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An Indexed Central Limit Theorem An Indexed Central Limit Theorem Bob Lowen (with B. Berckmoes, J. Van Casteren) University of Antwerp Coimbra July 2012 Workshop on Category Theory in honour of George Janelidze on the occasion of his 60th

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An Indexed Central Limit Theorem

An Indexed Central Limit Theorem

Bob Lowen (with B. Berckmoes, J. Van Casteren) University of Antwerp

Coimbra July 2012 Workshop on Category Theory in honour of George Janelidze on the

  • ccasion of his 60th birthday

Bob Lowen (with B. Berckmoes, J. Van Casteren) University of Antwerp An Indexed Central Limit Theorem

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An Indexed Central Limit Theorem

Standard triangular arrays

Standard triangular array (STA): a triangular array of real square integrable random variables ξ1,1 ξ2,1 ξ2,2 ξ3,1 ξ3,2 ξ3,3 . . . satisfying the following properties. (a) ∀n : ξn,1, . . . , ξn,n are independent (b) ∀n, k : E [ξn,k] = 0 (c) ∀n :

n

  • k=1

σ2

n,k = 1, where σ2 n,k = E

  • ξ2

n,k

  • (d)

n

max

k=1

σ2

n,k → 0

Bob Lowen (with B. Berckmoes, J. Van Casteren) University of Antwerp An Indexed Central Limit Theorem

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An Indexed Central Limit Theorem

The Lindeberg-Feller CLT

Theorem Given an STA (ξn,k)n,k and a normally distributed random variable ξ: if ∀ǫ > 0 :

n

  • k=1

E

  • ξ2

n,k; |ξn,k| ≥ ǫ

  • → 0

then

n

  • k=1

ξn,k

w

→ ξ.

Bob Lowen (with B. Berckmoes, J. Van Casteren) University of Antwerp An Indexed Central Limit Theorem

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An Indexed Central Limit Theorem

Lindeberg index

Lin ({ξn,k}) = sup

ǫ>0

lim sup

n→∞ n

  • k=1

E

  • ξ2

n,k; |ξn,k| ≥ ǫ

  • Bob Lowen (with B. Berckmoes, J. Van Casteren) University of Antwerp

An Indexed Central Limit Theorem

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An Indexed Central Limit Theorem

Lindeberg index

Lin ({ξn,k}) = sup

ǫ>0

lim sup

n→∞ n

  • k=1

E

  • ξ2

n,k; |ξn,k| ≥ ǫ

  • Fix 0 < α < 1, let β =

α 1−α and put

s2

n = (1 + β)n − β n

  • k=1

k−1 = n + β

n

  • k=1
  • 1 − k−1

P [ηα,n,k = −1/sn] = P [ηα,n,k = 1/sn] = 1 2

  • 1 − βk−1

P

  • ηα,n,k = −

√ k/sn

  • = P
  • ηα,n,k =

√ k/sn

  • = 1

2βk−1 Lin ({ηα,n,k}) = α

Bob Lowen (with B. Berckmoes, J. Van Casteren) University of Antwerp An Indexed Central Limit Theorem

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An Indexed Central Limit Theorem

Steps in the proof

K

  • η, η′

= sup

x∈R

  • P[η ≤ x] − P[η′ ≤ x]
  • Bob Lowen (with B. Berckmoes, J. Van Casteren) University of Antwerp

An Indexed Central Limit Theorem

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An Indexed Central Limit Theorem

Steps in the proof

K

  • η, η′

= sup

x∈R

  • P[η ≤ x] − P[η′ ≤ x]
  • Stein’s method (Stein 1986), Chen, Goldstein, Shao (Normal

approximation by Stein’s method, Springer 2011)

Bob Lowen (with B. Berckmoes, J. Van Casteren) University of Antwerp An Indexed Central Limit Theorem

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An Indexed Central Limit Theorem

Steps in the proof

K

  • η, η′

= sup

x∈R

  • P[η ≤ x] − P[η′ ≤ x]
  • Stein’s method (Stein 1986), Chen, Goldstein, Shao (Normal

approximation by Stein’s method, Springer 2011) H: all strictly decreasing functions h : R → R, bounded first and second derivatives and a bounded and piecewise continuous third derivative, lim

x→−∞ h(x) = 1 and lim x→∞ h(x) = 0.

