In random walks, a path may return to origin for the $r$-th time in $n$-th step ($n$ 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 [Feller's book][1] (on page 91) tells us an unexpected different story:
> $n$ is to be of the order of magnitude $r^2$.

If they are independent, $n$ will be of the order of magnitude $r$ instead of $r^2$. Thus they are not.

My question is: **What key point that we ignored led us to the intuitive but wrong conclusion?**

  [1]: http://sanghv.com/download/soft/Machine%20Learning,%20Artificial%20Intelligence,%20Mathematics%20eBooks/math/probability/an%20introduction%20to%20probability%20theory%20and%20applications%20(vol1,%203rd,%201968).pdf