In this paper, we obtain su ient onditions for the strong onsisten y of the least squares estimator βe of β under additional assumptionsonthenon-randomsequen ex1, x2

SUFFICIENT CONDITIONS FOR THE STRONG

CONSISTENCY OF LEAST SQUARES

ESTIMATOR WITH α^{-STABLE ERRORS}

João Tiago Mexia

Mathemati sDepartment, Fa ultyof S ien eandTe hnology

New Universityof Lisbon

Monte da Capari a2829516 Capari a, Portugal

e-mail: par rf t.unl.pt

and

JoAo˜ Lita da Silva

NewUniversity ofLisbon,Mathemati s Department

Fa ultyofS ien eandTe hnology

Quinta da Torre, 2825114 Monteda Capari a, Portugal

e-mail: jsf t.unl.pt

Abstra t

Let Yi = x^Ti β+ e^i, 1 6 i 6 n^, n >1 ^{bealinearregressionmodel}

andsupposethattherandomerrorse₁, e₂, . . .^areindependentandα^-

stable. In this paper, we obtain su ient onditions for the strong

onsisten y of the least squares estimator βe ^of β ^{under additional}

assumptionsonthenon-randomsequen ex₁, x₂, . . .^{ofrealve tors.}

Keywords: linearmodels,leastsquaresestimator,strong onsisten y,

stability.

2000Mathemati sSubje t Classi ation: 60F15.

1. Introdu tion

Thestrong onsisten yoftheLeastSquaresEstimator(LSE)hasbeenstud-

iedonthe last yearsbymanyauthors (seefor instante[6, 10,11,12,13,14℄

or [15 ℄). In the papers [10℄ and [11℄ are obtained ne essary and su ient

onditions for the LSE strong onsisten y. Nevertheless, the formulation

used by the authors of the papers quoted above assumes an i.i.d. random

errorsequen e withnullexpe tedvalueandnite absolutemoment oforder

1 6 r < 2^{. In a rst stage, our purpose will be to derive LSE strong on-}

sisten y from a formulation that is, insome sense, similar to the des ribed

previously. As a matter of fa t, we will assume that the errors e₁, e₂, . . .

arei.i.d. α^{-stableex luding any hypothesis ontheir absolute moments. Let}

us point out thatthe stability onditionfor theerrorsis introdu ed here to

makeusefulthesuitable'linearity' propertiesofthestabledistributions. On

a se ond stage, we generalized the above assumptions supposing that the

random variables e₁, e₂, . . .^areonly independent andα^{-stable, obtainingan}

extension of theresultspresentedin[10 ℄ and[11 ℄.

Consider the linearmodel,

(1.1) Y_i = x^T_i β+ e_i, 1 6 i 6 n, n >1,

where x₁, x₂, . . . ^{areknown non-random real} κ^{-ve tors,}

β:=



 β₁

.

.

.

β_κ





istheve tor of(unknown)parameters ande_i ^{istherandomerror inthe}i^-th

observation (i = 1, . . . , n)^.

Setting

S_n:= x1x^T

1 + . . . + xnx^T

n

wewillassumethatS⁻¹

n ^existsforn^{largeenough(forsmall}n^{su hthat}S⁻¹

n

doesnot exist,S⁻¹

n ^anbedened arbitrarily). DenotingtheLSEofβ ^byβe

we an write

(1.2) βe = β + S⁻¹_n

Xn

i=1

x_ie_i

sothat, βe ^{isstrongly onsistent ifand only if}S⁻¹_n Xn

i=1

x_ie_i −→ 0^{a.s. .}

2. Auxiliary tools

Thefollowing notational onventions willbe adoptedthroughout thepaper.

Given a ve tor x∈^{Rκ,the eu lidean ve tor norm will bedenoted by}

||x|| :=√ x^Tx

and the matrix normof anmatrix A∈ Mκ×κ(^R) ^{willbeindi ated by}

|||A||| := sup

x6=0

||Ax||

||x|| .

The tra e of A ∈ Mκ×κ(^R) ^{will be denoted by tr}(A)^{. For any matrix} A∈ M^κ×κ(^R)^{withrealeigenvalueswewillindi ateby}ν_max(A)^andν_min(A)

the maximum and minimum eigenvalue of A respe tively. Given two sym- metri matri esA, B∈ Mκ×κ(^R) ^{we write}A > B ^{to denotethat} A− B^is

nonnegative denite;wewillusethenotationA> B^{toindi atethat}A− B

ispositive denite.