Bob Lowen (with B. Berckmoes, J. Van Casteren) University of Antwerp An Indexed Central Limit Theorem

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An Indexed Central Limit Theorem

Steps in the proof

K

  • η, η′

= sup

x∈R

  • P[η ≤ x] − P[η′ ≤ x]
  • Stein’s method (Stein 1986), Chen, Goldstein, Shao (Normal

approximation by Stein’s method, Springer 2011) H: all strictly decreasing functions h : R → R, bounded first and second derivatives and a bounded and piecewise continuous third derivative, lim

x→−∞ h(x) = 1 and lim x→∞ h(x) = 0.

Step 1 If η is continuously distributed, then the formula lim sup

n→∞ K (η, ηn) = sup h∈H

lim sup

n→∞ |E [h(η) − h(ηn)]|

is valid for any sequence (ηn)n

Bob Lowen (with B. Berckmoes, J. Van Casteren) University of Antwerp An Indexed Central Limit Theorem

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An Indexed Central Limit Theorem

Step 2 Let h : R → R be measurable and bounded. Put fh(x) = ex2/2 x

−∞

(h(t) − E[h(ξ)]) e−t2/2dt

Bob Lowen (with B. Berckmoes, J. Van Casteren) University of Antwerp An Indexed Central Limit Theorem

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An Indexed Central Limit Theorem

Step 2 Let h : R → R be measurable and bounded. Put fh(x) = ex2/2 x

−∞

(h(t) − E[h(ξ)]) e−t2/2dt (Basic points of Stein’s method) (1) For any x ∈ R E [h(ξ)] − h(x) = xfh(x) − f ′

h(x).

(2) Moreover,

  • f ′′

h

  • ∞ ≤ 2
  • h′
  • ∞ ,

(3) If hz = 1]−∞,z] for z ∈ R, then for all x, y ∈ R

  • f ′

hz(x) − f ′ hz(y)

  • ≤ 1.

Bob Lowen (with B. Berckmoes, J. Van Casteren) University of Antwerp An Indexed Central Limit Theorem

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An Indexed Central Limit Theorem

Step 3 Let h ∈ H and put δn,k = fh  

i=k

ξn,i + ξn,k   − fh  

i=k

ξn,i   − ξn,kf ′

h

 

i=k

ξn,i   ǫn,k = f ′

h

 

i=k

ξn,i + ξn,k   − f ′

h

 

i=k

ξn,i   − ξn,kf ′′

h

 

i=k

ξn,i   Then E n

  • k=1

ξn,k

  • fh

n

  • k=1

ξn,k

  • − f ′

h

n

  • k=1

ξn,k

  • Bob Lowen (with B. Berckmoes, J. Van Casteren) University of Antwerp

An Indexed Central Limit Theorem

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An Indexed Central Limit Theorem

Step 3 Let h ∈ H and put δn,k = fh  

i=k

ξn,i + ξn,k   − fh  

i=k

ξn,i   − ξn,kf ′

h

 

i=k

ξn,i   ǫn,k = f ′

h

 

i=k

ξn,i + ξn,k   − f ′

h

 

i=k

ξn,i   − ξn,kf ′′

h

 

i=k

ξn,i   Then E n

  • k=1

ξn,k

  • fh

n

  • k=1

ξn,k

  • − f ′

h

n

  • k=1

ξn,k

  • =

n

  • k=1

E [ξn,kδn,k] −

n

  • k=1

σ2

n,kE [ǫn,k]

Bob Lowen (with B. Berckmoes, J. Van Casteren) University of Antwerp An Indexed Central Limit Theorem

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An Indexed Central Limit Theorem

Step 4 Let f : R → R have a bounded derivative and a bounded and piecewise continuous second derivative. Then for any a, x ∈ R

  • f (a + x) − f (a) − f ′(a)x
  • ≤ min
  • sup

x1,x2∈R

  • f ′(x1) − f ′(x2)
  • |x| , 1

2

  • f ′′
  • ∞ x2
  • Bob Lowen (with B. Berckmoes, J. Van Casteren) University of Antwerp

An Indexed Central Limit Theorem

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An Indexed Central Limit Theorem

Step 4 Let f : R → R have a bounded derivative and a bounded and piecewise continuous second derivative. Then for any a, x ∈ R

