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Generalization of problem 14. If $x_n$ varies with $n$ in such a way that $x_n \rightarrow \infty$ but $x_n / \sqrt{n} \rightarrow 0$, then $$ \mathbf{P}\left\{\mathbf{S}_n>x_n\right\} \sim\left[1-\mathfrak{N}\left(x_n\right)\right] e^{-f\left(x_n\right)} . $$ When $x_n^4 / n \rightarrow 0$ this reduces to (7.8). When $x_n^5 / n^{\frac{3}{2}}$ one may replace $f\left(x_n{ }^{\frac{3}{2}}\right)$ by the fourth-degree polynomial appearing on the right in (7.9), etc.

   Generalization of problem 14. If $x_n$ varies with $n$ in such a way that $x_n \rightarrow \infty$ but $x_n / \sqrt{n} \rightarrow 0$, then

$$
\mathbf{P}\left\{\mathbf{S}_n>x_n\right\} \sim\left[1-\mathfrak{N}\left(x_n\right)\right] e^{-f\left(x_n\right)} .
$$


When $x_n^4 / n \rightarrow 0$ this reduces to (7.8). When $x_n^5 / n^{\frac{3}{2}}$ one may replace $f\left(x_n{ }^{\frac{3}{2}}\right)$ by the fourth-degree polynomial appearing on the right in (7.9), etc.

An Introduction to Probability Theory and Its Applications: Volume 1

William Feller 1st Edition

Chapter 7, Problem 16

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Generalization of problem 14. If $x_n$ varies with $n$ in such a way that $x_n \rightarrow \infty$ but $x_n / \sqrt{n} \rightarrow 0$, then $$ \mathbf{P}\left\{\mathbf{S}_n>x_n\right\} \sim\left[1-\mathfrak{N}\left(x_n\right)\right] e^{-f\left(x_n\right)} . $$ When $x_n^4 / n \rightarrow 0$ this reduces to (7.8). When $x_n^5 / n^{\frac{3}{2}}$ one may replace $f\left(x_n{ }^{\frac{3}{2}}\right)$ by the fourth-degree polynomial appearing on the right in (7.9), etc.

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