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Which of the following is not a characteristic function ?

  • A.\(\phi_{1}(t) = \phi\left(\frac{t}{a}\right) \exp(ibt)\), \(a \neq 0\), \(b \in R\) and \(\phi\) is a characteristic function
  • B.\(\phi_{1}(t) = \phi(-t)\), where \(\phi\) is a characteristic function
  • C.\(\phi(t) = \frac{1}{1 + t^{4}}\), \(t \in R\)
  • D.\(\phi(t) = \sum_{j=0}^{\infty} a_{j}\phi_{j}(t)\), \(a_{j} \ge 0\) and \(\sum_{j=0}^{\infty} a_{j} = 1\), and \(\phi_{j}\) is a characteristic function, \(j = 0, 1, 2, \ldots\)

Answer: C

The function that is not a characteristic function is \(\phi(t) = \frac{1}{1 + t^{4}}\).

Near 0, \(\frac{1}{1 + t^{4}} = 1 - t^{4} + \ldots\), so \(\phi'(0) = \phi''(0) = 0\). If it were the characteristic function of X, then \(E(X^{2}) = -\phi''(0) = 0\), forcing \(X = 0\) almost surely, whose characteristic function is identically 1. That contradiction shows it is not a characteristic function.

The others are. \(\phi(t/a)e^{ibt}\) is the characteristic function of \(\frac{X}{a} + b\). \(\phi(-t)\) is that of \(-X\). And a mixture \(\sum a_{j}\phi_{j}\) with weights \(a_{j} \ge 0\) summing to 1 is the characteristic function of the corresponding mixture distribution.

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