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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 $B_n$?

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

  2. I really need some help to start.

Thanks in advance.

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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?

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  • $\begingroup$ So we have as a fact that $P[X_{N(n)} \le A_n+B_nx] = P[G_n \le (U_F(A_n+B_nx) - n)/\sqrt n]$ converges to $\Phi(g(x))$ as $n \to \infty$ if $(U_F(A_n+B_nx) - n)/\sqrt n \to g(x)$. So here we assumed that $g(x)$ exists. On the other hand, if we know that $P[X_{N(n)} \le A_n+B_nx]$ does converge, then $g(x)$ must exist. $T(x)$ must be nondegenerate, because it is a limit distribution? Why must $g(x)$ be finite on an interval? I also do not see why $g(x)$ needs to have two points of increase. And do I understand the statement, if I say that $A_n$ and $B_n$ need to exist if $\endgroup$ – numerion Oct 1 '14 at 15:01
  • $\begingroup$ $(U_F(A_n+B_nx)-n)/\sqrt n$ converges to some function $g(x)$ with exactly those $A_n$ and $B_n$? Does that mean, we only need to secure that $(U_F(A_n+B_nx)-n)/\sqrt n$ converges with those constants? I am still confused by the existence of $A_n$ and $B_n$. Do they exist in general? $\endgroup$ – numerion Oct 1 '14 at 15:03

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