Itiswell-knownthatarandomvariableZ ^(oritsdistribution) issaid to have a stable distribution if for any positive numbers c₁ ^and c₂^{, there is a}

positive numbera ^{anda realnumber} b^{su h that}

(2.1) c₁Z₁+ c₂Z₂ = a Z + b^d

where Z₁ ^and Z₂ ^are independent opies of Z^{.1 A random variable} Z ^is

alledstri tly stable if (2.1) holdswithb= 0 ^{(see [16℄).}

Equivalently (see [16 ℄), a random variable Z ^{is said to have a stable}

distribution ifthere are parameters 0 < α 6 2^, σ >0^, −1 6 λ 6 1 ^and µ

real su hthat its hara teristi fun tion hasthefollowing form,

(2.2)

E e^iZt

=



















exp − σ^α |t|^α



1 − iλ(^signt) tanπα 2

+ iµt

!

if α6= 1

exp − σ |t|



1 + iλ2

π (^sign t) log |t|

 + iµt

!

if α= 1 ,

where

signt:=









1 ^if t >0 0 ^if t= 0

−1 ^if t <0 .

Sin e(2.2) is hara terized byfour (unique) parameters

α∈]0, 2], σ >0, λ∈ [−1, 1], µ∈^R

respe tively,theindexofstability,thes aleparameter,theskewnessparame-

ter andtheshift parameter wewilldenotestabledistributionsbyS_α(σ, λ, µ)

and write

Z ∼ Sα(σ, λ, µ)

to indi ate that Z ^{has the stable} distribution S_α(σ, λ, µ)^{. A stable random}

variableZ ^{withindex ofstability} α ^{is alled}α^-stable.

Let us remark that a random variable Z on entrated at one point is always stable, i.e., a onstant µ ^{has degenerate} distribution S_α(0, 0, µ) ^for

1

ThenotationX = Y^{d meansthattherandomvariables}XandY agreeindistribution, thatis,F_X(x) = FY(x)^,x ∈R.

any 0 < α 6 2^{. This degenerate ase is of no spe ial interest and, unless}

stated expli itly, we always assume that Z ^is non-degenerate. In fa t, de- generate distributions have unusualproperties, for example, all moments of

a degenerate distribution arenite, whereas a non-degenerate stable distri-

bution with0 < α < 2 ^{hasinnite se ondorder moments.}

The distribution fun tions of α^{-stable random variables are absolutely}

ontinuous and their densitieshave derivatives of anyorder in every point,

but ex luding some spe ial ases, their representations an be expressed

onlythrough ompli atedspe ialfun tions. However,thereexistasymptoti

expansionsofα^{-stabledensitiesina}neighborhoodoftheoriginor ofinnity (see [9,17 ℄or [18℄). Themain spe ial aseis given by

S₂(σ, λ, µ) = N (µ, 2σ²) (σ > 0).

In this situation the skewness parameter λ ^{is totally irrelevant. Hen e, a}

random variable

Z ∼ S2(σ, λ, µ) (σ > 0)

hasprobabilitydensityfun tion

f_Z(z) = 1

2σ√πe^{−(z−µ)24σ2} , z∈^R.

The remaining spe ial ases of probability densities of α^{-stable random}

variables in losed form are: the Cau hy distribution S₁(σ, 0, µ) ^whose

densityis

f(x) = σ

π (x − µ)²− σ²
, x∈^R

and the LévydistributionS_1/2(σ, 1, µ) ^{whose densityis}

f(x) =

r σ

2π(x − µ)³e^{−2(x−µ)σ} , x > µ.

Somebasi properties ofstabledistributions are listedbelow: (see [16℄)

(1) Let Z₁ ^and Z₂ ^be independent random variables with Z_i ∼ S_α(σi, λ_i, µ_i)^,i= 1, 2^{. Then}

Z₁+ Z2∼ S^α



(σ^α₁ + σ^α₂)^1/α,λ₁σ^α₁ + λ2σ^α₂

σ^α₁ + σ^α₂ , µ₁+ µ₂

 .

(2) Let Z ∼ Sα(σ, λ, µ) ^{and let} a ^{be a real onstant. Then} Z + a ∼ S_α(σ, λ, µ + a)^.

