This example is from Stochastic Modelling and Applied Probability by Sören Asmussen (2010) p.358.
The setup is the following:
Let $\{Z_{t}\}$ be stochastic process on a Skorokhod space $D$ and a $\sigma$-field $\mathcal{F}$. Two such processes may be represented by measures $P$, $\tilde{P}$ on $(D,\mathcal{F})$.
One may study the Radon-Nikodym derivative $dP/d\tilde{P}$.
The author says, that this is too restrictive and gives the following example:
Consider the random walk of partial sums $\{S_{n}\}$ of $Z_{t}$ with incremental distribution $F,\tilde{F}$ with means $\mu\not=\tilde{\mu}$ and let $A:=\{S_{n}/n\rightarrow \mu\}$,$\tilde{A}:=\{S_{n}/n\rightarrow \tilde{\mu}\}$. The author states now, that $P$ is concentrated on $A$ and $\tilde{P}$ is concentrated on $\tilde{A}$, which are disjoint, which is finally excluding the absolute continuity.
Can someone explain me this more in detail? It looks like he is using the law of large numbers to argue that the measures are concentrated on the defined sets. (How?) But we need the iid case of some equidistant sequence $Z_{1},Z_{2},Z_{3},\ldots$ here for $S_{n}/n\rightarrow \mu$, which he hasn't mentioned explicitly in his example. But by saying that $\{S_{n}\}$ under $F,\tilde{F}$, has mean $\mu\not=\tilde{\mu}$ we can expect the iid case. Or am i wrong?
It is typical, at least in my experience, that "random walks" unless otherwise stated are sums of iid random variables. You can form two random walks with iid copies of normal distributions with means $\mu, \tilde{\mu}$ and variances $\sigma^2$ respectively. Then by the law of large numbers, the averages of their partial sums will converge to the their respective means.