If we know the direction of the sum of vectors $\frac{\sum_1^\infty w_i}{\|\sum_1^\infty w_i\|}=u$, how can we calculate the expection of the vectos? Given a set of vectors $w_i, i\in[1,\infty]$. If we know the direction of the sum of them as
$$\frac{\sum_1^\infty w_i}{\|\sum_1^\infty w_i\|}=u$$
How can we prove the expectation of vector $$\mathbb{E}\frac{w_i}{\|w_i\|}=u$$
Intuitively, the results seem to be. Because if the expectation of the direction is not $u$, the final sum will not be at the direction of $u$. But I don't know how can we rigorously prove this?
 A: To put the question rigorously, assume you have $W$ a random vector. $\left(W_n\right)_{n\ge1}$ are independent observations of $W$. If $\mathbb E\left\|W\right\|^2 < \infty$ then by the law of large numbers: $$\frac 1n \sum_{k=1}^n W_k = \mathbb E[W]$$ and the ratio that you are looking for:
$$\frac{\sum_{k=1}^\infty W_k}{\left\|\sum_{k=1}^\infty W_k\right\|} = \lim\limits_{n\to \infty} \frac{\frac1n\sum_{k=1}^n W_k}{\left\|\frac1n\sum_{k=1}^n W_k\right\|} = \frac{\mathbb E\left[W\right]}{\left\|\mathbb E[W]\right\|}.$$
This value in general is different from $\mathbb E\left[\frac{W}{\|W\|}\right]$. I would say that the answer at your question is NO.


*

*Studying when $\mathbb E\left[\frac{W}{\|W\|}\right] = \frac{\mathbb E\left[W\right]}{\left\|\mathbb E[W]\right\|}$:

Moreover, (thanks to the comment of @aschepler) using the fact that $u\mapsto \left\|u\right\|$ is strongly convex (I am assuming that we are working with $\left\|\cdot \right\|_2$). $$\left\|\mathbb E\left[\frac{W}{\left\|W\right\|}\right]\right\| \le \mathbb E\left[\left\|\frac{W}{\left\|W\right\|}\right\|\right] = 1$$
with equality holds if and only if $\frac{W}{\left\|W\right\|}$ is a constant vector. So $W = X \mathbf u$ with $X$ is a nonnegative random variable and $\mathbf u$ a unitary vector.
A: I am not certain if this answer is correct, but I believe it gets to what OP wanted, based on the OP?
I think I understand what is being asked here, so let me reveal what I believe the question intended to be.
Let's start with the finite case.
The sum of $n$ finite vectors $w_i,\ i\in \{1,\ldots, n\},\ $ is $\ \displaystyle\sum_{i=1}^n w_i.$ Since a vector has magnitude and direction, the vector $\ \displaystyle\sum_{i=1}^n w_i$ can be colloquially said to " have direction $\ \lambda\displaystyle\sum_{i=1}^n w_i,$ " where $\lambda$ can be any constant, because scalar multiplication of a vector doesn't affect the vector's direction.
The expectation of the vectors $w_i,\ i\in \{1,\ldots, n\},\ $ is by definition, their average, which is given by the formula: $$ \frac{\displaystyle\sum_{i=1}^n w_i}{n}. $$
So in the finite case, the direction and expectation of a finite set of vectors can be the same vector: just set $$\lambda = \frac{1}{n}. $$
Maybe the following is easier-to-understand summary of the result so far:

The expectation vector of a finite set of vectors $w_i,\ i\in \{1,\ldots, n\},\ $ is in the same direction as the sum of these $n$
vectors.

$$$$
Ok, so now we get to the OP, which I believe is simply asking about the countably infinite case:

Is it true that, for a countably infinite set of vectors $w_i,\ i\in\{1,\ldots, n\},\ $ the expectation vector is in the same direction as
the sum of these vectors?

An immediate problem arises with this question, as the sum of infinitely many vectors may not itself be a vector, because this infinite sum may not converge within the vector space. Furthermore, the expectation vector of infinitely many vectors might also not converge. Therefore, we must standardise the sum of the direction vectors, and use limits in order for the question to be meaningful. I believe this is what is meant from OP:
Is the direction of the sum of the vectors, as we add them up in the order they are given,
$$\lambda\lim_{n\to\infty} \frac{\displaystyle\sum_{i=1}^n w_i}{\left \vert \left \vert \displaystyle\sum_{i=1}^n w_i \right \vert \right \vert },$$
equal to the direction of the expectation of all the vectors:
$$\lim_{n\to\infty} \frac{\displaystyle\sum_{i=1}^n w_i}{n }.$$
Whilst it is true that neither of these limits necessarily converge, what I think is being sought after is, assuming they both do converge, when are they equal? (i.e. in the same direction)?
Surely this can only occur if there exists $c\in\mathbb{R}\setminus\{0\}$ such that
$$\lim_{n\to\infty} \frac{\left \vert \left \vert \displaystyle\sum_{i=1}^n w_i \right \vert \right \vert}{n} = c.$$