  • f (a + x) − f (a) − f ′(a)x
  • ≤ min
  • sup

x1,x2∈R

  • f ′(x1) − f ′(x2)
  • |x| , 1

2

  • f ′′
  • ∞ x2
  • Step 5

Let h ∈ H. Then for all x, y ∈ R

  • f ′

h(x) − f ′ h(y)

  • ≤ 1

Bob Lowen (with B. Berckmoes, J. Van Casteren) University of Antwerp An Indexed Central Limit Theorem

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An Indexed Central Limit Theorem

Step 6 |E[h(ξ) − h(

n

  • k=1

ξn,k)]| ≤ · · · ≤ · · · ≤ 1 2

  • f ′′

h

  • ∞

n

  • k=1

E

  • |ξn,k|3 ; |ξn,k| < ǫ
  • +
  • sup

x1,x2∈R

  • f ′

h(x1) − f ′ h(x2)

  • n
  • k=1

E

  • |ξn,k|2 ; |ξn,k| ≥ ǫ
  • +
  • sup

x1,x2∈R

  • f ′′

h (x1) − f ′′ h (x2)

  • n
  • k=1

σ2

n,kE [|ξn,k|]

≤ · · ·

Bob Lowen (with B. Berckmoes, J. Van Casteren) University of Antwerp An Indexed Central Limit Theorem

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An Indexed Central Limit Theorem

An inequality

Theorem Given an STA (ξn,k)n,k and a normally distributed ξ the inequality lim sup

n→∞ K

  • ξ,

n

  • k=1

ξn,k

  • ≤ Lin ({ξn,k})

is valid.

Bob Lowen (with B. Berckmoes, J. Van Casteren) University of Antwerp An Indexed Central Limit Theorem

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An Indexed Central Limit Theorem

An inequality

Theorem Given an STA (ξn,k)n,k and a normally distributed ξ the inequality lim sup

n→∞ K

  • ξ,

n

  • k=1

ξn,k

  • ≤ Lin ({ξn,k})

is valid.

?

Bob Lowen (with B. Berckmoes, J. Van Casteren) University of Antwerp An Indexed Central Limit Theorem

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An Indexed Central Limit Theorem

Enter approach theory

F: probability distributions Fc: continuous probability distributions ∗: convolution Bergstr¨

  • m’s direct convolution method

ηn → η (weak) ⇔ ∀ζ ∈ Fc : ηn ∗ ζ → η ∗ ζ (uniformly)

Bob Lowen (with B. Berckmoes, J. Van Casteren) University of Antwerp An Indexed Central Limit Theorem

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An Indexed Central Limit Theorem

Enter approach theory

F: probability distributions Fc: continuous probability distributions ∗: convolution Bergstr¨

  • m’s direct convolution method

ηn → η (weak) ⇔ ∀ζ ∈ Fc : ηn ∗ ζ → η ∗ ζ (uniformly) In Top ((F, Tw) → (F, TK) : η → η ∗ ζ)ζ∈Fc

Bob Lowen (with B. Berckmoes, J. Van Casteren) University of Antwerp An Indexed Central Limit Theorem

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An Indexed Central Limit Theorem

Enter approach theory

F: probability distributions Fc: continuous probability distributions ∗: convolution Bergstr¨

  • m’s direct convolution method

ηn → η (weak) ⇔ ∀ζ ∈ Fc : ηn ∗ ζ → η ∗ ζ (uniformly) In Top ((F, Tw) → (F, TK) : η → η ∗ ζ)ζ∈Fc In Met ((F, ?) → (F, K) : η → η ∗ ζ)ζ∈Fc

Bob Lowen (with B. Berckmoes, J. Van Casteren) University of Antwerp An Indexed Central Limit Theorem

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An Indexed Central Limit Theorem

Enter approach theory

F: probability distributions Fc: continuous probability distributions ∗: convolution Bergstr¨

  • m’s direct convolution method

ηn → η (weak) ⇔ ∀ζ ∈ Fc : ηn ∗ ζ → η ∗ ζ (uniformly) In Top ((F, Tw) → (F, TK) : η → η ∗ ζ)ζ∈Fc In App ((F, δw) → (F, K) : η → η ∗ ζ)ζ∈Fc

Bob Lowen (with B. Berckmoes, J. Van Casteren) University of Antwerp An Indexed Central Limit Theorem

