Central limit theorem kind of statement for records

Question

I am trying to prove the following statement, but I do not know how to go on:

Let $F(x)$ be an arbitrary continuous distribution function. Then there are constants $A_n, B_n > 0$ such that, as $n \to \infty$, \begin{align*} \lim P(X_{N(n)} < A_n + B_n x) =: T(x) \end{align*} exists and is nondegenerate if, and only if, \begin{align*} \lim_{n \to \infty} \frac{U_F(A_n + B_nx) - n}{\sqrt n} =: g(x) \end{align*} exists and is finite on an interval, where $g(x)$ has at least two points of increase. When it exists, $T(x) = \Phi(g(x))$.

Some remarks: Let $X_1, X_2, \ldots, X_n$ be i.i.d. random variables with continuous distribution $F(x)$, then $N(n)$ is defined to be \begin{align*} N(n) := \min\{j : j > N(n-1), X_j > X_{N(n-1)}\}. \end{align*} Furthermore it is \begin{align*} U_F(x) := -\log(1 - F(x)) \end{align*} and $\Phi(x)$ is the standard normal distribution function. I know that $F(X_j)$ is uniformly distributed on $(0,1)$ and that $U_F(X_j) \sim \text{Exp}(1)$.

I also know the following fact:

Let $F(x) = 1-e^{-x}$, $x > 0$. Let $X_{N(n)}$, $n > 1$ be the records and let $Y_1 = X_{N(1)} = X_1$, $Y_j = X_{N(j)} - X_{N(j-1)}$, $j \ge 2$. Then $Y_1, Y_2, \ldots$ are i.i.d. random variables and their common distribution function is $F(x)$ itself.

Since $X_{N(n)} = \sum_{k = 1}^n Y_k$, by the Central limit theorem it is \begin{align*} \lim_{n \to \infty} P(X_{N(n)} < n + x \sqrt n) = \Phi(x). \end{align*}

Questions: 1. How do I need to define the constants $A_n$ and $B_n$?

2. If I want to show that $T(x)$ is nondegenerate, do I need to show that $T'(x) \ge 0$ for all $x$?

3. I really need some help to start.

Thanks in advance.

Answer 1

Basically you already wrote the pieces yourself: if $(X_i)$ is i.i.d. with continuous $F$, then $Z_i=U_F(X_i)$ defines a sequence $(Z_i)$ of i.i.d. standard exponential random variables.

Furthermore, $Z_{N(n)}=U_F(X_{N(n)})$ hence, for every $(x_n)$, $$[X_{N(n)}\leqslant x_n]=[Z_{N(n)}\leqslant U_F(x_n)].$$ Finally, $Z_{N(n)}$ is distributed like the sum of $n$ i.i.d. standard exponential random variables hence, by the central limit theorem, $Z_{N(n)}=n+\sqrt{n}G_n$ where $G_n$ converges in distribution to the standard normal distribution.

Can you conclude?