ASYMPTOTIC THEORY 2

Chapter 9 ASYMPTOTIC THEORY 2

  We can now establish the convergence in distribution and the central limit theorem, which is of great importance.

  Definition

  A sequence of random variables T 1 ,T 2 , ... converges in distribution

  to a random variable T (denoted by T D

  n −→ T ) if lim P [T n ≤ x] = P [T ≤ x]

  n→∞

  at all points of continuity of F T (x) = P [T ≤ x].

  Convergence in distribution is weaker than in probability, i.e.,

  but not vice versa, except when the limit is nonrandom, i.e.,

  D T P

  n −→ α ⇒ T n −→ α.

  Theorem 23 D A sequence of random vectors (T

  n1 ,T n2 , ..., T nk ) −→ (T 1 ,T 2 , ..., T k ) iff

  1 ,c 2 , ..., c k ) 6= 0.

  This is known as the Cram´ er − Wold device. The main result is the following:

  Asymptotic Theory 2

  Theorem 24 Central Limit Theorem of Lindemberg-Lévy. Let X 1 ,X 2 , ..., X n be i.i.d.

  with E (X 2

  i ) = μ, V ar (X i )=σ < ∞. Then

  The vector version: Let X 1 ,X 2 , ..., X n be i.i.d. with E (X i )=μ ,E (X i − μ) (X i − μ) =

  Σ where 0 < Σ < ∞. Then

  A modern proof of the result is based on characteristic functions. Example. If X

  i v N (μ, σ ), then

  for all n a result which is trivial. Suppose instead that

  We know that T D

  etc. The Binomial distribution gets closer and closer to normal.

  Asymptotic Theory 2

  We can now approximately calculate for example

  CLT for non-identically distributed random variables.

  Theorem 25 (Lyapunov) Suppose that X 1 ,X 2 , ..., X n are independent random vari-

  ables with

  and additionally

  e.g. if

  The Lindeberg-Feller CLT is even weaker. Theorem 26 (Lindeberg-Feller) Let x i be independent with mean μ i and variance

  i , and distribution functions F

  i . Suppose that B n = i=1 σ i satisfies

  σ 2

  2 → 0, B n → ∞, as n B → ∞.

  n

  n

  Then

  1 P n

  P n

  n ( i=1 x i − i=1 μ i ) D

  £ 1

  ¤ 12 −→ N (0, 1)

  2 B 2

  n

  n

  Asymptotic Theory 2

  if and only if the Lindeberg condition

  P n R

  i=1 |t−μ |>εB n (t −μ i ) dF i i (t)

  2 → 0, n → ∞, each B ε>0

  n

  is satisfied.

  The key condition for the above CLT is the Lindeberg condition. Which basically ensures that no one term is so relatively large as to dominate the entire sample, in the limit. The following CLT gives some more transparent conditions that are sufficient for the Lindeberg condition to hold.

  − 2

  Theorem 27 Let x i be independent with mean μ i and variance σ 2 i , and let σ n =

  max 1≤i≤n E |x i |

  The above condition although sufficient is not necessary. To see this assume

  1 ¡

  1 ¢

  that y t =0 with probability 2 1 − t 2 , 0 with the same probability and t with

  probability 1 t 2 . In this case y t tends to a Bernoulli random variable, and the CLT

  certainly applies in this case. Yet the condition of the above theorem is not satisfied.

  Furthermore, let assume that y 1

  t =0 with probability 1 − t 2 and t with prob-

  1 ability −

  t 2 . Then E (y t ) = 1t → 0, and V ar (y t )=1 − t 2 → 1. Hence σ n → 1.

  Despite this, it is clear that y t is converging to a degenerate random variable which takes the value of 0 with probability 1 (in fact is x t =y t − E(y t ) that is degenerate).

  However, it is verified that E |x i |

  =O t δ+2 and consequently for any δ > 0

  the condition of the above theorem must fail for n large enough.

  Dependent random variables CLT’s are available too.

  Combination Properties

  9.1 Combination Properties

  D Theorem 28 P (Slutsky’s) Suppose that andX

  n −→ X and Y n −→ c. Then

  n Y n −→ Xc if c nonzero

  Application: Suppose that we look at

  when s X is the sample variance. The CLT tell us that

  The LLN and CMT say that

  Theorem 29 Continuous Mapping Theorem II. Suppose that

  T D

  n = (T n1 ,T n2 , .., T nk ) −→ T = (T 1 ,T 2 , .., T k )

  and g : R q →R . Then

  but notice that the assumption requires the joint convergence of (Y n ,X n ).

  Asymptotic Theory 2

  Theorem 30 d (Cramer) Assume that X

  n → N (μ, Σ), and A n is a conformable ma-

  trix with plimA ¢

  d ¡

  n = A. Then A n X n →N μ, AΣA

  Notice that if

  9.2 Delta Method. Suppose that

  where X has a cdf F and θ is a p × 1 vector. Suppose that g : R q →R . Then

  The proof is by the mean value theorem, i.e.,

  where ∂g θ lies between θ

  −

  ∧

  0 and θ . Now for ∂θ continuous at θ 0 , we have

  as required.

  For example, sin X when μ = 0. Now (sin x) D = cos x . Hence n sin X −→ N (0, 1) .

  √

  In fact we can state the following theorem.

  Delta Method.

  Theorem 31 Suppose that X n is asymptotically distributed as N (μ, σ 2 n ), with σ n →

  0. Let g be a real valued function differentiable m (m ≥ 1) times at x = μ, with

  m

  g j (μ) 6= 0 but g (μ) = 0 for j < m. Then

  For example, let X n be asymptotically N (0, σ 2 n ) , with σ n → 0. Then

  To see this apply the above theorem with g (x) = log 2 (1 + x) , m = 2 and μ = 0.

  Asymptotic Theory 2