In the derivation of the renewal function, I have seen two different formulas which differ by $1$ as follows. Let $N_t$ be a pure renewal process with jump distribution $F$ and $S_n=\sum^n X_i$, then \begin{align} \mathbb EN_t &= \sum_{n=0}^\infty\mathbb P(N_t >n) \\ &=\sum_{n=0}^\infty \mathbb P(N_t \geq n)-\mathbb P(N_t=n)\\ &= \sum_{n=0}^\infty \mathbb P(S_n \leq t) -1 \\ &=\sum_{n=0}^\infty F^{\star n}(t)-1 \end{align}
But also from Asmussens book \begin{align} \mathbb EN_t &= \sum_{n=0}^\infty\mathbb P(N_t >n) \\ &=\sum_{n=0}^\infty \mathbb P(S_n\leq t)\\ &=\sum_{n=0}^\infty F^{\star n}(t) \end{align}
And in Liao's book $$\mathbb E N_t=\sum_{n=1}^\infty \mathbb P(S_n\leq t)=\sum_{n=1}^\infty F^{\star n}(t)$$
The first has the problem that $\mathbb P(N_t \geq n)=\mathbb P(S_n< t)$ which is fixed in Asmussens book and the third I can make sense of if I set $F^{\star 0}=1$ to agree with the first. How do I make sense of Asmussens one in relation to the other 2.
Define $m(t) = \mathbb E[N_t]$ where $N_t = \sum_{n=0}^\infty \mathsf 1_{[0,t]}(S_n)$. Then $$m(t) = \mathbb E\left[\sum_{n=0}^\infty \mathsf 1_{[0,t]}(S_n)\right] = \sum_{n=0}^\infty \mathbb P(S_n\leqslant t) = \sum_{n=0}^\infty F^{n*}(t),$$ where the interchange of summation with expectation is justified by Tonelli's theorem (all summands are nonnegative). Note that $S_n\leqslant t$ iff $\mathsf 1_{[0,t]}(S_n)=1$, so $\mathbb E[\mathsf 1_{[0,t]}(S_n)] = \mathbb P(S_n\leqslant t)$.
Asmussen's text states \begin{align} \mathbb E[N_t] &= \sum_{n=0}^\infty \mathbb P(N_t>n)\\ &= \sum_{n=0}^\infty \mathbb P(S_n\leqslant t)\\ &= \sum_{n=1}^\infty \mathbb P(Y_1+\cdots+Y_n\leqslant t)\\ &= \sum_{n=0}^\infty F^{*n}(t) = m(t), \end{align} which is consistent with $S_0=Y_0=0$ a.s.