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 '16 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 '16 at 1:13

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

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.


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