Types of conditional convergence

U N I V E R S I T A T I S M A R I A E C U R I E - S K Ł O D O W S K A L U B L I N – P O L O N I A

VOL. LIX, 2005 SECTIO A 97–105

WIOLETTA NOWAK and WIESŁAW ZIĘBA

Types of conditional convergence

Abstract. The aim of this paper is to investigate relations between different types of conditional convergence. Results presented in this paper generalize theorems obtained by P. Fernandez [2] and A. R. Padmanabhan [5].

1. Introduction. Let (Ω, A, P ) be a probability space, and let F be a sub-σ-field contained in A. We denote by E^FX conditional expectation of X with respect to F . Let L⁺ (L⁺(F )) be the set of nonnegative random variables (F -measurable). For X ∈ L⁺we can define the conditional expec- tation E^FX as in [4]. For X = X⁺−X⁻, where X⁺= max(X, 0) and X⁻= max(−X, 0), we define the conditional expectation E^FX = E^FX⁺−E^FX⁻ if min(E^FX⁺, E^FX⁻) < ∞ a.s. If max(E^FX⁺, E^FX⁻) < ∞ a.s., then we say that X ∈ L¹(F ).

We will denote by F_X(x) = P {ω : X(ω) < x} the distribution function for X, by ζF_X the set of continuity points of F_X(x), that is ζF_X = {x : FX(x) = F_X(x+)}.

Definition 1.1. The conditional distribution function for random variable X given σ-field F we will call a process F (x, ω) = E^FI_[X<x](ω) such that F (x, ω) is left-continuous and nondecreasing.

2000 Mathematics Subject Classification. Primary 60E05, Secondary 62E20.

Key words and phrases. Convergence in probability, conditional expectation.

Obviously lim_x→+∞F (x, ·) = 1 a.s. and limx→−∞F (x, ·) = 0 a.s. Note that if x ∈ ζ_FX, then lim_ε→0F (x − ε, ·) = F (x, ·) a.s.

We will say that random variables X and Y have the same conditional distribution if for each x ∈ R

E^FI_[X<x] = E^FI_{[Y <x]} a.s.

Note that if random variables X and Y have the same conditional dis- tribution, then these variables have the same distribution, because for each x ∈ R we have

FX(x) = P (X < x) = EI_[X<x] = E(E^FI_[X<x]) = E(E^FI_{[Y <x]}) = FY(x).

The following example shows that the opposite implication is not true.

Example 1. Let (Ω, A, P ) = ([0, 1], B([0, 1]), µ), where µ denotes Lebesgue measure, F = A,

X(ω) =

(0, ω ∈ [0,¹₂), 1, ω ∈ [¹₂, 1], and Y (ω) = 1 − X(ω). Then F_X(x) = FY(x), but

I_[X<¹

2](ω) = (

1, ω ∈ [0,¹₂), 0, ω ∈ [¹₂, 1], and

I_{[Y <}¹

2](ω) = (

0, ω ∈ [0,¹₂), 1, ω ∈ [¹₂, 1].

Then I_[X<x] 6= I_{[Y <x]} for ω ∈ Ω and x ∈ [0, 1], hence these variables have the same distribution, but do not have the same conditional distribution.

Definition 1.2. We say that a sequence {X_n, n ≥ 1} of r.v. F -conditionally converges in distribution to the r.v. X if for each x ∈ ζ_FX

E^FI_[Xn<x]−→ E^FI_[X<x] a.s., n → ∞.

This fact will be denoted by X_n^{F −D}−→ X.

Note that if X_n^{F −D}−→ X, then X_n−→ X, since for each x ∈ ζ^D _FX we have

n→∞lim Fn(x) = lim

n→∞EI_[Xn<x]= lim

n→∞E E^FI_[Xn<x]
= E lim

n→∞E^FI_[Xn<x]

= EE^FI_[X<x] = F (x).

The following example shows that the opposite implication is false.

Example 2. Let (Ω, A, P ) be defined as in the previous example, F = σ([0,¹₃), [¹₃,²₃), [²₃, 1]), for even values of n

Xn(ω) =







1

4, ω ∈ [0,¹₃), ω, ω ∈ [¹₃,²₃),

2

3, ω ∈ [²₃, 1], while for odd values of n

Xn(ω) =







2

3, ω ∈ [0,¹₃), ω, ω ∈ [¹₃,²₃),

1

4, ω ∈ [²₃, 1].

Then X_n −→ X, however for x =^{D 1}₃ we have E^FI_[X

n<¹₃] = E^FI_An = I_An, where

A_n=

([0,¹₃) for n = 2k,

[²₃, 1] for n = 2k + 1, k ∈ N.

