Define $W_{r}=\min\{t:N_{t}\geq r\}(r=1,2,...)$ as the waiting time until the $r$th arrival in a Poisson process $\{N_{t}:t \geq 0\}$ with rate $\lambda$. The question is to find $Cov(W_{1},W_{r})(r\geq 2)$.
Here is my attempt.
Since $W_{1}\sim Exp(1/\lambda)$ and $W_{r}\sim Gamma(r,1/\lambda)$, I will separate $W_{r}$ as $$W_{r}=X_{1}+...+X_{r}$$ , where $X_{1}=W_{1}$, $X_{i}\sim Exp(1/\lambda)$, and the distributions are independent.
Then since $Cov(X,Y+Z)=Cov(X,Y)+Cov(X,Z)$, $$Cov(W_{1},W_{r})=Cov(X_{1},X_{1})+...+Cov(X_{1},X_{r})=Var(W_{1})=1/\lambda^{2}$$ .
My question is, can I guarantee the existence of such $X_{i}$'s from the beginning?
Your work is fine. The $X_i$ have a very tangible meaning in the context of a Poisson process: they are the "inter-arrival times" between consecutive arrivals, and are independent. In fact, this is how one typically proves the distribution of $W_r$ is gamma in the first place.