ALMOST NONE OF THE

Internal. J. Math. & Math. Scl.

VOL. 13 NO. 4 (1990) 775-778

ALMOST NONE OF THE

^SEQUENCES

OF O’s AND

l’s

ARE ALMOST CONVERGENT

JEFFCONNOR

Dcpartnmnt

ofMathematics

OhioUniversity Athens, Ohio45701

(Received July 21, 1989 and in revised form February 5, 1990)

775

Abstract. Weestablishthat, in thesenseoftileLawofLargeNumbers, almostnoneofthe sequences ofO’sand l’sareassignedthesamevalue by everyBanachlimit.

KEYWORDS AND PHRASES.Banach limit, Mmostconvergence 1980

AMS SUBJECT CLASSIFICATION CODE.

40G99

The result established in this noteis precisely the result promised in the title. To place the theoremin perspective, however, it will behelpful to recall afew definitions andafundaanental resultof probabilitytheory.

First werecall anextremely useful extension of the usualnotion of convergence.

A

sequence x

(x,,)

is said to be Cesaro summable to s provided lira,,n -a

.= ^x

^{s. If x is Cesaro}

summable tos,wewriteC-lira^a’=

Banachlimitsprovide the first step in developing another extension of the usual definitionof convergence.

DEFINITION. A

realvaluedfunctionfdefinedon thebounded realnumbersequencesisaBanach limitprovided

(1) f(ax +

^by)

a.f(x) +

^by(y),

(2) f(x) >_

Oif x,

>_

0 n=1,2,3

(3) f(x) f(Tx)

whereT(.z.,,x,.,.a )=(.,’,xa

(4) f(e)

^{1 wheree} (1,1,...

forallboundedrealsequencesx

(xn),

y (y,) ad real numbers a, b.

The existence of Banach limitscan be establishedbyacorollaryoftheHahn-Banach theorem

[1]. G.G. Lorentz

usedthesefunctionals togivemeaning tothe phrase "almost convergent to

s."

DEFINITION.

A

bounded rea/sequencex isalmost convergent to sprovided

f(x)

^s forevery Banachlimit

f

The notions of Cesarosummability andahnost convergence both extend the usual concept of convergenceinanon-trivial fashion. Straightforward applicationsof thedefinitions yieldthat

C-

limx limx

f(x)

^{for every}convergent sequence x and everyBanach limit

f.

^{It can alsobe}

F76 JEFF CONNOR

readilyestablished(front thedefinitions) ^th;,t the s.quence0,1,0, 1,... isboth Cesarosummable andalmost convergent^to

1/2.

Lorentz

also characterized the almost ^c,mvergent sequcnccs as being the ’uniformly’

Cesaro

summablesequences.

TIIEOREM

[4].

^A bounded realsequencex=(.r,,)isahnost convergent tosifand onlyif

k

lira^k k

- ^E

^{t’-I a’,,+, s}

uniformlywithrespect ^ton.

An

elegant proofofLorentz’sthcoremwhich alsoyicldsthe existenceof Banachlimits isgiven by

G.

Bennett andN. Kaltonin

[2].

^{Observethat ifa} sequence isalmost convergent tos thenit must also beCesarosummable tos.

Wenowestablish theframeworkforcomputing the promisedprobability.

We let ^f/

{0,1} v,

denote the r -field of subsetsgenerated by the coordinate projections and Pdenote the natural ’fair coin’probabilitymeasuredefinedonZ.

Nowlet

(X,)

be the sequence of

{0,1}-w,

dued random variablesdefinedon ftby

X,,(w)

w,,;

(X,,)

^{is a}sequence of independent identically distributed randomvariables, eachwith expected value

1/2.

^Observethat ifweset

S,, = ^Xt,

^theLawof

^Large

Numbers yeilds that

P[w

^{gt" li,n}

S,,(w)ln 1/2]

¹

orequivalently

P[., e

^{ft C-hm,o,,}

/2]

Thisearlyversionofthe lawoflargenttmberswasknown toEmileBorel

[3]. In

moreconventional language^wehave establishedthat alrnost allofthe sequencesofO’sand l’sareCesarosummable to

1/2.

Borel’s

Law

of

Large

Numbers indicates that the

Cesaro

method is, in the senseofmeasure, extremelyeffective onft. Wenowshow that themethod ofalmost convergenceis notnearly^as effective.

TIIEOREM. Almostnoneof thesequences ofO’sand ’sare

Mmost

convergent.

PROOF" Borel’s theorem togetherwith Lorentz’scriteriontells usthat almost all of thew’sinfl thatarealmost convergent arealmost convergent to

1/2.

Lorentz’s

criterionalso tellsusthat ifwq f satisfiesthe condition thatfor eachk

>_

2 thereis annsuchthat

.X’.+,

(,o)

+... + ^X.+(w)

^k,

thenwisnot almostconvergent to

1/2.

Alternatively, ifw (/ flis almostconvergent to

1/2,

^the

thereisak

>_

2 forwhich,regardlessofn, ^wehave

X,t.+() +-.. + X,+,(w) <

^k

ALHOST NONE OF THE SEQUENCES OF Ors AND lfs 777 With this inmind,let k

>

2 and

Ak N ^[":

^{E i?"X.k+(w)}

^{+... +} X..+.(w) < k].

Nowobservethat theindependenceofthe sequence

(X.)

implies that of thesequence (N,,t+t +."

+ X,,+),,>"

correspondingly, given

2,[ a. x,,,.+,() +... + x,+,() < l

n>l

II ^{p[ e} .

x,,.+,(,)+...

+ x,,+() < 1

=(1 2

-

⁾

sinceeach event

[w

^{E ft"}

X,,,+(w) + +

X,,t.+t.(w)

< k]

has probability1 2

-t’.

Since

A, c (’l ^{[ e n.} x,,,.+,() +... + x,+() < 1

n>l

it follows that

P(At.) <_

(1

2-’p

forall j,i.,;.,P(At,) 0, andso

P(l.Jt> A)

0.

Nowset

F

fl- Ok>2A, and notethat P(F} 1.

By

construction, ifw

F

then for cach k

>

2 thereisannsuch that

X,,k+(w)

+-.-+ X,,+(w)

k, orequivalently,

(w,,,+t +... + ^w,,,+,)/k

^1.

Thisshows usthatwisnotalmost com,ergent to

1/2.

^Since

F

C

{w

f’w^isnot a.lmost convergent ^to

1/2},

wehave establishedthat

P[w

^{gt wisalmost}

convergent]

0. 1 REFERENCES

1.S. Banach, "Theorie des operations limaires," MonografieMat.,PWN,Warsaw,1932, pp. 29-33.

2.G. BennettandN.J. Kalton, Conststency Theoremsfor^AlmostConvergence, Trans.Amer. Math. Soc.198 (1974),^23-43.

3.E. Borel, Sur^/esprobabthltes denombrablesetleursapphcationsarthmetques,Rend.Circ.Mat.Palermo

(1909),247-271.

4.G.G.Lorentz,A Contrtbutwn to the Theoryof^DivergentSequences, Acta Math80(1945),^167-190.