Characteristic function characterizes probability distributions.

When you study probability theory at the university level, you almost always come across the term “characteristic function.”

What is a characteristic function? The answer is obvious: the characteristic function of a probability distribution F is the Fourier transform of F.

The concept of characteristic function is very important and includes the following theorem:

Theorem: When the characteristic functions Φ and Φ’ of probability distributions F and G are equal, F = G.

From this we can see that studying probability distributions and studying characteristic functions are equivalent.

Furthermore, if we replace the concept that the random variable X_n converges to a random variable in terms of a characteristic function, we get

This comes down to the concept of simple convergence of the characteristic function.

From the above, the concept of the characteristic function plays a very important role in studying the convergence of probability distributions, and in particular, the central limit theorem is proved by the convergence of the characteristic function.

[日本語訳]

大学レベルの確率論を勉強すると, 必ずと言っていいほど『特性関数』という言葉が出てきます.

特性関数とは何か? 答えは明白で, 確率分布 F の特性関数とは, F のフーリエ変換のことです.

特性関数の概念は非常に重要で, 次の定理があります:

定理: 確率分布 F, G の特性関数 Φ, Φ’ が等しい時, F = G である.

このことから, 確率分布を研究するのと特性関数を研究するのは同値であることがわかります.

さらに, 確率変数 X_n が確率変数に収束するという概念も, 特性関数の言葉で置き換えると,

特性関数の単純収束という概念に帰着されます.

以上のことから, 特性関数の概念は, 確率分布の収束を研究する際に, 非常に重要な役割を果たしており, 特に中心極限定理の証明は, 特性関数の収束を以て証明されます.