9.2 Characteristic functions

Characteristic functions are powerful tools in probability theory, linking random variables to their moments. They're closely related to moment-generating functions but exist for all distributions, making them more versatile for analyzing probability distributions.

Characteristic functions have unique properties that make them useful for various probability calculations. They can be used to compute moments, determine independence, and even recover probability distributions through inversion formulas. Understanding these functions is crucial for advanced probability analysis.

Characteristic Functions

Characteristic function and MGF relationship

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• Definition of characteristic function (CF) $\phi_X(t) = E[e^{itX}]$ where $i = \sqrt{-1}$ is imaginary unit and $t$ is real number
• CF is moment-generating function (MGF) $M_X(t) = E[e^{tX}]$ evaluated at $it$: $\phi_X(t) = M_X(it)$
• If MGF exists, CF also exists

Computation of characteristic functions

• Discrete random variable $X$ with probability mass function $p_X(x)$, CF is $\phi_X(t) = \sum_{x} e^{itx} p_X(x)$
• Continuous random variable $X$ with probability density function $f_X(x)$, CF is $\phi_X(t) = \int_{-\infty}^{\infty} e^{itx} f_X(x) dx$
• Examples of CFs for common distributions
  • Bernoulli distribution: $\phi_X(t) = 1 - p + pe^{it}$ (coin flip)
  • Binomial distribution: $\phi_X(t) = (1 - p + pe^{it})^n$ (number of successes in $n$ trials)
  • Poisson distribution: $\phi_X(t) = \exp(\lambda(e^{it} - 1))$ (number of events in fixed interval)
  • Normal distribution: $\phi_X(t) = \exp(i\mu t - \frac{1}{2}\sigma^2 t^2)$ (bell curve)

Properties of characteristic functions

• Existence: if $E[|X|] < \infty$, CF $\phi_X(t)$ exists for all $t$
• Uniqueness: if two random variables $X$ and $Y$ have same CF, they have same distribution
• Continuity: CF $\phi_X(t)$ is uniformly continuous on $\mathbb{R}$
• Other properties
  • $\phi_X(0) = 1$
  • $|\phi_X(t)| \leq 1$ for all $t$
  • $\phi_{aX+b}(t) = e^{itb} \phi_X(at)$ for constants $a$ and $b$ (linear transformation)
  • If $X$ and $Y$ are independent, $\phi_{X+Y}(t) = \phi_X(t) \phi_Y(t)$ (convolution)

Inversion formula for probability distributions

• Continuous random variable $X$ with CF $\phi_X(t)$, probability density function $f_X(x)$ recovered using inversion formula: $f_X(x) = \frac{1}{2\pi} \int_{-\infty}^{\infty} e^{-itx} \phi_X(t) dt$
• Discrete random variable $X$ with CF $\phi_X(t)$, probability mass function $p_X(x)$ recovered using inversion formula: $p_X(x) = \frac{1}{2\pi} \int_{-\pi}^{\pi} e^{-itx} \phi_X(t) dt$
• Cumulative distribution function (CDF) $F_X(x)$ obtained from probability density function or probability mass function by integration or summation