Does anyone know if the Gumbel can occur as a limit distribution for such a sum?
When we have $n$ exponential distributed variables $X_i \sim Exp(\gamma = i)$, (with expectation $1/i$ and variance $1/i^2$) then the sum
$$S = \sum_{i=1}^n (X_i - 1/i)$$
approaches a Gumbel distribution.
There is a connection between this sum and the maximum order statistic.
We can see this sum as the waiting time for filling $n$ bins when the filling of the bins is a Poisson process.
- Approach with the sum. The waiting time between the filling of bins bin is exponential distributed. For waiting until one bin is filled, since all bins are empty the rate is $n$. The waiting time for a second bin to be filled is when $n-1$ bins are empty and the rate will be $n-1$, and so on...
- Approach with the maximum. We can consider the waiting times for filling each individual bin. The waiting time to fill all bins is equal to the maximum of the individual waiting times.
The distribution of the maximum of exponential distributed variables approaches a Gumbel distribution. Therefore the expression in terms of a sum, which has an equal distribution, will also approach the Gumbel distribution.
See also Intuition about the coupon collector problem approaching a Gumbel distribution on Cross Validated.
This is of course not general.
If we use $X_i = N(\mu = 1/i, \sigma^2 = 1/i^2)$ then a (properly scaled) sum will approach a normal distribution.
That is a trivial example but there are more cases that will converge to a normal distribution. The relevant condition that needs to be fulfilled is the Lyapunov condition.
$\alpha$ need not be a random variable.
The most natural choice for $(\Omega, \mathcal{F}, P)$ is product space: let $\Omega = [0,1]^{[0,1] \cup \{2\}}$ and for $A \subset [0,1] \cup 2$, let $\pi_A : \Omega \to [0,1]^A$ be the projection map. (For $a \in [0,1] \cup 2$, we let $\pi_a$ denote $\pi_{\{a\}} : \Omega \to [0,1]^{\{a\}} = [0,1]$.) Let $\mathcal{F}$ be the product $\sigma$-field on $\Omega$, which is by definition the smallest $\sigma$-field that makes all $\pi_a : \Omega \to [0,1]$ measurable. It is then not hard to show that every $B \in F$ is of the form $B = \pi_A^{-1}(C)$ for some countable $A \subset [0,1] \cup \{2\}$ and some $C \subset [0,1]^A$ which is measurable with respect to the product $\sigma$-field on $[0,1]^A$. That is, a measurable subset of $\Omega$ can only look at countably many coordinates.
Now for each countable $A \subset [0,1] \cup \{2\}$, let $\mu_A$ be the measure on $[0,1]^A$ which is the infinite product of Lebesgue measure, and for $B = \pi_A^{-1}(C) \in \mathcal{F}$, set $\mu(B) = \mu_A(C)$. It's easy to verify that $\mu$ is well defined and countably additive (note that $\bigcup_n \pi_{A_n}^{-1}(C_n)$ is of the form $\pi_A^{-1}(C)$ where $A = \bigcup_n A_n$ is countable). Moreover, $\mu$ is a probability measure and, under $\mu$, the $\pi_a$ are iid $U(0,1)$ random variables. So $(\Omega, \mathcal{F}, \mu)$ satisfies the hypotheses, taking $\xi_a = \pi_a$ and $u = \pi_2$.
We then define $\alpha$ as you say, via $\alpha(\omega) = \pi_{\pi_2(\omega)}(\omega)$. Then $\alpha$ is certainly not a random variable. If it were, then $\alpha^{-1}([0,1/2])$ would be of the form $\pi_A^{-1}(C)$ for $A$ countable and $C \subset [0,1]^A$. Let $b \in [0,1] \setminus A$. Define $\omega$ via $\omega(2)=b$, $\omega(b)=1$, and $\omega(a) = 0$ for all $a \in [0,1] \setminus \{b\}$. Define $\omega'$ similarly but with $\omega'(b)=0$. Then $\alpha^{-1}([0,1/2])$ contains $\omega'$ but not $\omega$, whereas $\pi_A^{-1}(C)$ must contain both of $\omega,\omega'$ or neither.
This doesn't rule out the possibility of being able to choose some more exotic $(\Omega, \mathcal{F}, P)$ (which should perhaps be left to the set theorists). But even if you could, I agree with fedja that no good can come of it. For example, working formally, you might observe that for each finite $A \subset [0,1]$, $\alpha$ is independent of $\{\xi_a : a \in A\}$ (since almost surely $u \notin A$), whence $\alpha$ is independent of $\{\xi_a : a \in [0,1]\}$, which appears to be absurd.
Best Answer
The fundamental issue here is bounding the distribution of the supremum of a collection of random variables. The book "Upper and Lower Bounds for Stochastic Processes" by Michel Talagrand is largely devoted to this issue, https://link.springer.com/book/10.1007/978-3-642-54075-2 as is his earlier related book "the generic chaining".