I am working through this paper on Vasicek's portfolio loss distribution.

On page 3 he mentions that by the law of large numbers,

$$\lim_{n\to\infty}\sum_{k=0}^{\lfloor nx \rfloor} \binom{n}{k}s^k(1-s)^{n-k} = 1_{\{x \ge s \}}$$ where $s \in (0,1)$,

Here $x$ is some real number, $k$ is an integer and $\binom{n}{k}$ is the binomial coefficient.

but I can't quite work out why this is the case.

I have tried some sort of binomial distribution argument but haven't been very successful!

Any help would be greatly appreciated.

  • $\begingroup$ Please explain what the symbol mean in your formula. Also please rephrase the tile as a question. $\endgroup$
    – SRKX
    May 18, 2016 at 5:11
  • $\begingroup$ It's better, but we still can't really understand what the meaning is. Just in a few words you could explain what the paper is about and what that formula is supposed to represent... $\endgroup$
    – SRKX
    May 19, 2016 at 1:13

1 Answer 1


$$\begin{split}\sum_{k=0}^{[nx]}\binom{n}{k}s^k(1-s)^{n-k}& =\sum_{k=0}^n \mathbb{1}_{k\leq [nx]} \binom{n}{k}s^k(1-s)^{n-k} \\ & = \sum_{k=0}^n \mathbb{1}_{k\leq nx} \binom{n}{k}s^k(1-s)^{n-k} \\ & = \mathbb{P}(\mathcal{B}(n,s)\leq nx)\\ & = \mathbb{P}\left(\frac{\mathcal{B}(n,x)}{n}\leq x\right)\end{split}$$

where $\mathcal{B}(n,s)$ is a binomial of parameters $(n,s)$

using the law of large number you get $\frac{\mathcal{B}(n,s)}{n}\to_{n\to\infty} s$ and you can conclude.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.