Capital de Trabajo

This section is almost entirely an application of the subindependence of coordi­ n ate slabs proved in the previous section. Our aim is to prove a sort of Central

Lim it Theorem for the tp balls. On the probability space of the normalized Ç

ball, say ÜT, with probability measure the Lebesgue measure in ÜT, we define th e random variables x t—> (x, ^), for each 6 € 5 ”“ ^. We prove th at the average

of their densities is very close to a Gaussian.

More precisely, if we denote by these densities, we prove th a t as n

tends to infinity,

for all t G R .

Here, cr is the rotation invariant probability measure on the sphere 5 ”“ ^ =

The proof is composed of two m ain steps. Firstly, we prove th at

1 n \

2 \ x p j

where the symbol is used here in the usual way:

f n ^ 9n <=> lim — = 1.

9 n

Then, using the subindependence of the coordinate slabs and standard prob­ abilistic arguments, we prove th at “m ost” of the mass of K is inside a shell

where ^ is approximately Applying this to the integral on the right hand

side of the above relation, we get what is required.

6.1

P relim inaries

Before stating our Theorem we make a few remarks about these R.V.s and calculate their densities ge and their variances.

• First of all we notice th a t for each fixed n and p, they all have the same

variance = Ik

Indeed, write 6 = ^lei + . . . + ^nCn, where E R s.th. 6} = 1.

Since K is coordinate symmetric, (cc,ei) (a;,ej) = 0 when i ^ j , and /^ (x ,e i)^ does not depend on i. Therefore, if we put g^ = (cc,ei)^, we have:

/

^ 2 Y ^ e i 6 j { x , e i ) { x , e M

i=l

2

= g

• The sequence {gn}%=i converges to a number say g. (This is the number

th a t appears in the statem ent of the Central Limit Theorem above). Indeed, if we put A for the radius of ÜT, we get:

= J ^ x l = 2 J u^V oln-i{K n {xi =

= 2 r - vF prV ol„ _i{B ^-^)d u

= 2u„_i,p

r

- v ^ f f ^ X d v

Jo

= 2A"+^i;„_i,p J ' ’ - v ^ ) " ^ d v

Using the fact th at Un,p = ^^r(i+n/pj^ , and Stirling’s formula, we get

th a t ^^"vf+2/n converges to a constant Cp, which depends only on p.

Actually, this Cp is bounded in “p” , a piece of information which will be of use later.

But

lim TIP J v^(l — v ^ ) ~ d v = jim J ^ dy

• The densities are the functions: ge{t) = Voln-i (^K fl ^^^^e

J

ge{u)du

=

P { { x ,6) < t)

= J Voln-i

(^K D (^{6) '^

-]r

du

6.2

The basic approxim ation

We start by proving the critical approximation mentioned before, for the inte­ gral /5T1 - 1 ge{t)do-{9) th at interests us. So, we want to prove th at:

We shall prove this for t > 0. A similar argument can be applied for t < 0.

We first recall th a t if u is a unit vector in R ’^, then.

So we have:

f ge{t)da{9) = f \ i m \ v oln{K D {t < (x,9) < t 6})da{9) JS”-~^ JS'^~^6 —*■0o

= 6 Jk

= lim ^ / O ' <

6 - ^ 0 6 J k V \x\ \ he /

We need only notice now th at

n —3

1 f{t+s)/\x\ 1 A \ ^ 1 ( P n '

and that

2j ^ ( i - u r - ^ d u ~ ^

to complete the proof of (6.1).

6.3

T h e m ain T heorem

The above integral over K is an average of Gaussian densities of different

variances. The key idea is th at most of these Gaussians have about the same variance because ^ is typically very close to Qni i.e. th at the set where ^

is “far” from Qn, has small probability. More precisely, it is proved in Lemma

6 .3, th a t for all positive numbers r,

35

X12

n - 9 n -

Both Lemma 6.1, and the Corollary following Lemma 6.2 below, are used in the proof of Lemma 6.3. Our aim in these two statem ents, is to bound from

above, expressions of the form . . . ccf”'* by an appropriate expression

involving just the familiar Jiçx\ where Xi denotes (æ,ei).

