Question

Let \( X \) be a random variable such that \[ P \left( \frac{a}{2\pi} X \in \mathbb{Z} \right) = 1, \quad a>0, \] where \( \mathbb{Z} \) denotes the set of all integers. If \( \phi_X(t), t \in \mathbb{R} \), denotes the characteristic function of \( X \), then which of the following is/are true?

• A. \( \phi_X(a) = 1 \)
• B. \( \phi_X(\cdot) \) is periodic with period \( a \)
• C. \( |\phi_X(t)|<1 \text{ for all } t \neq a \)
• D. \( \int_0^{2\pi} e^{-itn} \phi_X(t) dt = \pi P\left(X = \frac{2\pi n}{a}\right), \, n \in \mathbb{Z}, \, i = \sqrt{-1} \)

Hint:

If a random variable takes discrete values that are integer multiples of a constant, its characteristic function is periodic with a period equal to the reciprocal of that constant.

Answer 1

The Correct Option is A, B

Step 1: Understanding the given condition.
The condition \[ P\left(\frac{a}{2\pi} X \in \mathbb{Z}\right) = 1 \] means that \( X \) takes values only at discrete points of the form \( X = \frac{2\pi n}{a} \), where \( n \) is an integer. Hence, \( X \) is a discrete random variable supported on multiples of \( \frac{2\pi}{a} \).

Step 2: Characteristic function definition.
The characteristic function of \( X \) is defined as: \[ \phi_X(t) = E[e^{itX}] = \sum_{n=-\infty}^{\infty} p_n e^{it \frac{2\pi n}{a}}, \] where \( p_n = P\left(X = \frac{2\pi n}{a}\right) \).

Step 3: Compute \( \phi_X(a) \).
Substituting \( t = a \), we get: \[ \phi_X(a) = \sum_{n=-\infty}^{\infty} p_n e^{i a \frac{2\pi n}{a}} = \sum_{n=-\infty}^{\infty} p_n e^{i 2\pi n} = \sum_{n=-\infty}^{\infty} p_n (1) = 1. \] Thus, \( \phi_X(a) = 1 \), which makes statement (A) true.

Step 4: Checking periodicity.
Now, consider: \[ \phi_X(t + a) = \sum_{n=-\infty}^{\infty} p_n e^{i (t + a) \frac{2\pi n}{a}} = \sum_{n=-\infty}^{\infty} p_n e^{i t \frac{2\pi n}{a}} e^{i 2\pi n} = \sum_{n=-\infty}^{\infty} p_n e^{i t \frac{2\pi n}{a}} = \phi_X(t). \] Thus, \( \phi_X(t) \) is periodic with period \( a \), proving statement (B) true.

Step 5: Analysis of other options.
(C) \( |\phi_X(t)|<1 \) for all \( t \neq a \) is not necessarily true because for a discrete distribution, \( \phi_X(t) \) may attain 1 at other multiples of \( a \).
(D) The integral expression does not hold in this general setting—it is unrelated to the periodicity condition and characteristic function form.

Step 6: Conclusion.
Therefore, both statements (A) and (B) are correct.