(3) LetZ ∼ Sα(σ, λ, µ)^{and let}a^{bea non-zero real onstant. Then} aZ∼ Sα(|a| σ,^sign (a)λ, aµ) ^if α6= 1 aZ∼ S1



|a| σ,^sign(a)λ, aµ − 2

πσλalog |a|



if α= 1.

(4) LetZ ∼ Sα(σ, λ, µ) ^withα6= 1^{. Then}Z ^{is stri tly stableifand only}

ifµ= 0^.

(5) Z ∼ S1(σ, λ, µ)^{isstri tly stableifand onlyif}λ= 0^.

(6) Z ∼ Sα(σ, λ, µ)^{issymmetri about} µ^{ifand onlyif}λ= 0^.

(7) LetZ ^have distribution S_α(σ, λ, 0) ^withα <2^{. Thenthere exist two}

i.i.d. randomvariablesA ^andB ^{with ommon} distributionS_α(σ, 1, 0)

su hthat

Z =^d

1 + λ 2

1/α

A−

1 − λ 2

1/α

B ^if α6= 1

Z =^d

1 + λ 2

 A−

1 − λ 2

 B+

+σ

1 + λ π

 log

1 + λ 2



−

1 − λ π

 log

1 − λ 2



if α = 1.

The next statement is a powerful tool in matrix analysis and generalizes a

result whi h is a sine qua non for probability theory: the s alar version of

the Krone kerlemma.

Lemma 2.1 Krone ker's lemma (matrix ase). Let x_i, i = 1, 2, . . . ^{be a}

sequen e ofrealκ^{-ve tors,}A_i asequen e ofreal symmetri nonsingularκ×κ

matri es with A_i+1 > A_i > O ^and P_i a sequen e of nonsingular κ × κ

matri es. Suppose that r_i=P∞ j=iA⁻¹

j x_j exists (and isnite). Then:

(i) If A_∞:= lim

n→∞

A_n existsandis nite, then lim

n→∞

A⁻¹

n

Xn

i=1

x_i existsand

isnite.

(ii) If lim

n→∞

1

tr(An) = 0 ^then lim

n→∞

1

tr (An) Xn

i=1

x_i= 0^.

(iii) If lim

n→∞

A⁻¹

n = O^, lim

n→∞

P_nr_n= 0^and

lim sup

n→∞

Xn

i=1

A⁻¹_n A_i− Ai−1

P⁻¹

i

 < ∞ ^then lim

n→∞A⁻¹

n

Xn

i=1

x_i = 0^.

P roof. ^{Theproof an be founded in[1 ℄on page 2.7.}

Remark 1. The ondition lim sup_n→∞P_n

i=1

A⁻¹_n A_i− Ai−1

P⁻¹

i

  < ∞

isimplied bythemu h stronger ondition

lim sup

n→∞

ν_max(A_n) ν_min(A_n) <∞.

For some remarkson amatrix versionof Krone ker's lemma, see[2℄.

To nish this se tion, we present an important result that gives us the

almost sure onvergen e for sums of independent random variables X_i ∼ S_α(σ_i, λ_i, µ_i)^.

Proposition 2.1. Let {Xi, i > 1} ^be independent random variables X_i ∼ S_α(σ_i, λ_i, µ_i)^,0 < α 6 2^{. Then,}

X∞

i=1

X_i ^{onverge a.s.} ⇐⇒

X∞

i=1

σ^α_i <∞ ^and X∞

i=1

µ_i ^onverge.

P roof. ^{We establish the proof only for} 0 < α < 2 ^{sin e the ase} α = 2

is obvious in view of the form of hara teristi fun tions of normal law.

We show rst that if

P∞

i=1σ^α_i <∞ ^and P∞

i=1µ_i ^{onverge then} P∞ i=1X_i

onverges almostsurely. Setting,

a_n:= (σ^α₁ + . . . + σ^α_n)^1/α, b_n:= λ₁σ^α₁ + . . . + λ_nσ^α_n σ^α₁ + . . . + σ^α_n

and c_n:= µ₁+ . . . + µ_n

we get, forall n∈^N,

Xn

i=1

X_i ∼ Sα(a_n, b_n, c_n)

provided thatX_i∼ Sα(σ_i, λ_i, µ_i)^.