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An Indexed Central Limit Theorem

Distance δw δw (η, D) = sup

F0⊂Fcfinite

inf

ψ∈D

sup

ζ∈F0

K (η ∗ ζ, ψ ∗ ζ)

Bob Lowen (with B. Berckmoes, J. Van Casteren) University of Antwerp An Indexed Central Limit Theorem

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An Indexed Central Limit Theorem

Distance δw δw (η, D) = sup

F0⊂Fcfinite

inf

ψ∈D

sup

ζ∈F0

K (η ∗ ζ, ψ ∗ ζ) Limit associated with δw λw(ηn)(η) = sup

ζ∈Fc

lim sup

n→∞ K (η ∗ ζ, ηn ∗ ζ)

Bob Lowen (with B. Berckmoes, J. Van Casteren) University of Antwerp An Indexed Central Limit Theorem

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An Indexed Central Limit Theorem

Distance δw δw (η, D) = sup

F0⊂Fcfinite

inf

ψ∈D

sup

ζ∈F0

K (η ∗ ζ, ψ ∗ ζ) Limit associated with δw λw(ηn)(η) = sup

ζ∈Fc

lim sup

n→∞ K (η ∗ ζ, ηn ∗ ζ)

A sidestep concerning the naturality of δw

Bob Lowen (with B. Berckmoes, J. Van Casteren) University of Antwerp An Indexed Central Limit Theorem

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An Indexed Central Limit Theorem

Distance δw δw (η, D) = sup

F0⊂Fcfinite

inf

ψ∈D

sup

ζ∈F0

K (η ∗ ζ, ψ ∗ ζ) Limit associated with δw λw(ηn)(η) = sup

ζ∈Fc

lim sup

n→∞ K (η ∗ ζ, ηn ∗ ζ)

A sidestep concerning the naturality of δw Tightness: a collection D of probability distributions is said to be tight if for every ε > 0 there exists a constant M > 0 such that for all F ∈ D F(−M) ∨ (1 − F(M)) ≤ ε

Bob Lowen (with B. Berckmoes, J. Van Casteren) University of Antwerp An Indexed Central Limit Theorem

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An Indexed Central Limit Theorem

Prohorov’s theorem D is relatively compact in the weak topology if and only if it is tight.

Bob Lowen (with B. Berckmoes, J. Van Casteren) University of Antwerp An Indexed Central Limit Theorem

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An Indexed Central Limit Theorem

Prohorov’s theorem D is relatively compact in the weak topology if and only if it is tight. Index of compactness in approach spaces: χc(A) := sup

U∈U(A)

inf

x∈A

λU(x)

Bob Lowen (with B. Berckmoes, J. Van Casteren) University of Antwerp An Indexed Central Limit Theorem

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An Indexed Central Limit Theorem

Prohorov’s theorem D is relatively compact in the weak topology if and only if it is tight. Index of compactness in approach spaces: χc(A) := sup

U∈U(A)

inf

x∈A

λU(x) Tychonoff : χc(

  • j

Xj) = sup

j

χc(Xj)

Bob Lowen (with B. Berckmoes, J. Van Casteren) University of Antwerp An Indexed Central Limit Theorem

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An Indexed Central Limit Theorem

Prohorov’s theorem D is relatively compact in the weak topology if and only if it is tight. Index of compactness in approach spaces: χc(A) := sup

U∈U(A)

inf

x∈A

λU(x) Tychonoff : χc(

  • j

Xj) = sup

j

χc(Xj) Kuratowski-Mrowka : χc(X) = 0 ⇔ ∀Z : prZ : X × Z → Z closed

Bob Lowen (with B. Berckmoes, J. Van Casteren) University of Antwerp An Indexed Central Limit Theorem

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An Indexed Central Limit Theorem

Prohorov’s theorem D is relatively compact in the weak topology if and only if it is tight. Index of compactness in approach spaces: χc(A) := sup

U∈U(A)

inf

x∈A

λU(x) Tychonoff : χc(

  • j

Xj) = sup

j

χc(Xj) Kuratowski-Mrowka : χc(X) = 0 ⇔ ∀Z : prZ : X × Z → Z closed (Functional approach to topology, M.M. Clementino, E. Giuli, W. Tholen, 2003, Cambridge University Press)

