Central limit theorems

Liapounov's theorem

D3.1.1 For each `n >= 1`, let `{X_(n_1), X_(n_2), ..., X_(n_k)}` be a collection of rvs on `(Omega_n, F_n, P_n)` such that `X_(n_1), ...` are independent, where `k_n -> oo` as `n->oo`. Then `{X_(nj), 1 <= j <= k_n}_(n>=1)` is called a double array (DA). If `k_n = n`, called a triangular array.

• `S_n = sum_(j=1)^(k_n) X_(nj)`
• `alpha_(nj) = E[X_(nj)]`, `alpha_n = sum a_(nj)`
• `sigma_(nj)^2 = var(X_(nj))`, `sigma_n^2 = sum sigma_(nj)^2`
• `gamma_(nj) = E|X_(nj) - a_(nj)|^3`, `gamma_n = sum gamma_(nj)`

Want to establish `(S_n - a_n)/(b_n) ->^d N(0,1)` for `{a_n}, {b_n} sub RR`.

L3.1.1 Let `{theta_(nj)}` be a DA of complex number such that as `n -> oo`:

• `max_(1<= j <= k) |theta_(nj)| -> 0`
• `S_n <= M < oo quad AA n`
• `S_n -> theta < oo`

then `prod (1 + theta_(nj)) -> e^theta`

Proof: Use fact that `log(1 + z) = sum (-1^(m-1) z^m)/(m!)`. Show that `|log(1 + theta_n) - theta_n|` bdd uniformly by 1, hence `log(1+ theta_j) < theta_(nj) + k|theta_(nj)|^2`. Look at sum, and note first part `-> theta` and second part `-> 0`

T3.1.2 (Liapounov) For a DA `{x_(nj)}`, `gamma_n < oo AA n`, if `(gamma_n)/(sigma_n^3) -> 0` then `(S_n - alpha_n)/(sigma_n) ->^d N(0,1)`

Proof: Let `phi_X(T)` be cdf of `(X_(nj) - alpha_(nj))/(sigma_n)`. Show `phi_X(t) - 1` meets assumptions of L3.1.1, and hence converges to `e^(-t^2/2)`

T3.1.3 `{X_n}_(n>=1)` a sequence of rvs, `gamma_n < oo`, if `(sum gamma_j)/(sigma_n^3) -> 0` then `(S_n - sum alpha_j)/(sigma_n) -> N(0,1)`

C3.1.4 For a DA `{X_(nj)}` if `|X_n - alpha_j| <= M_(nj)` wp1 and `lim_(n->oo) max M_(nj) = 0` then `|(S_n - E(S_n))/(delta_n)| ->^d N(0,1)`.

D3.1.2 A null array saticcies one of the following conditions:

• `AAj lim_(n->oo) P(|X_(nj) - alpha_(nj)| > epsi sigma_n) = 0`
• `lim_(n->oo) max_j P(|X_(nj) - alpha_(nj)| > epsi sigma_n) = 0`
• `lim_(n->oo) P(max_j |X_(nj) - alpha_(nj)| > epsi sigma_n) = 0`
• `lim_(n->oo) sum_j P(|X_(nj) - alpha_(nj)| > epsi sigma_n) = 0`
• `lim_(n->oo) max_j |phi_(nj)(t) - 1| = 0`

Lindeburg-Feller CLT

D3.2.1 A DA satisfies the Lindeburg condition (LC) if `lim_(n->oo) 1/(sigma_n^2) sum_(j=1)^k E(X_(nj)^2 I(|X_(nj)| > epsi sigma_n^2)) = 0 quad AA epsi > 0`

L3.2.1 Let `u(m,n): NN xx NN -> RR` st `AA m lim_(n->oo) u(m,n) -> 0`. Then there exists a `uarr` sequence `{M_n}_(n>=1) -> oo` st `lim_(n>oo) u(M_n,n) = 0`

T3.2.1 (Lindeburg-Feller). For a DA. Assume `Var(X_(nj)) < sigma_(nj)^2 < oo AA n`, `alpha_(nj) = 0`. If LC then:

• `(S_n - alpha_n)/(sigma_n) ->^d N(0,1)`
• the DA is a null array

D3.2.1 A sequence of rv `{X_n}` is m-dependent if `EE m in NN => AA >=1, j >=m => X_n+j "is independent of " F_n=sigma{X_j:1<=j<=n}`. If `m = 0` then `{X_n}` is independent.

T3.2.3 Let `{X_n}` be a sequence of `m`-depedendent uniformly bounded random variables st `(sigma_n)/(n^(1/3)) -> oo` then `(S_n - alpha_n)/(sigma_n) ->^d N(0,1)`.

Proof. Split up into large and small blocks. Show `S_n = S_n^' + S_n^('') + S_n^(''')`, show `(S_n^(''))/(sigma_n) "and" (S_n^('''))/(sigma_n) -> 0`, then `(sigma_n^')/(sigma_n) ->0 " and" (S_n^')/(sigma_n^') - >^d N(0,1)` as LC condition fulfilled.

Functional central limit theorem

D3.3.1 The Weiner measure is a probability measure on `(C, ccC)` corresponding to a stochastic prcoess `X_t quad t in [0,1]` having two properties:

• `W(X_j <= alpha) = 1/sqrt(2 pi t) int_RR e^(-y^2/(2t)) dy` ie ~ N(0,t)
• For a finite number of times `0 <= t_1 < t_2 < ... < t_k = 1` the random variables `X_(t_1) - X_(t_2), ..., X_(t_(k-1))-X_(t_k)` are independent under W (independent increments)

Remarks:

• the stochastic `X = {X_t}_(t>=1)` is called a Weiner process or Brownian motion, commonly denoted `W = {W_t}_(t>=1)`
• `W` is the analogue of `Z`
• a stationary normal proceess has the following properties:
  • any finite dimenional distribution `(N_(t_1), ..., N_(t_k))` is k-variate norma
  • `mu = E(N_t) AA t`, `sigma^2(h) = Cov(N_t, N_(t+h))`
• `W_t ~ N(0, t)`, `(W_(t_1), ..., W_(t_k)) ~ N(0, Sigma_k)`, `Sigma_k = ((min{t_i, t_j}))_(k xx k)`

T3.3.1 Let `{P_n}_(n>=1)` and `P` be probability measures. If all finite dimensional distributions of `P_n` converge to `P` and `{P_n}` is tight, then `P_n ->^d P`

T3.3.2 Consider `{X_n}` and `X` continous random functions on `(C, ccC)` and if

• `(X_(nt_1), ..., X_(nt_k) ->^d (X_(t_1), ..., X_(t_k))`
• `{X_n}` is tight

then `X_n ->^d X`

T3.3.3 If `{X_n0}_(n>=1)` is tight and `AA epsi, eta > 0 EE delta in (0,1) "and" n_0 "st" AA n >= n_0 1/delta P(sup_(t <= s <= t+delta) |X_n(s) - X_n(t)| > epsi) < eta AA t in [0,1]` then `{X_n}` is tight.

C3.3.4 Let `{X_n}` and `X` be random functions on `(C, ccC)`, if

• `(X_(nt_1), ..., X_(nt_k)) ->^d (X_(t_1), ..., X_(t_k))`
• `AA epsi, eta > 0 EE delta in (0,1) "and" n_0 "st" AA n >= n_0 1/delta P(sup_(t <= s <= t+delta) |X_n(s) - X_n(t)| > epsi) < eta AA t in [0,1]`

then `X_n ->^d X`

Donska. `Z_n ->^d W`. Mapping can be extended to `(C, ccC)`