What is the meaning of the following expressions? Let $A_1,A_2,\ldots,A_n,\ldots$ be a sequence of events. What is the meaning of these events:
$$A^*=\bigcap_{k=1}^{\infty}\bigcup_{n=k}^{\infty}A_n$$
$$A_*=\bigcup_{k=1}^{\infty}\bigcap_{n=k}^{\infty}A_n$$
My attempt: I think $A^*$ means that $$\forall N\in \mathbb{N}: \exists n > N: A_n=1$$ and $A_*$ means that $$\exists N\in\mathbb{N}: \forall n>N: A_n=1$$
Am I right? Maybe there are some logical explanation of these expressons beside that only mathematical formula meaning

I suspect you mean something like:
$$
A^* = \{x \mid (\forall N\in\mathbb{N})(\exists n\ge N)\,x\in A_n\}
$$
and
$$
A_* = \{x \mid (\exists N\in\mathbb{N})(\forall n\ge N)\,x\in A_n\}.
$$
These sets are commonly called $\limsup_n A_n$ and $\liminf_n A_n$, respectively. It can be useful to describe these sets in words:
$\limsup_n A_n$ (which is $A^*$) is the set of all points that are in $A_n$ for infinitely many $n$
and
$\liminf_n A_n$ (which is $A_*$) is the set of all points that are in $A_n$ for all but finitely many $n$.
There are many analogues to the limit superior and inferior of sequences of real numbers that justify this notation. Scroll down on this wiki page see more about them.