Therefore,we an write

(2.3)

Xn

i=1

X_i= a^d _nY_n+ c_n ^if α6= 1

and

(2.4)

Xn

i=1

X_i= a^d _nY_n+ 2

π b_na_n log a_n+ c_n ^if α= 1

where {Yn, n > 1} ^{is a sequen e of random variables satisfying} Y_n∼ Sα(1, b_n,0)^{. Hen e, there exist two i.i.d. random variables} A_n ^and B_n ^{with ommon} distribution S_α(1, 1, 0) ^{su h that,}

(2.5) Y_n=^d

1 + bn

2

1/α

A_n−

1 − bⁿ 2

1/α

B_n ^if α6= 1

and

Y_n=^d

1 + b_n 2

 A_n−

1 − bn

2



B_n+1 + b_n π log

1 + b_n 2



−

−1 − bn

π log

1 − bn

2



if α= 1.

(2.6)

Sin e

P∞

i=1σ^α_i <∞ ^and P∞

i=1µ_i ^{is onvergent we obtain the}

onvergen e inlawof Y_n ^{sin e} A_n ^and B_n ^haveboth distribution S_α(1, 1, 0)

for all n ∈ ^{N (let us note that} b_n ^{is a onvergent sequen e). A ording}

to (2.3) and (2.4) , the series

P_n

i=1X_i ^{onverges in law whi h implies}

the existen e of a (proper) random variable S ^{su h that} Pn

i=1X_i −→ S^a.s.

(see [4℄).

Wenowshowthat,if

P∞

i=1X_i ^{onvergesalmostsurelythen}P∞

i=1σ^α_i <∞

and

P∞

i=1µ_i ^{onverge. Sin e}

|bn| =  

λ₁σ^α₁ + . . . + λ_nσ^α_n σ^α₁ + . . . + σ^α_n

61, ∀n ∈^N

thereexistasubsequen e b_ηn

onvergingtoapointof[−1, 1]^{whi himplies}

the onvergein lawofthe following random sequen es:

1 + b_ηn 2

1/α

A_ηn−

1 − bη_n

2

1/α

B_ηn ^if α6= 1

and

1 + b_ηn 2

 A_ηn−

1 − bη_n

2



B_ηn+1 + b_ηn π log

1 + b_ηn 2



−

−1 − b^ηn

π log

1 − b^ηn

2



if α= 1 .

Therefore,thealmost sure onvergen e of

P∞

i=1X_i ^{leads to}

n→∞lim

η_n

X

i=1

σ^α_i <∞

(see Theorem14.2 of[3℄ inpage 193)sothat,

X∞

i=1

σ^α_i <∞

sin e this isa seriesof non-negative terms. Thus,

λ₁σ^α₁ + . . . + λ_nσ^α_n+ . . .

σ^α₁ + . . . + σ^α_n+ . . . ^onverge,

X∞

i=1

µ_i ^onverge.

3. The strong onsisten y of LSE

Fromthesixties,thestrong onsisten yofLSEhasattra tedmu hattention

frommanystatisti ians. Inthemiddleandlateofseventiestheproblemwas

satisfa torily solved for the ase where the random errors possesses a nite

varian e. In fa t, a paper by Lai, Robbins and Wei (1979) ^{showed that if}

the e₁, e₂, . . . ^{arei.i.d. with}E e1

= 0^and 0 < E e²₁

<∞ ^{thena su ient}

onditionfor the strong onsisten y ofβe ^is,

S⁻¹_n = Xn

i=1

x_ix^T_i

!−1

−→ O

as n → ∞^{. Hen e, the parti ular ase of the linear regression model (1.1)}

givenbyi.i.d. stri tlystableerrorswithindexofstabilityα= 2^{is ompletely}

under overbytheLai,Robbins &Wei'sresult.

Theorem 3.1. Consider the linear regression model (1.1) . If

(a) the random variables e₁, e₂, . . . ^{are i.i.d.} S₂(σ, λ, 0),

(b) lim

n→∞S⁻¹_n = O,

then βe ^{is strongly} onsistent.

Remark 2. Letusnotethatthene essityofthe onditionlim_n→∞S⁻¹_n = O

on the LSE strong onsisten y it had been proved earlier by H. Drygas

(1976) ^onthesameassumptions fortheerror sequen e: e₁, e₂, . . . ^{i.i.d. with} E e1

= 0 ^and 0 < E e²₁

<∞^.

