# Special case of distribution function

I want an example for a distribution function $F$ with those two conditions for all $\lambda>0$ $$\liminf_{x\to\infty}\frac {1-F(x)} {e^{-\lambda x}}=0$$ $$\limsup_{x\to\infty}\frac {1-F(x)} {e^{-\lambda x}}=\infty$$ I want one example of a distribution function where those two conditions hold. I may use a $F$ with $F(t)=f_k$ with $t\in [x_k,x_{k+1}]$ and choose fitting $f_k,x_k$, but i have no clue how to go on. Thanks Nordmann

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Got something from an answer below? – Did Feb 8 '13 at 7:32

If you pick positive $c_1 < c_2 < \dots$ and $a_1<a_2<\dots$ such that $$F(x) =\begin{cases} 0, &\qquad \text{if } x < a_1,\\[6pt] 1-e^{-c_k}, &\qquad \text{if } a_k\le x < a_{k+1},\text{ for } k=1, 2, \dots, \end{cases}$$ then to satisfy these conditions it's enough to have $$c_k/a_k\to\infty, \qquad c_k/a_{k+1}\to 0.$$ So, pick $c_k:=e^{(2k+1)^2}$, $a_k:=e^{4k^2}$.

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You don't seem to be naive in the use of TeX like so many people here, but you use the antiquated \ {\rm blah\ blah\ blah\ } instead of \text{ blah blah blah } and the complicated {array} instead of the far simpler {cases} when it's about cases (i.e. piecewise definition). Do you prefer those for a reason? – Michael Hardy Jan 31 '13 at 1:26
I guess it just shows you how long ago I learned TeX. – David Moews Jan 31 '13 at 1:27
Im new to Tex, but i appreciate every advice – nordmann Jan 31 '13 at 1:28

Define $a_n=\exp(3^n)$ and $x_n=a_n^2+1$ and assume that $\mathbb P(X=x_n)=c\mathrm e^{-a_n}$ for every $n\geqslant1$, where $c$ is chosen such that $c\sum\limits_n\mathrm e^{-a_n}=1$. Note that $x_n\ll a_{n+1}$ and that, for every $n$, $\mathrm e^{-a_n}\leqslant\sum\limits_{k\geqslant n}\mathrm e^{-a_k}\leqslant2\mathrm e^{-a_{n}}$.

Then $1-F(a_n^2)\gt c\mathrm e^{-a_n}$ hence $\mathrm e^{\lambda a_n^2}(1-F(a_n^2))\gt c\exp(\lambda a_n^2-a_n)\to+\infty$. And $1-F(x_n)\leqslant2c\mathrm e^{-a_{n+1}}$ hence $\mathrm e^{\lambda x_n}(1-F(x_n))\leqslant2c\exp(\lambda x_n-a_{n+1})\to0$.

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Hint: Take $F(t)$ to be constant on intervals $(x_k, x_{k+1})$ with a jump at each $x_k$ from a value where $\frac{1-F(x)}{e^{-\lambda x}}$ is large (say $k$) to one where it is small (say $1/k$).

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Sure. But why does this yield a nondecreasing F? – Did Feb 8 '13 at 7:32
If $\dfrac{1-F(x_k-)}{e^{-\lambda x_k}} = k$ and $\dfrac{1-F(x_k+)}{e^{-\lambda x_k}} = 1/k$ with $k > 1$, you must have $F(x_k+) > F(x_k-)$. – Robert Israel Feb 8 '13 at 8:16
This does not answer my question. One would also need that $\mathrm e^{\lambda (x_{k+1}-x_k)}\geqslant k^2$ for every $k$, a condition which seems difficult to achieve for every positive $\lambda$. – Did Feb 8 '13 at 10:24
Just take $x_{k+1} \ge x_k + \log(k^2)/\lambda$. – Robert Israel Feb 8 '13 at 18:51
Quote: difficult to achieve for every positive λ. Unquote. – Did Feb 8 '13 at 20:22