Since lim_n→∞I_An does not exist, the sequence {X_n, n ≥ 1} is not condi- tionally convergent in distribution.

Definition 1.3. We say that a sequence {X_n, n ≥ 1} of r.v. F -conditionally converges in probability to the r.v. X if for every ε > 0

E^FI_{[|Xn−X|>ε]} −→ 0 a.s., n → ∞, which will be denoted by X_n^{F −P}−→ X.

It is easily seen that if X_n^{F −P}−→ X, then X_n−→ X, because^P

n→∞lim P (|Xn− X| > ε) = lim

n→∞EI_{[|Xn−X|>ε]}= lim

n→∞E E^FI_{[|Xn−X|>ε]}

= E

n→∞lim E^FI_{[|Xn−X|>ε]}

= 0.

Theorem 1.4. If X_n ^{F −P}−→ X, then for every F -measurable r.v. η > 0 a.s.

E^FI_{[|Xn−X|>η]} −→ 0 a.s., n → ∞.

Proof. Choose δ > 0. Then

E^FI_[|Xn−X|>η]≤ E^FI_[|Xn−X|>δ, η≥δ]+ E^FI_[|Xn−X|>η, η<δ] a.s.

Therefore

n→∞lim E^FI_{[|Xn−X|>η]} ≤ lim

n→∞E^FI_{[|Xn−X|>δ]}+ E^FI_[η<δ]= E^FI_[η<δ]−→ 0 a.s., if δ → 0. Since δ is arbitrary, we have proved the result. 

Let F and G be sub-σ-fields contained in the σ-field A and F ⊂ G. In such a case if X_n^G−D−→ X, then X_n^{F −D}−→ X, as for each x ∈ ζ_FX we have

n→∞lim E^FI_[Xn<x]= lim

n→∞E^FE^GI_[Xn<x]= E^F lim

n→∞E^GI_[Xn<x]

= E^FE^GI_[X<x] = E^FI_[X<x] a.s.

Similarly, if X_n^G−P−→ X, then X_n^{F −P}−→ X. Indeed, for every ε > 0 we have

n→∞lim E^FI_{[|Xn−X|>ε]}= lim

n→∞E^FE^GI_{[|Xn−X|>ε]}

= E^F lim

n→∞E^GI_[|Xn−X|>ε] = 0 a.s.

The opposite implications are not true.

Example 3. Let (Ω, A, P ) be defined as in the previous examples, G = σ([0,¹₂), [¹₂, 1]), F = {∅, Ω}. If

X_n(ω) =

(1, ω ∈ [0,¹₂), 0, ω ∈ [¹₂, 1],

for even values of n and X_n(ω) = 1 − X_n−1(ω), for odd values of n then

n→∞lim E^FI_[Xn<x]= lim

n→∞P [Xn< x] = P [X < x] = E^FI_[X<x] for x ∈ ζ_FX. However E^GI_[Xn<¹

2]= E^GIAn= IAn, where A_n(ω) =

([0,¹₂) for n = 2k + 1, [¹₂, 1] for n = 2k, k ∈ N and lim_n→∞I_An does not exist.

Example 4. Let (Ω, A, P ) be defined as in the previous example. Every integer can be written in the form n = 2^k+ s, where k = max{l : 2^l≤ n}, s = 0, 1, ..., 2^k− 1. If G = B, F = {∅, Ω} and

X₂k+s(ω) =

(1, ω ∈ [₂^sk,^s+1₂k ), 0, otherwise, then for every ε > 0,

E^FI_[|Xn|>ε]= P [|Xn| > ε] = 1

2^k −→ 0, n → ∞.

However for every ε > 0, E^GI_[|Xn|>ε] = I_[|Xn|>ε]. Thus for ε = ¹₂ we have lim sup_n→∞I_[|Xn|>ε] = 1 a.s. but lim infn→∞I_[|Xn|>ε] = 0 a.s. Hence it is not true that X_n^G−P−→ X, but X_n^{F −P}−→ X.

For F = A convergence X_n ^{F −P}−→ X implies a.s. convergence. Indeed, E^FI_{[|Xn−X|>ε]} = I_{[|Xn−X|>ε]} a.s. and

lim sup

n→∞

I_[|Xn−X|>ε] = lim

n→∞sup

k≥n

I_[|Xk−X|>ε]= lim

n→∞I^S∞

k=n[|Xk−X|>ε]= 0 a.s.

Hence

n→∞lim P

∞

[

k=n

[|Xk− X| > ε]

!

= 0,

which is equivalent to a.s. convergence. On the other hand, the last state- ment implies the previous one, as can be easily seen. Thus a continuous link between convergence in probability and a.s. convergence is established.