Bob Lowen (with B. Berckmoes, J. Van Casteren) University of Antwerp An Indexed Central Limit Theorem

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An Indexed Central Limit Theorem

Index of relative compactness: χrc(A) := sup

U∈U(A)

inf

x∈X

λU(x)

Bob Lowen (with B. Berckmoes, J. Van Casteren) University of Antwerp An Indexed Central Limit Theorem

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An Indexed Central Limit Theorem

Index of relative compactness: χrc(A) := sup

U∈U(A)

inf

x∈X

λU(x) Index of tightness: e(D) = inf

M>0

sup

F∈D

F(−M) ∨ (1 − F(M))

Bob Lowen (with B. Berckmoes, J. Van Casteren) University of Antwerp An Indexed Central Limit Theorem

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An Indexed Central Limit Theorem

Index of relative compactness: χrc(A) := sup

U∈U(A)

inf

x∈X

λU(x) Index of tightness: e(D) = inf

M>0

sup

F∈D

F(−M) ∨ (1 − F(M)) Indexed Prohorov theorem In (F, δw) for any D ⊂ F: χrc(D) = e(D)

Bob Lowen (with B. Berckmoes, J. Van Casteren) University of Antwerp An Indexed Central Limit Theorem

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An Indexed Central Limit Theorem

Back to the CLT j(η): the supremum of the discontinuity jumps of the distribution

  • f η

Bob Lowen (with B. Berckmoes, J. Van Casteren) University of Antwerp An Indexed Central Limit Theorem

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An Indexed Central Limit Theorem

Back to the CLT j(η): the supremum of the discontinuity jumps of the distribution

  • f η

λw(ηn)(η) = sup

ζ∈Fc

lim sup

n→∞ K (η ∗ ζ, ηn ∗ ζ)

Bob Lowen (with B. Berckmoes, J. Van Casteren) University of Antwerp An Indexed Central Limit Theorem

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An Indexed Central Limit Theorem

Back to the CLT j(η): the supremum of the discontinuity jumps of the distribution

  • f η

λw(ηn)(η) = sup

ζ∈Fc

lim sup

n→∞ K (η ∗ ζ, ηn ∗ ζ)

Step 7 For η ∈ F and (ηn)n in F λw (ηn)(η) ≤ lim sup

n→∞ K (η, ηn) ≤ λw (ηn)(η) + j(η)

Bob Lowen (with B. Berckmoes, J. Van Casteren) University of Antwerp An Indexed Central Limit Theorem

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An Indexed Central Limit Theorem

Back to the CLT j(η): the supremum of the discontinuity jumps of the distribution

  • f η

λw(ηn)(η) = sup

ζ∈Fc

lim sup

n→∞ K (η ∗ ζ, ηn ∗ ζ)

Step 7 For η ∈ F and (ηn)n in F λw (ηn)(η) ≤ lim sup

n→∞ K (η, ηn) ≤ λw (ηn)(η) + j(η)

Consequence For η ∈ Fc and (ηn)n in F λw (ηn)(η) = lim sup

n→∞ K (η, ηn)

Bob Lowen (with B. Berckmoes, J. Van Casteren) University of Antwerp An Indexed Central Limit Theorem

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An Indexed Central Limit Theorem

The Indexed Lindeberg-Feller CLT

Theorem Given an STA (ξn,k)n,k and a normally distributed ξ the inequality λw(

n

  • k=1

ξn,k)(ξ) ≤ Lin ({ξn,k}) is valid.

Bob Lowen (with B. Berckmoes, J. Van Casteren) University of Antwerp An Indexed Central Limit Theorem

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An Indexed Central Limit Theorem

References

Berckmoes R., Lowen R., van Casteren J. A., Approach theory meets probability theory. Topology and its Applications 158 (2011) 836-852. Berckmoes R., Lowen R., van Casteren J. A., Distances on probability measures and random variables. Journal of Mathematical Analysis and Applications 374:2 (2011) 412-428. Berckmoes R., Lowen R., van Casteren J. A., An isometric study of the central limit theorem via Stein’s method., submitted Lowen R., Index Calculus: Approach Theory at Work 450 pp, Springer Verlag, to appear 2013

Bob Lowen (with B. Berckmoes, J. Van Casteren) University of Antwerp An Indexed Central Limit Theorem