This is done in two steps:

Firstly, in Lemma 6.1 we prove a subindependence property for the coor­ dinate R.V.s Xi. i.e. that:

Then in the Corollary to Lemma 6.2, we prove th a t each term of the form

Sk can be bounded from above by a m ultiple of the m -th power of xf.

i.e. th at:

As was mentioned above, Lemma 6.1 uses the subindependence of the com­

plements of the coordinate slabs of the ^ -b a ll, while Lemma 6 . 2 uses standard

results concerning log-concave functions.

L e m m a 6 . 1 With K a normalized ip-ball as above,

Where the mi are positive integers and 1 < I < n.

Proof: We shall give the proof only for 1 = 2. The general case is similar (and in fact we only need the special case). In this case, we have to prove the inequality: f < / x l ^ ^ f Jk Jk Jk Notice that:

/

xl^^xT^ =

Jh ^ 2mi ^ 2m2 _ /Cn{xi>0,X2>0} = / (2m i (2m.2 r v ^ ”'^ -'^ d v ) J K n { x i > 0 , X 2 > 0 } \ J o J \ J o J = 477117712 /

(f

_ 1 ^ 2 m 2 - 1 ^ VKn{xi>0,X2>0} ^ 4771x7712 / , ( / ^l{xi>u,X2>v}

]

J k \ \ J K n { x i >0,X2>0} - - y = 477117712 y* 2 ^P{xi > U , X2> v)dudv

Now, by the subindependence of complements of coordinate slabs,

P{x\ > u ^ X2> v ) < P {x i > u ) P [ x2 > v)

and therefore,

4mim2

J

^ P {xi > u , X2> v)dudv <

< 4 7 7 1 1 7 7 1 2 / ^ 1^27712 > u ) P { x2 > v)dudv

JB.\

= {^j P {xi > u)dv}j {^j 2m2V^^^~^P{x2 > v)dv

Using Fubini again, /q°° 2m i7i^”‘^“ ^P(a;i > u)du = j^n{%i>o} and

277i2n^”^^“ ^F(a;2 > n)du = JKn{x2>o} so the following inequality holds:

/ a;2mi^2m2 < f ^2m, f ^2m2

./fCn{xi>o,x2>o} v/K’n { ii> o } ./jK’n{x2>o}

This clearly suffices for the proof.

The following Lemma, is based on standard results for log-concave func­ tions, whose origins date back to the works of Schur and Ostrowski.

L e m m a 6 . 2 I f f is a decreasing log-concave function, then for any positive

integer m the following relation holds:

f{x)dx'^ x ^ ^ f { x ) d x < ^ --3^ ^ x^f {x) dx^

Proof: We shall use the two following inequalities which are true for all

decreasing log-concave functions, and k positive integer:

and

Uo ^ ^f {x) dx (6.3)

We firstly prove the following inductive relation:

£ x ^ f { x) dx < Jo (6.4)

Indeed,

J

x^f { x) dx =

J

f { x ) kt^ ^dt^ dx

= (j^°° / ( l ) d x ) dt

and in case th a t A; > 2 we can apply (6.4) A: — 2 successive times, to get:

j ( x'‘f { x ) d x < k { k - l ) - - - 3 x ^ f { x) dx

kl ( I o ° ° f ( x ) d x

2

1

m .

