Characteristic functions

Definition and basic properties

Let `X` be a random variable on `(RR, B(RR))` with probability measure `mu` and distriubtion function `F`

D2.1.1 The characteristic function of X is: `phi_X(t) = E(e^(itX)) = int_Omega e^(i t omega) dP(omega) = int_R e^(itx) dmu(x) = int_(-oo)^(oo) e^(itx) dF(x) AA t in RR`

• `phi_X(t)` is an integral of a complex function over the real domain
• if `g(x) = g_1(x) + ig_2(x)` then `int g(x) = int g_1(x) + i int g_2(x)`
• every characeristic function has an associated probability measure and random variable

Properties:

• `|phi_X(t)| <= phi_X(0) = 1 quad AA t in RR `
• `bar(phi_X(t)) = phi_X(-t) quad AA t in RR `
• `phi_X` is uniformly continuous on `RR`
• if `{phi_i}` are cf's, and `sum p_i = 1` then `sum p_i phi_i` is a cf
• if `phi_X` is a cf, then so is `bar(phi_X), |phi_X|^2, Re(phi_X)`

P2.1.2 `|phi_X(t_0)| = 1 iff X "is a lattice rv"`

P2.1.3 If `F` is absolutely continuous, then `lim_(|t|->oo) phi_X(t) = 0`

Inversion formula

• `AA y > 0 0 <= sgn(x) int_0^y sin (alpha x) / (x) dx <= int_0^oo (sinx)/x dx`
• `int_0^oo sin (alphax)/x = pi/2 sgn(alpha)`
• `int_0^oo (1-cos (alphax))/(x^2) dx = pi/2 alpha`

T2.2.1 `mu(x_1, x_2) + 1/2(mu{x_1} + mu{x_2}) = lim_(T->oo) int_(-T)^T (e^(-itx_1) - e^(-itx_2))/(it) phi_X(t) dt`

• `int_(-oo)^oo (e^(-itx_1) - e^(-itx_2))/(it) phi_X(t) dt` will only exist if `phi_X` is integrable, in which case we can replace the limit above

C2.2.2 If `x_1, x_2 in C(F)` then `mu(x_1, x_2) = lim_(T->oo) int_(-T)^T (e^(-itx_1) - e^(-itx_2))/(it) phi_X(t) dt`

T2.2.3 (Uniqueness theorem) If 2 pm's `mu_1, mu_2` have the same cf's `phi_1, phi_2` then `mu_1 = mu_2`.

T2.2.4 If `phi_X in L^1` then `F` is absolutely continuous with density `f(x) = 1/(2pi) int_R e^(-itx) phi_X(t) dt) < oo`.

T2.2.6 `AA x_0 in R mu{x_0} = lim_(T->oo) int_(-T)^T e^(itx_0) phi_X(t) dt`.

C2.2.7 `sum_(x in RR) mu{x}^2 = lim_(T->oo) 1/(2T) int_(-T)^T |phi_X(t)|^2 dt`

Convergence theorems and applications

Let `{X_n}_(n>=1), X` be random variables on `(RR, B(RR))` with characteristic functions `{phi_n}_(n>=1), phi)` respectively.

T2.3.1 If `X_n ->^d X` then `phi_n -> phi` uniformly for any finite interval, ie. `spr_(|t| < K) |phi_n(t) - phi(t)| -> 0`

L2.3.1 `AA delta > 0 mu({x | x delta > 2}) <= 1/delta int_(-delta)^delta |1 - phi(x)| dx`

L2.3.2 If `phi_n(t) -> phi(t) < oo AA |t| <= delta_0`, and `phi` is continuous at 0, then `{mu_n}` is tight.

T2.3.4 (Levy-Cramer) if `lim_(n->oo) phi_n(t) = phi(t) in R` and `phi` is continuous at 0, then:

• `mu_n ->^d mu`, `mu` a pm
• `phi` is the characteristic function of `mu`

T2.3.5 `If E|X|^r < oo, r in NN` then `phi_x` is r-times continuously differentiable and `phi_X^((r))(t) = E( (iX)^r e^(itx))`. Conversely, if `phi_X^((r))` exists for even integer `r` then `X` has finite rth order (absolute) moments.

T2.3.6 If `E|X|^r < oo`, `r > 1`, then `phi_X` admits the follow Tayloring expansion around `t=0`:

• `phi_X(t) = sum_(j=0)^r ((it)^j)/(j!) EX^j + o(|t|^r)`
• `phi_X(t) = sum_(j=0)^(r-1) ((it)^j)/(j!) + theta^r/(r!) E|X|^r t^r`, `theta in (0,1)`

P2.3.7 Suppose `{c_n} sub CC -> c ` then `lim_(n->oo) (1+(c_n)/n)^n = e^c`.

Characteristic functions in `RR^k`

D2.4.1 Let `X = (X_1, ..., X_k)` be a random vector on `RR_k` with pm. The characteristic function of `X` is `phi_X(vec t) = E e^(it^T X) = int e^(it^T x) mu(dx_1, ..., dx_2)`.

• inversion formula generalises
• uniqueness theorem generalixses
• multivariate version of convergence is valid

P2.4.1 A pm `mu` on `(RR^k, B(RR^k))` is determined by its values assigned to `ccH = {H_(ac) | H_(ac) = {X in R^k, a'X <= c, AA a in RR^k, c in RR}}`

T2.4.2 (Cramer-Wald). Let `{X_n}` be a sequence of random vectors and `X` an rvec on `(RR^k, B(RR^k))` then `X_n ->^d X iff a'X_n ->^d a'X quad AA a in RR^k`.