2. Main results.

Theorem 2.1. If X_n^{F −P}−→ X, then for each x ∈ ζ_FX E^F

I_[Xn<x]− I_[X<x]

−→ 0 a.s., n → ∞.

Proof. If X_n^{F −P}−→ X, then for every ε > 0, lim_n→∞E^FI_{[|Xn−X|>ε]}= 0 a.s.

Moreover E^F

I_[Xn<x]− I_[X<x]

= E^FI_([Xn<x]4[X<x])

= E^FI_([Xn<x]4[X<x])I_{[|Xn−X|≥ε]}+ E^FI_([Xn<x]4[X<x])I_{[|Xn−X|<ε]}

= E^FI_([Xn<x]4[X<x])I_{[|Xn−X|≥ε]}+ E^FI_([Xn<x]4[X<x])I_{[X−ε<Xn<X+ε]}

≤ E^FI_{[|Xn−X|≥ε]}+ E^FI_([Xn<x]\[X<x])I_{[X−ε<Xn<X+ε]}

+ E^FI([X<x]\[Xn<x])I_{[X−ε<Xn<X+ε]}

≤ E^FI_{[|Xn−X|≥ε]}+ E^FI[x≤X<x+ε]+ E^FI[x−ε≤X<x]

= E^FI_{[|Xn−X|≥ε]}+ F_X^F(x + ε) − F_X^F(x) + F_X^F(x) − F_X^F(x − ε) a.s.

Thus for every ε > 0 lim sup

n→∞

E^F

I_[Xn<x]− I_[X<x]

≤ F_X^F(x + ε) − F_X^F(x − ε) a.s.

Since ε is arbitrary, we have lim sup_n→∞E^F

I_[Xn<x]− I_[X<x]

= 0 a.s. and hence lim_n→∞E^F

I_[Xn<x]− I_[X<x]

= 0 a.s. for x ∈ ζ_FX.  Corollary 2.2. If Xn

F −P−→ X, then X_n^{F −D}−→ X.

Theorem 2.3. limn→∞E^FI_([Xn<x]4[X<x]) = 0 a.s., x ∈ ζFX if and only if for every D ∈ A and each x ∈ ζ_FX

n→∞lim E^FI_[Xn<x]I_D = E^FI_[X<x]I_D a.s.

Proof. Necessity. If lim_n→∞E^FI_([Xn<x]4[X<x]) = 0 a.s. then by the previ- ous theorem for every x ∈ ζ_FX we have

E^FI_[Xn<x]ID − E^FI_[X<x]ID

 =

E^F I_[Xn<x]− I_[X<x]
I_D 

≤ E^F

I_[Xn<x]− I_[X<x]

= E^FI_([Xn<x]4[X<x]) −→ 0 a.s., n → ∞.

Sufficiency. If D = [X < x] and x ∈ ζ_FX, then lim_n→∞E^FI_[Xn<x]I_[X<x] = E^FI_[X<x]. Hence

n→∞lim E^FI_([Xn<x]4[X<x])

= lim

n→∞E^F(I_[Xn<x]− I_[Xn<x][X<x]+ I_[X<x]− I_[Xn<x][X<x])

= lim

n→∞E^FI_[Xn<x]− lim

n→∞E^FI_[Xn<x][X<x]

+ E^FI_[X<x]− lim

n→∞E^FI_[Xn<x][X<x]= 0 a.s.

 Note that lim_n→∞E^FI_([Xn<x]4[X<x]) = 0 a.s. for x ∈ ζ_FX is equivalent to lim_n→∞E^FI_{([Xn≤x]4[X≤x])} = 0 a.s. for x ∈ ζ_FX.

Theorem 2.4. If limn→∞E^FI_([Xn<x]4[X<x]) = 0 a.s. for each x ∈ ζFX, then X_n^{F −P}−→ X.

Proof. Let B(x, r) = {y : |x−y| < r} and ε > 0. Let {xi}_i=1,2...be a count- able dense subset of R. Select γ such that 0 < γ < ^ε₂ and P [|X −x_i| = γ] = 0.

Then E^FI_[|X−xi|=γ,i=1,2,... ] = 0 a.s. It is clear thatS∞

i=1B(x_i, γ) = R, there- fore by continuity of measure P , we have lim_t→∞PX ∈ S^t_s=1B(xs, γ) = 1, hence

t→∞lim E^FI[^X∈Sts=1B(xs,γ)] = 1 a.s.

For 0 < δ < 1 we define a random variable N (ω) = infn

t : E^FI[X∈St

s=1B(xs,γ)] > 1 − δ o

. Note that A_n= [N (ω) = n] ∈ F and P [S∞

n=1An] = 1, i.e. N < ∞ a.s.