Moregenerally,

Theorem 3.2. Consider the linear regression model (1.1) . If

(a) the random variables e₁, e₂, . . . ^{are i.i.d.} S₂(σ, λ, µ) ^withµ6= 0,

(b) lim

n→∞S⁻¹_n = O^, lim sup

n→∞

Xn

i=1

S⁻¹_n x_ix^T_i  

 < ∞^and      

X∞

i=1

S⁻¹

i x_i      <∞,

then βe ^{is strongly} onsistent.

P roof. ^{From (1.2)weget}

βe = β + S⁻¹_n Xn

i=1

x_i(ei− µ) + µS⁻¹n

Xn

i=1

x_i.

Sin e e_i − µ ∼ S2(σ, λ, 0) ^{the thesis is a} onsequen e of Theorem 3.1 and Krone ker's lemma (matrix ase).

The next theorem an be interpreted as an alternative path to the results

presented in[10℄ and [11℄. Indeed, the authors of thepapers quoted above

supposean i.i.d. error random sequen ewhere ea h term hasnull expe ted

value and nite absolute moment of order 1 6 r < 2^{. Our results will}

establishes thestrong onsisten y of the LSE assuming only that the error

random sequen eisi.i.d. α^-stable(α < 2).

Theorem 3.3. Let 0 < α < 2^, α 6= 1 ^{and onsider the linear regression}

(a) the random variables e₁, e₂, . . . ^{are i.i.d.} S_α(σ, λ, µ),

(b) lim

n→∞S⁻¹_n = O^, lim sup

n→∞

Xn

i=1

S⁻¹_n x_ix^T_i  

<∞ ^and X∞

i=1

 S⁻¹_i x_i

^{min (α,1)}<∞,

then βe ^{is strongly} onsistent.

P roof. ^{Theidentity (1.2)} guarantees,

βe = β + S⁻¹_n Xn

i=1

x_ie_i = β + S⁻¹_n Xn

i=1

x_ie^∗_i + µ S⁻¹_n Xn

i=1

x_i

withe^∗_i ∼ Sα(σ, λ, 0)^{. Ontheonehand,by}Krone ker's lemma(matrix ase) we have µ S⁻¹_n Pn

i=1x_i −→ 0 ^as n → ∞ ^{provided that}  

P^∞_i=1S⁻¹

i x_i 

 < ∞

and lim sup_n→∞Pn i=1

S⁻¹_n x_ix^T

i

 < ∞^{. On the other hand, ea h ompo-}

nent ofthe ve tor

Pn i=1S⁻¹

i x_ie^∗_i ^hasdistribution

S_α







 σ

Xn

i=1

|aij|^α

!1/α

, λ

Xn

i=1

sign(a_ij) |aij|^α Xn

i=1

|a^ij|^α

,0









, j= 1, . . . , κ

where a_ij ^isthej^{-th omponent of theve tor} a_i := S⁻¹_i x_i. Sin e

X∞

i=1

|a^ij|^α 6 X∞

i=1

 S⁻¹_i x_i

 ^α

we obtain the almost sure onvergen e of

P∞ i=1S⁻¹

i x_ie^∗_i ^{through the}

Proposition2.1and Krone ker's lemma (matrix ase) permitus to on lude

S⁻¹

n

P_n

i=1x_ie^∗_i −→ 0^{a.s. .}

On the next result we will use the notation |x|min := min |x1| , . . . , |xκ|

where xis annon-random κ^{-ve tor.}

Theorem 3.4. Consider the linear regression model (1.1) . If

(a) the random variables e₁, e₂, . . . ^{are i.i.d.} S₁(σ, λ, µ),

(b) lim

n→∞S⁻¹_n = O^, lim sup

n→∞

Xn

i=1

S⁻¹_n x_ix^T_i 

 < ∞^and

X∞

i=1

 S⁻¹_i x_i

  log

S⁻¹_i x_i

min ^onverges,

then βe ^{is strongly} onsistent.

