Find a formula for forecasting of the sum of $n$ iid processes

Question

Let $X=(X_t)_{t\in \mathbb N}$ be a infinite time series. We know that the forecasting or prediction of $X_{T+h}$ if we observe $X_1,...,X_T$ is $$P_{X_1,..,X_T}(X_{T+h})= E[X_{T+h} | X_1,..., X_T ]$$ Now, suppose $X^j=(X_t^j)_{t\in \mathbb N}$, $j=1,...,n$ iid coppies of $X$. Define: $$Y_t = X_t^1 + ...+ X_t^n$$ Suppose that we only observe $Y_1,.., Y_T$. We know that: $$P_{Y_1,..,Y_T}(Y_{T+h})=E[Y_{T+h}|Y_1,...,Y_T]=\sum_{j=1}^n E[X^j_{T+h}| Y_1,...,T_T ]$$ But I want to know if there is any formula for $P_{Y_1,..,Y_T}(Y_{T+h})$ involving $P_{X_1,..,X_T}(X_{T+h})$. I think I can do this: $$P_{Y_1,..,Y_T}(Y_{T+h})=\sum_{j=1}^n E[X^j_{T+h}| Y_1,...,T_T ]=nE[X_{T+h}| Y_1,...,T_T ]= n P_{Y_1,..,Y_T}(X_{T+h})$$ But I don't know if this equal to $nP_{X_1,..,X_T}(X_{T+h})$.