An analog of Wald's identity for random walks with infinite mean

We deduce an analog of the classical Wald's identity $\mathbf ES_\tau=\mathbf E\tau\mathbf E\xi$ in the case of the infinite mean of summands. We find the conditions on $\tau$ under which $\mathbf E\min(S_\tau,x)\sim\mathbf E\tau\mathbf E\min(\xi,x)$ as $x\to\infty$.