P roof. ^{Therelation (1.2) givesus}

βe = β + S⁻¹_n Xn

i=1

x_ie_i = β + S⁻¹_n Xn

i=1

x_ie^∗_i + µ S⁻¹_n Xn

i=1

x_i

with e^∗_i ∼ S1(σ, λ, 0)^{. Ea h omponent of the ve tor} Pn i=1S⁻¹

i x_ie^∗_i ^has

distribution

S₁







 σ

Xn

i=1

|aij| , λ

Xn

i=1

sign (a_ij) |aij| Xn

i=1

|aij|

,−2σλ π

Xn

i=1

a_ijlog |aij|









, j= 1, . . . , κ

where a_ij ^isthej^{-th omponent of} a_i := S⁻¹_i x_i. Therefore,we have

S⁻¹

n

Xn

i=1

x_ie^∗_i −→ 0^a.s.

whi h is a onsequen e of the Proposition 2.1 and the Krone ker's lemma

(matrix ase) provided that lim_i→∞S⁻¹

i x_i = 0 ^{(note that our} assumptions guarantees

S⁻¹_i x_ix^T

i

 < ∞ ^as i→ ∞^).

Sin e

X∞

i=1

S⁻¹

i x_i     

<∞ and lim sup

n→∞

Xn

i=1

S⁻¹_n x_ix^T_i  

  < ∞

we get µ S⁻¹_n Xn

i=1

x_i −→ 0 ^as n→ ∞ ^{and thethesisis} established.

Remark 3. If the random errors e₁, e₂, . . . ^{are i.i.d. stri tly stable with}

index of stability 0 < α < 2 ^{then the strong} onsisten y of the LSE is still valid assuming lim_n→∞S⁻¹_n = O^, lim sup_n→∞P_n

i=1

S⁻¹_n x_ix^T

i

 < ∞

and

P∞ i=1

 S⁻¹_i x_i

^α<∞^.

The previous results an be extended to the ase where therandom varia-

bles e₁, e₂, . . . ^{are (only)} independent and e_i ∼ Sα(σ_i, λ_i, µ_i)^{. In this sense,}

we nish this work with a possible generalization where the assumption of

identi aldistributionforrandomerrorse₁, e₂, . . .^{is ompletelyex luded. For}

abetterpresentationofourideasandtheirasso iatedresultsletusstartwith

the ase0 < α 6 2^,α6= 1^.

Theorem 3.5. Let 0 < α 6 2^, α 6= 1 ^{and onsider the linear regression}

model (1.1) . If

(a) the random variables e₁, e₂, . . . ^are independent su h that e_i ∼ S_α(σi, λ_i, µ_i),

(b) lim

n→∞S⁻¹_n = O^,lim sup

n→∞

Xn

i=1

S⁻¹_n x_ix^T_i  

  < ∞^,

X∞

i=1

 σ_i

 S⁻¹_i x_i

 α

<∞

and

X∞

i=1

µ_iS⁻¹

i x_i      <∞,

then βe ^{is strongly} onsistent.

P roof. ^{From(1.2)weonlyhavetoprovethealmostsure} onvergen eofthe series

P∞ i=1S⁻¹

i x_ie_i ^{sin e the} Krone ker's lemma (matrix ase) establishes the thesis. Ea h omponent of theve tor

Pn i=1S⁻¹

i x_ie_i ^hasdistribution,

S_α







 Xn

i=1

σ_i|aij|
α

!1/α

, Xn

i=1

σ_i|aij|
α

λ_i^sign(a_ij) Xn

i=1

σ_i|aij|
α

, Xn

i=1

µ_ia_ij









, j= 1, . . . , κ

wherea_ij ^isthej^{-th omponentof}a_i:= S⁻¹_i x_isothat,fromProposition2.1 we obtain thealmostsure onvergen eof

P∞ i=1S⁻¹

i x_ie_i^.

The ase α= 1 ^{isdes ribed next.}

Theorem 3.6. Consider the linear regression model (1.1) . If

(a) the random variables e₁, e₂, . . . ^are independent and su h that e_i ∼ S₁(σ_i, λ_i, µ_i),

(b) lim

n→∞S⁻¹_n =O^,lim sup

n→∞

Xn

i=1

S⁻¹_n x_ix^T_i  

 <∞^,

X∞

i=1

σ_i 

S⁻¹_i x_i 

 log S⁻¹_i x_i

min

onverges and

X∞

i=1

µ_iS⁻¹

i x_i      <∞,

then βe ^{is strongly} onsistent.