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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 Feller's book (on page 91) tells us an unexpected different story:

$n$ is to be of the order of magnitude $r^2$.

If they are independent, $\frac{n}{r}$ should not be relevant to $r$, but it surely is from the above statement. Thus they are not.

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

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 Feller's book (on page 91) tells us an unexpected different story:

$n$ is to be of the order of magnitude $r^2$.

If they are independent, $\frac{n}{r}$ should not be relevant to $r$, but it surely is from the above statement. Thus they are not.

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

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 Feller's book (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?

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 Feller's book (on page 91) tells us an unexpected different story:

$n$ is to be of the order of magnitude $r^2$.

If they are independent, $\frac{n}{r}$ should not be relevant to $r$, but it surely is from the above statement. Thus they are not.

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

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 (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?

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 Feller's book (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?