- #1

MagentaCyan

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- TL;DR Summary
- I have the characteristic function of a probability distribution but I'm having difficulty obtaining its derivative.

## Problem summary

I have the characteristic function of a probability distribution but I'm having difficulty obtaining its derivative.## Background

I am reading the following paper: Schwartz, Lowell M. (1980). On round-off error.*Analytical Chemistry*,

**52**(7), 1141-1147. DOI:10.1021/ac50057a033.

The early part of the paper is concerned with deriving the characteristic function for a distribution that relates to ##n## quantized (rounded) samples drawn from a Gaussian distribution with mean, ##\mu##, standard-deviation ##\sigma##, where the width of the quantization interval is##q##. The characteristic function is stated as being:

$$

G_{\bar{y}}(t) = \left(\sum_{k=-\infty}^{\infty} \frac{\sin \pi \left( \frac{t}{n \phi} + k \right)} {\pi \left( \frac{t}{n \phi} + k \right)} \:\, \exp\left[ -\frac{1}{2}(\frac{t}{n} + k \phi)^2 \sigma^2 + i \, \mu (\frac{t}{n} + k \phi) \right]\right)^n,

$$

where ##\phi=2\pi/q##.

Up to that point in the paper, I'm OK. I can follow Schwartz's argument, and do my own derivation of his results, including the derivation of ##G_{\bar{y}}(t)##.

Beyond that, however, I'm stuck.

After presenting the characteristic function, Schwartz states that "

*the first moment*

*is found after*to be

**lengthy but straightforward**manipulations"$$

M_1 = E(\bar{y}) = \mu + \frac{q}{\pi} \sum_{k=1}^{\infty} \frac{(-1)^k}{k} \, X \, \sin\left( \frac{2 \pi k \mu}{q} \right),

$$

with ##X= \exp(-2 k^2 {\pi}^2 {\sigma}^2/q^2)## ... this being how Schwartz himself present the result). In calculating ##M_1##, Schwartz relies on the fact that the ##r##-th moment (if it exists) for a PDF ##f(z)## can be obtained from its characteristic function, ##g_z(t)## as follows:

$$

M_r = (-i)^r \left[ \frac{d^r g_z(t)}{ d t^r} \right]_{t=0}

$$

However, I haven't been able to obtain the result for ##M_1## myself because I have not been able to get a formula for the derivative of ##G_{\bar{y}}(t)##.

## My approach to finding the derivative

The first thing that struck me about the formula Schwartz gives for ##M_1## is that the summation is shown as being from ##k=1##, rather than from the ##k=-\infty## that appears in the formula for ##G_{\bar{y}}(t)##. That makes me think that there has been some kind of simplification based on symmetry, but I'm not sure what ... or how it has been used.The second observation is that the formula inside the summation can be treated as the product of a ##\mathrm{sinc}## function and an exponential. I can obtain the derivative with respect to $t$ of both those parts, and use the chain rule to obtain the derivative of the product, which I have done ... and verified the result using Mathematica.

However, I'm not sure how then to proceed. Specifically, I don't know how deal with the summation over the product, and the subsequent raising to the power ##n##. Nor do I see what I would then do to progress to a formula with the changed lower bound on the summation.