In random walks, a path may return to origin for the $r$-th time in $n$-th step ($r$ is given but $n$ is not given), and under this condition, these $n$ steps can be split into $r$ sections where each section ends with an $i$-th return ($i=0,1,\ldots,r$). We denote the length of each section as $l_1,l_2,\ldots,l_r$ ($l_1+l_2+\ldots+l_r=n$).
What I concern is the independence between $l_1,l_2,\ldots,l_r$.
In the obvious sense the random walk starts from scratch every time when the path returns to the origin. Thus one may draw a conclusion that they are independent of each other. But the following excerpt from Feller's book (on page 91) tells us an unexpected different story (Because if they are independent, $n/r$ will be irrelevant to $r$ instead of being proportional to $r$).
The result in the above words totally comes from calculation.
My question is: What key point that we ignored led us to the intuitive but wrong conclusion?