Weak Convergence (= Convergence in Distribution):

Billingsley sections 25, 26 pp. 327ff

  1) Def & motivation
  2) ChF's
  3) 
  4)

Examples:

  What should it mean for Xn => X ?

  a) fn(x) -> f(x) for all x?    (no... e.g. Pr[Xn = i/n] = 1/n, 0<=i<n)
  b) Pr[ Xn in A ] -> Pr[ X in A ] for every Borel set?  (no.. same e.g.)
  c) Fn(x) -> F(x) for every x?  (same as above, for A = (-oo,x] ), i.e.
     E[ phi(Xn) ] -> E[ phi(X) ] for phi = 1_{(-oo, x]} (no... e.g. Xn = 1/n)

  d) TFAE:
     i) Fn(x) -> F(x) wherever F(x) = F(x-)
    ii) E[ phi(Xn) ] -> E[ phi(X) ] for all Cb phi(.)      (OK in R^n etc)
   iii) E[ phi(Xn) ] -> E[ phi(X) ] for all C^oo_b phi(.)
    iv) E[ phi(Xn) ] -> E[ phi(X) ] for phi(x)=exp(i w.x)


Note that could have different OFP's (e.g. X_n ~ Bi(n,p) => X ~ Po(lambda))
so Xn=>X cannot imply Xn -> X in ANY other sense

Partial converse: If Xn => X, then there exists a probability space OFP
and random variables Xn, X on OFP such that Xn->X a.s. (for EVERY w, in fact).

Cool result:

   Helly's Thm:  For any sequence \mu_n of PD's there exists a
   nonnegative measure \mu s.t. \int \phi d\mu_n -> \int \phi d\mu
   for all Cb \phi.

   N&S Condition that \phi be a PROBABILITY measure:

   "Tightness":  For all eps>0 there exists a compact set K_eps with 
                 \mu_n(K_eps) > 1-eps  for all n.

   In R^1: For all eps there is k<oo s.t.  Pr[ |Xn| <= k ] >= 1-eps

-------

MGF:
   M(t) = E[e^{t X}]

   Properties: M(0) = 1, M'(0) = mu, M''(0) = sig^2 + mu^2, ...
               (take log for easier formulas)

   Examples:
       Bi(n,p):       (p e^t +q)^n
       No(mu,sig^2):  exp(mu t + sig^2 t^2/2)
       Po(lam):       exp(lam * (e^t-1))
       Ga(a,b):       (1-t/b)^(-a)          [ b = RATE param ]
       Cau(m,s):      oo

ChF's:          i w X
   phi(w) = E[ e      ] = E[ cos(w X) ] + i E[ sin(w X) ]

   Properties: 
     * phi(0)=1, phi'(0) = i mu, phi''(0) = -sig^2, ...
     * Indep -> phi_{X+Y}(w) = phi_X(w) phi_Y(w)
     * phi(w) -> 0 as w->oo IF phi has a density function
     * Inversion: f(x) = (1/2pi) int phi(w) exp(-i w x) dw

   Examples:
       Bi(n,p):       (p e^{i w} +q)^n
       No(mu,sig^2):  exp(i mu w - sig^2 w^2/2)
       Po(lam):       exp(lam * (e^{i w}-1))
       Ga(a,b):       (1-i w/b)^(-a)
       Cau(m,s):      exp(i w m - |w s|)

   Note:
       Bi(n, lam/n): (1 + (1/n)lam(e^{i w}-1))^n -> exp(lam * (e^{i w}-1))


Note:  Let X1 X2 ... be iid Cauchy, and let Y be their average.  What is
       the distribution of Y???