There is apparently a result in probability theory saying:

If $A(z)$ is any cumulative distribution function, $\alpha(t)$, the corresponding characteristic function and $\Phi(z) = \int_{-\infty}^{z}e^{-\frac{t^{2}}{2}}\mathrm{d}t$ is the cumulative distribution of the normal distribution, then, for any $T > 0$:

$$|A(z) - \Phi(z)| \leq \int_{-T}^{T}\mathrm{d}t\left|\dfrac{\alpha(t) - e^{-\frac{t^{2}}{2}}}{t}\right| + \dfrac{24}{T \pi \sqrt{2 \pi}}$$

Reference: Eq. 4, Page 11 of http://www.glassonion.org/ecc.pdf

Could anyone tell me what name this theorem goes by ? I am unable to find any in the above form.

Could anyone tell what parameter of $\Phi(z)$ is specific to the CDF function $A(z)$ ?

I presume that for a different CDF, say, $A^{\prime}(z)$ satisfying the above result, the corresponding $\Phi(z)$ would be different.

  • $\begingroup$ I don't understand the last part of your question. There is only one function $\Phi$ and the inequality presumable holds for any $z$. $\endgroup$ – Hagen von Eitzen Jan 28 '13 at 21:52
  • $\begingroup$ I meant that, in the above case, $\Phi(x)$ is the CDF of a normal distribution with mean 0 and variance 1. I was asking how the mean and variance values depend on the particular CDF function $A(z)$. $\endgroup$ – Pavithran Iyer Jan 28 '13 at 21:59

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