I am reading a book "Statistics of directional data" which deals with probability density functions $f(\theta)$ where $\theta$ represents the angle around the circle and $f(\theta)$ is periodic with period $2\pi$.

When defining the characteristic function as

$\phi(t) = E[e^{it\theta}]=\int_0^{2\pi}e^{it\theta}f(\theta)d\theta$,

the author argues that because of the periodicity of $f(\theta)$, we have

$\int_0^{2\pi}e^{it\theta}f(\theta)d\theta = \int_0^{2\pi}e^{it(\theta+2\pi)}f(\theta)d\theta$

from which he concludes $e^{2\pi it} = 1$ and therefore that the characteristic function should only be defined for integer values of $t$. It seems to me that this is reversed logic as the last equation only holds if $t$ is an integer, unless I'm missing something?

Then, as if aware of this issue, the author goes on by saying that "In fact, the theory of Fourier series for periodic functions shows that it is sufficient to take $t$ as an integer.", and so he defines the characteristic function $\phi(t)$ for integer t only.

To me, in this case the Fourier series is only an analogy, since while the characteristic function correspond to the coefficients of a Fourier series, no use is made of the actual series. Yet I would like to know if someone has any interpretation to offer for the Fourier series in this case, and also if any interpretation could be given to the characteristic function $\phi(t)$ for non integer values of $t$, since a priori $\phi(t)$ can be defined for any real $t$?


1 Answer 1


Just some thoughts about my question:

About the interpretation of $\phi(t)$ for non-integer values of $t$, if $\phi(t)$ is regarded as the coefficients of a Fourier series, then $\phi(t)$'s for non-integer values of $t$ are simply not part of the series since the associated functions $e^{it\theta}$ do not satisfy the periodic boundary condition of $f(\theta)$.

Therefore the question really is, is there any useful interpretation that can be given to $\phi(t) = E[e^{it\theta}]=\int_0^{2\pi}e^{it\theta}f(\theta)d\theta$ for non-integer $t$, which do not correspond to coefficients of the Fourier series of $f(\theta)$? So far, my understanding is that $\phi(t)$ for $t = 0,1,2...$ are used to derive certain properties of random distributions such as the mean direction ($t=1$) and skewness ($t=2$).

For example, for $t=1$,

$\phi(1) = E[e^{i\theta}] = E[cos(\theta)] + i E[sin(\theta)]$

and we have the interpretation of $\phi(1)$ as the expected/mean vector sum (in the complex plane) where each value of $\theta$ in the random distribution corresponds to a vector on the unit circle.

Then for $t = 2$,

$\phi(2) = E[cos(2\theta)] + i E[sin(2\theta)]$

but more useful is a kind of centralized characteristic function that can be defined around the mean direction $\theta_0$,

$\phi'(2) = E[cos(2(\theta-\theta_0)] + i E[sin(2\theta-\theta_0)]$

and it is seen that $E[sin(2\theta-\theta_0)]$ gives a measure of the distribution skewness (asymmetry of the distribution around the mean value). At least that's what the book I referred to in my question does. However it seems to me that for any $t > 1$, $\phi'(t)$ would give a measure of skewness, and that using only integer values of $t$ is only for the sake of convention in defining meaningful measures. Unless I'm missing some deeper insight that can be gained from using integer values of $t$?


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