Empirical Characteristic Function

The empirical characteristic function (ecf) of a random sample {, , ...} from a statistical distribution is defined by

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In this representation, each random variable can be envisioned as a particle orbiting the unit circle in the complex plane. The ecf is the expected orbit or mean of the random variable orbits. For large , the ecf converges to the distribution characteristic function. The graphic shows the orbit of a standardized stable distribution with parameters and in blue. The orbit of the ecf of 500 random variables with the same parameters is shown in red and the position, at , of each random variable on the unit circle is shown as a blue dot. The red dot is the mean of these positions. Each time you change the or slider a new random sample is generated. When or , the distribution will be symmetric about zero and the characteristic function will be confined to the real line, the axis in this Demonstration.