Jo

/

X f { x ) d s

So, for k = 2m

U

oo f { x ) d x ]\ m—1 f oo/ x ^ ^ f { x ) d x <

Now using (6.3) for A: = 3 and taking the (m — l)-th power, we get:

U

oo \ 3m—3 / roo \ m—1

f ( x ) d x j < ( / (0)) 3 " - ' ( j [ x ^ f { x ) d x j (6.6)

C o ro lla ry 6 .2 . 1 For all positive integers m ,

L

’ ■ - ‘ ( / « ■ O ' (6.7)

Proof: It is easy to check th at for all positive integers m,

j o ( l - y ^ ) ^ d u

and then the relation we want to prove becomes:

(6 .8) n —1 \ m X / o ' ( l - u » ) ' ^ d « 2 / o ' ( l - « ! > ) ^ d u ] <=> f / \ l - u '’) ' ^ d u ) f u " " ( l - v F ) " ^ d u < < { ^ j \ \ l - v F p ^ d v f "

W hich is tru e by Lemma 6.2, since the function f { u) = ( 1— is decreasing

and log-concave.

L e m m a 6.3 For all positive numbers r,

P

n - Qr > ’■) < r z ? e n

Proof: We first prove th a t |a;|^ is close to namely th at,

( > + ! ) .:

(6.9)

The first inequality is obvious by Cauchy-Schwartz. For the second one, we have:

Using the Corollary of Lemma 6.2, x | < 36 x |) = 36p^. Thus,

/ |x|^ < 36np^ + n (n -

From this we can conclude th at the integral is small, and

therefore th at ^ is close to Indeed,

-

? Aw-!': A1-1'+':

= k j

Jk k r - g i

The last inequality is true by (6.9).

Finally by Chebychev’s inequality we have:

\x\

n - Qr > r \ T = P

\x\ _{2 \ 2} n - Q n > r ^ \ r

Lemm a 6.3 deals comfortably with the possibility th a t |cc| might be too large, b u t for technical reasons we need a stronger estim ate from below. For-

tunately, this can be deduced from Lemma 6.3 using the logarithmic concavity of the function

s Voln{K n B{sy/n))

where B{s^yn) is the Euclidean ball of radius

L e m m a 6.4 F o rr >

-

©

Proof: Let Qn, be such th a t P = 1/2. It is easy to see by Lemma 6.3 th a t |^>n — ^n| ^ 2^ ^ ' Combining this with Lemma 6.3 again, we get th at

when n is large enough,

^

+ i ) - Ï

So, we have a point at which the log-concave function f [ s ) = P < 5)

takes the value 1/2, and a point just a bit further on, where it takes a value close to 1. So, using the log-concavity of the function, we have an estim ate for

its value at points before Qn. Indeed, take 5 < ^ , and A = • Then,

= A (^n + + (1 -

and thus by log-concavity

f i ê n ) > f ( ê „ + • f ~ \ s )

But since /(p n ) = 1/2 and / (^Qn + > 3/4 the above relation implies th at ■ 2N i/(i-A)

and hence that

Now we need only notice th at for s < g^ — < Qn,

^ V v ^ - - (3 ) - (3 )

and then put r = Qn — s io get the required result.

T h e o re m 6 . 1 I f ge is the density of the marginal in direction 9, of the ip-hall,

then

fo r each t, uniformly in p.

as n 00

Proof: By (6.1), it is sufficient to prove th at

/.

1 ^yn ( P n \ 1 /

2' w j ^

for each t, uniformly in p. For this, we shall divide the set K into two

subsets and use appropriate techniques in each one. These are: K i = K C\

{ |;|^ ~ ^^1 ^ K2 = K n { | ^ - Qn\ > ^ } .

We shall show th a t the mass of K is concentrated in the first set, where the integrated function is pretty smooth. By applying a Lipschitz estim ate, we shall see th at the limit of the integral there, is the required Gaussian.

Although the integrand is not particularly well-behaved on K2, the measure

Our aim is to prove the following two statem ents: 1 y/n y/2^ \x\ 1 y/n k l lim lim

/.

/.

exp exp t n T j ï j2 n T b P Qy/ÔÆexp = 0 (6 . 11) (6 . 12)

For the first one, as was already mentioned, we shall use the fact th a t the integrand is smooth in this set.

