Characteristic Function Of Cauchy Distribution

Let X follows a Cauchy distribution with a parameterμ andλ i.e. X ~ (μ,λ ). To find the characteristic function of X, we first find the characteristic function of the stsndard Cauchy variate

$$Z \ = \ \frac{X \ - \ \mu}{\lambda}$$

The probability function of Z is given by

$$f(x) \ = \ \frac{1}{\pi} \ . \ \frac{1}{1+z^2} \ \ \ \ ; \ -\infty \ < \ z \ < \ \infty$$

The characteristic function of Z is given by

$$\phi_{z}(t) \ = \ E({e^i}^{tz})$$

$$= \ \int_{-\infty}^{\infty}\ {e^i}^{tz} \ f(z) \ dz$$

$$= \ \frac{1}{\pi} \ \int_{-\infty}^{\infty}\ \frac{1}{1+z^2} \ dz$$

To find Φ_z(t), we consider stsndard Laplace distribution given by the probability density function

$$f_1(z) \ = \ \frac{1}{2} \ {e^-}^{|z|}, \ \ \ \; \ -\infty \ < \ z \ < \ \infty$$

Then, the characteristic function of the standard Laplace variable is given by

$$\phi_1 (t) \ = \ \frac{1}{2} \ \int_{-\infty}^{\infty} \ {e^i}^{tz} \ {e^-}^{|z|} \ dz$$

$$= \\frac{1}{2} \ \int_{-\infty}^{\infty} \ (Coz \ tz \ + \ i \ sin \ tz) \ {e^-}^{|z|} \ dz$$\

$$ = \ \frac{1}{2} \ \int_{-\infty}^{\infty} \ Coz \ tz \ . \ {e^-}^{|z|} \ dz \ \ \ \ \ \ \ \ \ [\therefore \ The \ second \ integral \ being \ an \ odd \ function \ vanishes]$$

$$= \ \int_{o}^{\infty} \ Cos \ tz \ . \ {e^-}^{|z|} \ dz$$

Integrating by parts, we get,

$$ = \ \left [ \ -Cos \ tz \ . \ {e^-}^{z} \ \right ]_{0}^{\infty} \ -t \int_{o}^{\infty} \ sin \ tz \ . \ {e^-}^{z} \ dz$$

$$= \ 1 \ - \ t \ \int_{o}^{\infty} \ sin \ tz \ . \ {e^-}^{z} \ dz$$

$$ \ = \ 1 \ - \ t \ \left [ -{e^-}^{z} \ sin \ tz \ \mid_{0}^{\infty} \ + \ t \ \int_{0}^{\infty} \ coz \ tz \ {e^-}^{z} \ dz \ \right ]$$

$$= \ 1 \ - \ t^2 \ \int_{0}^{\infty} \ coz \ tz \ {e^-}^{z} \ dz$$

$$\therefore \int_{0}^{\infty} \ coz \ tz \ {e^-}^{z} \ dz = \ 1 \ - \ t^2 \ \int_{0}^{\infty} \ coz \ tz \ {e^-}^{z} \ dz$$

$$\Rightarrow \ \int_{0}^{\infty} \ coz \ tz \ {e^-}^{z} \ dz \ = \ \frac{1}{1 + t^2}$$

$$Hence, \phi_1(t) \ =\ \frac{1}{1 + t^2}.$$

This characteristic functionΦ₁(t) is absolutely integral in (-∞,∞). Thus, by inversion theorem, the corresponding density function is

$$f_1 \ (z) \ = \ \frac{1}{2 \pi} \ \int_{-\infty}^{\infty} \ {e^-}^{itz} \ \phi_1 (t) \ dt$$

$$ or, \ \ \ \frac{1}{2} \ {e^-}^{|z|} \ = \ \frac{1}{2 \pi} \ \int_{-\infty}^{\infty} \ {e^-}^{itz} \ \frac{1}{1+t^2} \ dt$$

$$ or, \ \ \ {e^-}^{|z|} \ = \ \frac{1}{\pi} \ \int_{-\infty}^{\infty} \ \frac{{e^-}^{itz}}{1+t^2} \ dt \ \ \ \left [ \ \therefore \ Changing \ t \ to \ -t \ which \ does \not \ affect \ the \ value \ of \ the \ integral \ \right ]$$

On changing t and z, we get

$${e^-}^{|t|} \ = \ \frac{1}{\pi} \ \int_{-\infty}^{\infty} \ \frac{{e^-}^{itz}}{1+t^2} \ dz$$

$$ = \ \phi_z (t)$$

Hence, the characteristic function of standard Cauchy variate Z is

$$ \phi_z (t) \ = \ {e^-}^{|t|}$$

Then the characteristic function of Cauchy distribution X is given by

Φ_x(t) = E [e^itX]

= E [e^it(μ+λz)]

= e^itμ E (e^itλz)

= e^itμΦ_z(tλ)

=e^{itμ .}e^-|tλ|

∴Φ_x(t) =e^itμ-λ|t| ;λ > 0

Further, since |t| is not differentiable at t = 0, theΦ_x(t) is not differentiable at t = 0. Since r^th moment of X is given by

$$\mu_{r}^{'} \ = \ \frac{1}{i^r} \ \left [ \frac{d^r \ \phi_x(t)}{dt^r} \ \right ]_{t=0}\ = \ \frac{\phi_x(0)}{i^r}$$

the moments of any ordher do not exist. Hence the mean and variance of Cauchy distribution do not exist. This property is a very peculiar one which is not found in other distributions discussed so far.

Additive Property of Cauchy Distribution

If X₁ and X₂ are two independent Cauchy varaites with parameters (μ₁,λ₁) and (μ₂,λ₂) respectively, then X₁ + X₂ is a Cauchy variate with parameters (μ₁ +μ₂,λ₁ +λ₂)

proof:

We have the characteristic function of X₁ and X₂ are given by

Φ_x1 (t) = e^{itμ₁ -}λ₁ ^|t|

and Φ_x2(t) = e^{itμ_{2 -λ2} |t|}

Since X₁ and X₂ are independent, the characteristic function of X₁ + X_{2 is given by}

Φ_{x1 + x2} (t) =Φ_{x1 (t) .}Φ_x2(t)

∴Φ_{x1 + x2} (t) = e^{it(μ1 +μ2) - (λ₁ +λ2) |t|}

which is the characteristic function of a Cauchy function of a Cauchy random variable with parameter(μ₁ +μ₂,λ₁ +λ₂). Hence by uniqueness theorem of characteristic functionX₁ + X₂~(μ₁ +μ₂,λ₁ +λ₂). Thus the Cauchy distribution follows additive property.

Bibliography

Sukubhattu N.P. (2013). Probability & Inference - II. Asmita Books Publishers & Distributors (P) Ltd., Kathmandu.

Larson H.J. Introduction to Probability Theory and Statistical Inference. WileyInternational, New York.