter. Its coefficients are computed by matching the common innovation $Z_{t-j}$: \begin{align*} V_t - V_{t-1} &= \sum_{j=0}^{\infty}\phi_j Z_{t-j} - \sum_{j=0}^{\infty}\phi_j Z_{t-1-j} \\ &= \theta_0 Z_t + \sum_{j=1}^{\infty}\theta_j Z_{t-j} \\ &= \phi_0 Z_t + \sum_{j=1}^{\infty}(\phi_j-\phi_{j-1})Z_{t-j} \\ &= \bigl(\psi_0-\Psi(1)\bigr)Z_t + \sum_{j=1}^{\infty}\psi_j Z_{t-j}. \end{align*} Adding the permanent component $\Psi(1)Z_t$ restores the original linear process: \begin{align*} \Psi(1)Z_t + V_t - V_{t-1} = \sum_{j=0}^{\infty}\psi_j Z_{t-j}. \end{align*} Therefore \begin{align*} Y_t = \mu + \Psi(1)Z_t + V_t - V_{t-1}. \end{align*} [/guided] [/step] [step:Sum the first differences and telescope the stationary component] For every $t \in \mathbb{N}$, the definition of the difference operator gives \begin{align*} \Delta X_t = X_t - X_{t-1} = Y_t. \end{align*} Therefore \begin{align*} X_t - X_0 &= \sum_{k=1}^{t} Y_k \\ &= \sum_{k=1}^{t}\left(\mu + \Psi(1)Z_k + V_k - V_{k-1}\right) \\ &= \mu t + \Psi(1)\sum_{k=1}^{t}Z_k + V_t - V_0. \end{align*} Thus \begin{align*} X_t = X_0 + \mu t + \Psi(1)\sum_{k=1}^{t}Z_k + V_t - V_0. \end{align*} Set $C_t := V_t$ for every $t \in \mathbb{Z}$. Since $(V_t)$ is a stationary causal linear process, $(C_t)$ is a stationary causal linear process, and the preceding identity becomes \begin{align*} X_t = X_0 + \mu t + \Psi(1)\sum_{k=1}^{t}Z_k + C_t - C_0. \end{align*} This is the corrected Beveridge–Nelson decomposition. [guided] We now sum the one-period identity. For every $t \in \mathbb{N}$, the definition of the difference operator gives \begin{align*} \Delta X_t = X_t - X_{t-1} = Y_t. \end{align*} Repeated addition from $k=1$ through $k=t$ gives \begin{align*} X_t - X_0 &= \sum_{k=1}^{t} Y_k. \end{align*} Substitute the decomposition of $Y_k$ obtained above: \begin{align*} X_t - X_0 &= \sum_{k=1}^{t}\left(\mu + \Psi(1)Z_k + V_k - V_{k-1}\right) \\ &= \mu t + \Psi(1)\sum_{k=1}^{t}Z_k + \sum_{k=1}^{t}(V_k - V_{k-1}). \end{align*} The last sum is telescoping: every intermediate $V_k$ with $1 \leq k \leq t-1$ appears once with a plus sign and once with a minus sign. Hence \begin{align*} \sum_{k=1}^{t}(V_k - V_{k-1}) = V_t - V_0. \end{align*} Therefore \begin{align*} X_t = X_0 + \mu t + \Psi(1)\sum_{k=1}^{t}Z_k + V_t - V_0. \end{align*} Finally define the process $(C_t)_{t \in \mathbb{Z}}$ by $C_t := V_t$ for every $t \in \mathbb{Z}$. Since $(V_t)$ is a stationary causal linear process, so is $(C_t)$. With this notation, \begin{align*} X_t = X_0 + \mu t + \Psi(1)\sum_{k=1}^{t}Z_k + C_t - C_0, \end{align*} which is the Beveridge–Nelson decomposition for every $t \in \mathbb{N}$ in the stated form. [/guided] [/step]