Sanity check for Laplace-Stiletjes transform of compound process

Question

I have a counting process $Y(t)=\sum_{i=1}^{N(t)}X_i$, where $X_1,X_2,\ldots$ are iid, and $N(t)$ is a non-negative integer random variable independent of $X_i$. I want to calculate the Laplace-Stieltjes transform $\mathcal L$ of $P(N(t) \geq n)$ and want to check the following reasoning. I know the MGF $M_{Y(t)}(s)=\mathcal G_{N(t)}(M_{X_1}(s))$, where $G_{N(t)}$ is the probability generating function and $M_{X_1}(s)$ is the MGF of $X_1$, and I'm trying to use the property here that states ¹: $$ s\mathcal{L}\{F_X(x)\}=M_X(-s), \text{ for distribution function } F_X. $$

Can I then claim:

\begin{align} \mathcal L\{P(Y(t)\geq n)\}(s) &=\mathcal L\{1-F_{Y(t)}(n)\}(s)\\ &=\frac{1}{s}-\mathcal L\{F_{Y(t)}(n)\}(s)\\ &=\frac{1}{s}-\frac{1}{s}\mathcal G_{N(t)}\left(M_{X_1}(-s)\right)? \end{align}

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Footnotes

1. From the appendix of the book Fundamentals of Queueing Theory by John F. Shortle, James M. Thompson, Donald Gross, Carl M. Harris

Answer 1

After thinking this over, I believe the error arises from a simple and obvious mistake, namely $F_{Y(t)}(n)\neq P(Y(t)<n)$. More specifically in my above argument,

$$ \begin{align} \mathcal L\{P(Y(t)\geq n)\}(s) &=\mathcal L\{1-P(Y(t)<n)\}(s)\\ &\color{red}{\neq}\frac{1}{s}-\mathcal L\{F_{Y(t)}(n)\}(s)\\ &=\frac{1}{s}-\frac{1}{s}\mathcal G_{N(t)}\left(M_{X_1}(-s)\right). \end{align} $$