Let F{y) = exp ( —^ ) - The derivative of F , is bounded by

Therefore, if t is large, say t > this gives an upper bound for the

derivative of order When t is small, the function F can have large

derivative, but only where y is small. More precisely, for y > 2t, the derivative is decreasing, so, one can check th a t in case th at t < ^ , the

derivative in a range near Qn, has again a bound of order Thus, for

y > gn — we have a Lipschitz property:

l^(y ) - < c \ y - gn\

where c is of order g~^. This applied in the integrals, gives:

r

JKi \ y / ^ < - L <-

L

1 y/n f F n d - 2 1^ exp < 1 y/n y / ^ cc < y / ^ log 72, exp - Qi t n T lx P _{Q n \ / ^} exp — ^ 0 y / n

Since Qn — ^ Q and Vol {Ki) — ^ 1 (by Lemma 6.3), we have th a t

which completes the proof of (6.11).

To prove (6.12), we notice th a t it is enough to prove:

lim

f

—j- = 0 (6.13)

ti-*ooJk^ |œ|

We divide K2 into three subsets. The first one has a small radius. This is

-K’2 , 1 = where I is chosen such th at Vnl^\/n = 1, where Vn

is the volume of the unit Euclidean ball. So, I is like a constant tim es •\/n.

The other two subsets of K2 are: J^2 , 2 = - K ' h | ^ < ^ ^ Qn — and

For the first one we have:

lim / ^ < lim / ^

= lim Vnn^/ri

f

/

— vT'~^du da(6)

n-^oo Jo u

= lim Vnl^~^ y /n r

n—foo n — 1

= 0

For the second one, we shall use Lemma 6.4. We have:

lim / ^ < Urn ^ V o I „ ( K 2,2)

*

( I ) -

= 0

Finally, for the last one, we shall use Lemma 6.3:

/ v _ 1

1 35pl

< lim

gn + log n/n^/® n^/3 log'^ n

= 0

(The choice of log n in the above argument is not crucial: we need a function

of n which tends to infinity, to handle the case of K2,2, but more slowly than

to handle the case of Ki . )

R em a rk

1. In the statem ent of Theorem 6.1 we write th at the convergence is uni­ formly in “p” . As the proof stands, this would be clear, if all the p„’s where uniformly bounded.

B ut this is not difficult to see: From the discussion in the beginning of

this Chapter, we mentioned th at there is a constant Cp depending only

on p, such th a t as n — > co.

As it can be observed by Stirling’s formula, these Cp’s are bounded in p. And since the integral is also bounded in p, we have what we want.

A ppendix

Using the notation introduced in Chapter 3, we prove in the Lemma below,

th a t the function 1 — is very much like the function ( 1 — , a property

used in the proof of Theorem 3.1.

Its proof uses standard inequalities for log-concave functions.

L e m m a A . l For p > 1 and 0 < t < 1,

Proof:

*1 —1

We shall use the form: 1 —

l o ^ n —

For the first inequality, we have:

yP - t P \ ^ du

Now substituting we get:

dx dx < ( 1 - «")V + i - x*>)V-,

< (1 -

J o

W hich completes the proof of the first inequality.

For the second one, we shall use the following inequality which holds for a decreasing log-concave function / , and for p > 1:

x^~'^f{x)dx < T{p + 1) f { x) dx^ (A.l)

n —1

Applying this for f { x ) = ( 1 — x^) p , we get that:

T{p + l ) ( j \ l - v F ) ' ^ d v j > £ p u ^ - \ l - u ”) '^ d u > (1 - f \ l - x f f ^ d x Jt P = ( 1 - tp) — I— ( 1 _ ^ ^ 7 1 - 1 + P = ( 1 - fP)” ^ 71 — 1 4- p But then, i/p i - i / p £ { l - y P f i ^ d u >

(l-fP)"/^f

J

(r(p + i))-

On the other hand,

1 / 1 n - l

/ ’\ l - u ”)V -d u = i . B ( l , l + —

Jo P \ P P

_

r ( i + i ) r ( i + ^ )

Applying Lemma A.2 with z = ^ and a = ^ we get

r ( i + f)

\ p J

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