I have a (seemingly) simple question. How can I see (rigorously) that \begin{equation} \prod_{i=1}^n (1-p_i) = \exp{\left(-\sum_{i=1}^n p_i\right)} + O\left(\sum_{i=1}^n p_i^2\right) \end{equation} for $\max_{i}p_i\rightarrow 0$ and $p_i\geq 0,\, \forall i$. For $n=1$, the above is a just a Taylor approximation. But, for $n>1$, I have trouble combining the individual error terms (for the individual $i$'s) in the product to obtain the $O(\sum p_i^2)$ error term. Any ideas? Every hint/help is much appreciated. Thanks in advance!
How to show $\prod_{i=1}^n (1-p_i) = \exp{(-\sum_{i=1}^n p_i)} + O(\sum_{i=1}^n p_i^2)$
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0The clause "$\max_i p_i\to 0$, as $n\to\infty$" doesn't make sense. – 2012-12-02
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0yes, I agree. sorry for the confusion. the $n\rightarrow \infty$ piece comes from the larger context of the problem. in the above formulation, I was just cutting out the piece that I am having trouble with. – 2012-12-03
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0Don't you need a $1$ on the right side? Otherwise, LHS is looking like $1$, RHS like $0$. – 2012-12-03
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0of course there was a mistake, thanks for pointing it out. there was an "exp" missing. sorry about that! – 2012-12-03
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0Still looks funny --- LHS $\gt0\gt$ RHS. – 2012-12-03
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0You should correct the statement, see my answer. – 2012-12-04
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0As explained in my answer, I don't think it should be corrected. – 2012-12-04
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0sorry for any confusion that a faulty statement might have caused. my apologies! but I don't understand the comment "LHS > 0 > RHS". Why RHS < 0? – 2012-12-05
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0but many thanks for all the help, it is much appreciated! – 2012-12-05
2 Answers
I think that the problem is now correct.
Suppose $\forall i,\;|1-p_i| \leq 1$ (for instance if $\forall i,\;0\leq p_i \leq 1$). We have $$ \left|\prod_{i=1}^n (1-p_i) - \prod_{i=1}^n e^{-p_i}\right| \leq \sum_{i=1}^n |1 - p_i - e^{-p_i}| \leq \frac{1}{2} \sum_{i=1}^n p_i^2. $$
Hence, $$ \prod_{i=1}^n (1-p_i) = \exp\left(-\sum_{i=1}^n p_i\right) + O\!\left(\sum_{i=1}^n p_i^2\right). $$
The first inequality comes from the fact that if $a_1,\dots,a_n$ and $b_1,\dots,b_n$ are complex numbers with modulus lesser than $1$, then $|\prod a_i - \prod b_i| \leq \sum |a_i - b_i|$ (this follows from an easy recurrence).
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0great answer, many thanks! I really appreciate your support! – 2012-12-05
Consider
$$ \log \prod_{i=1}^n (1-p_i) = \sum_{i=1}^n \log(1-p_i).$$
Since $\max_i p_i \to 0$, use the Taylor series expansion of $\log(1-p_i)$, then it follows that
$$ \log \prod_{i=1}^n (1-p_i) = -\sum_{i=1}^n p_i + O\left( \sum_{i=1}^n p_i^2 \right). $$
Take exponents of both sides and use $e^x=1+ O(x)$, $x$ around $0$, to obtain
$$ \prod_{i=1}^n (1-p_i)= exp(-\sum_{i=1}^n p_i) + 1 + O\left( \sum_{i=1}^n p_i^2 \right). $$
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0Funny how much easier it is to do a problem when the problem statement is actually correct. – 2012-12-04
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0Your solution is not sharp enough : when taking $p_i=0$, it writes $1 = 2 + O(0)$. – 2012-12-04