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Takács (1959, Example 2) states:

Let $\{ \xi_n \}$ be a sequence of independent and identically distributed positive random variables, with $ \xi_0 = 0 $.

Let $\zeta_n = \xi_0 + \xi_1 + \dots + \xi_n$ (for $n = 0, 1, 2, \dots$).

Suppose that there exists a positive constant $A$ such that $$\lim_{x \to \infty} x^{1/2} \, \mathbb{P}\{\xi_n > x\} = A $$ Then $$ \lim_{n \to \infty} \mathbb{P}\left\{ \dfrac{\zeta_n}{A^2n^2} \leq x \right\} = F_{1/2}(x) $$ where $$ F_{1/2}(x) = \begin{cases} 2\left[ 1 - \Phi\left( \left( \dfrac{\pi}{2x} \right)^{1/2} \right) \right] & \text{if}\ x \geq 0 \\ 0 & \text{if}\ x < 0 \end{cases} $$ and $\Phi(x)$ is the distribution function for the standard normal distribution $$ \Phi(x) = \frac{1}{(2\pi)^{1/2}} \int_{-\infty}^{x} e^{-y^2/2} \, dy $$

Why is this statement true?

Update: Revised the question title after learning about fat-tailed random variables. Also noted this question about central limit theorems for heavy-tailed random variables.

Reference

Takács, L. (1959). On a sojourn time problem in the theory of stochastic processes. Trans. Amer. Math. Soc. 93, 531–540

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  • $\begingroup$ Interesting! Does the first limit hold for all $n=0,1,\dots$? $\endgroup$ Commented Jul 2 at 6:16
  • $\begingroup$ I think so. At least on the grounds that the variables $\xi_n$ are identically distributed (and independent). $\endgroup$ Commented Jul 2 at 6:23
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    $\begingroup$ Sounds like a strong condition, so it makes sense. I would try to understand what we can say about the moments $E[\xi^k]$ and therefore $E[\zeta^k]$ $\endgroup$ Commented Jul 2 at 6:32

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The statement appears to be an application of a Generalised Central Limit Theorem. The essence is that if $x^{\alpha} \mathbb{P}\{\xi_n>x\} \to c$ for $\alpha \in (0,2]$ then $\zeta_n$ will tend to a Lévy $\alpha$-stable distribution.

In more detail, the following is derived as a specific case from Nolan (2020, Theorem 3.1.2: Explicit Form of the Generalized Central Limit Theorem).

Let $X_1, X_2, \dots$ be independent and identically distributed copies of $X$, where $X$ has cumulative distribution function $F(x)$ and satisfies the tail conditions $x^\alpha F(-x) \to 0$ and $x^\alpha \left(1- F(x)\right) \to c$ as $x \to \infty$. Let $$a_n = \frac{2}{\pi c^2 n^2}$$ Then as $n \to \infty$, $a_n(X_1 + X_2 + \dots + X_n)$ converges in distribution to a Lévy distribution with scale parameter 1, shift parameter 0.

(This statement comes from applying Nolan (2020, Theorem 3.1.2) with the values $c^- = 0$, $c^+ = c$, $\alpha = \frac{1}{2}$, noting that $\Gamma(\frac{1}{2}) = \sqrt{\pi}$.)

Therefore $$ \lim_{n \to \infty} \mathbb{P}\left\{ \dfrac{2\zeta_n}{\pi A^2n^2} \leq x \right\} = L(x) $$ where $L(x)$ is the cumulative distribution function for the Lévy distribution with scale parameter 1, shift parameter 0. $$ L(x) = 2 - 2\Phi\left( \left( \frac{1}{x} \right)^{1/2} \right) $$ The correspondence with Takács (1959, Example 2) follows immediately, from considering $L\!\left( \dfrac{\pi x}{2} \right)$.

Reference

Nolan, John P. (2020). Univariate Stable Distributions: Models for Heavy Tailed Data. Springer Nature Switzerland.

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