Moreover, if K_t=St

s=1B(xs, γ), then

E^FI_{[|Xk−X|>ε]}= E^FI_{[|Xk−X|>ε,}^S∞

n=1An]=

∞

X

n=1

E^FI_{[|Xk−X|>ε]}I_An

=

∞

X

n=1

E^FI_{[X /∈KN,|Xk−X|>ε]}I_An+

∞

X

n=1

E^FI_{[X∈KN,|Xk−X|>ε]}I_An

≤

∞

X

n=1

E^FI_{[X /}∈K_N]IAn+

∞

X

n=1

E^FI_[X∈KN,|Xk−X|>ε]IAn

≤ δ +

∞

X

n=1

E^FI(^SN_s=1[X∈B(xs,γ)]∩[|Xk−X|>ε])I^An .

Thus, since

N

[

s=1

[X ∈ B(x_s, γ)] ∩ [|X_k− X| > ε]

⊂

N

[

s=1

[xs− γ < X < x_s+ γ] ∩ [|X_k− X| > ε]

⊂

N

[

s=1

[x_s− γ < X < x_s+ γ] ∩ [x_s− γ < X_k < x_s+ γ]^C

⊂

N

[

s=1

[x_s− γ < X < x_s+ γ] ∩ [X_k< x_s+ γ]^C

!

∪

N

[

s=1

[xs− γ < X < x_s+ γ] ∩ [xs− γ < X_k]^C

!

⊂

N

[

s=1

[X < xs+ γ] ∩ [Xk< xs+ γ]^C

!

∪

N

[

s=1

[x_s− γ < X] ∩ [x_s− γ < X_k]^C

!

⊂

N

[

s=1

[X < xs+ γ] ∩ [X_k< xs+ γ]^C

!

∪

N

[

s=1

[Xk≤ x_s− γ] ∩ [X ≤ x_s− γ]^C

! ,

we have

E^FI_{[|Xk−X|>ε]}

≤ δ +

∞

X

n=1

E^F

N

X

s=1

I_{([X<xs+γ]4[Xk<xs+γ])}IAn

!

+

∞

X

n=1

E^F

N

X

s=1

I_{([Xk≤xs−γ]4[X≤xs−γ])}I_An

! a.s.

Hence lim sup

k→∞

E^FI_{[|Xk−X|>ε]}

≤ δ + lim

k→∞

∞

X

n=1

E^F

N

X

s=1

I_{([X<xs+γ]4[Xk<xs+γ])}IAn

+

N

X

s=1

I_{([Xk≤xs−γ]4[X≤xs−γ])}I_An

!

= δ +

∞

X

n=1 n

X

s=1

k→∞lim I_AnE^FI_{([X<xs+γ]4[Xk<xs+γ])}

+

∞

X

n=1 n

X

s=1

k→∞lim IAnE^FI_([Xk≤x_s−γ]4[X≤x_s−γ])= δ a.s.

Since δ is arbitrary, we have proved the result.  Theorem 2.5. Let X be F -measurable random variable. If Xn

F −D−→ X, then X_n^{F −P}−→ X.

Proof. If Xn

F −D−→ X, then for each x ∈ ζ_FX we have lim_n→∞E^FI_[Xn<x]= E^FI_[X<x] = I_[X<x] a.s. and lim_n→∞E^FI_[Xn<x]I_[X<x]= I_[X<x]. Hence

n→∞lim E^FI_([Xn<x]4[X<x]) = lim

n→∞E^F(I_[Xn<x]− I_[Xn<x]I_[X<x]) + lim

n→∞E^F(I_[X<x]− I_[Xn<x]I_[X<x]) = 0 a.s.

Therefore by the previous theorem we have that X_n^{F −P}−→ X.  Note that if F is the trivial σ-field (F = {∅, Ω}), then we obtain the following well-known result.

Corollary 2.6. Let C be a random variable such that P [C = c] = 1 for some c ∈ R. Then Xn

−→ C if and only if XD _n−→ C.^P

Acknowledgments. We thank the referee for valuable comments.

References

[1] Billingsley, P., Convergence of Probability Measures, John Wiley & Sons, Inc., New York–London–Sydney, 1968.

[2] Fernandez, P., A note on convergence in probability, Bol. Soc. Brasil. Mat. 3 (1972), 13–16.

[3] Majerek D., W. Nowak and W. Zięba, On uniform integrability of random variables, Stat. Probab. Lett. 74 (2005), 272–280.

[4] Neveu, J., Discrete-Parameter Martingales, North-Holland Publishing Company, Amsterdam–Oxford, 1975.

[5] Padmanabhan, A.R., Convergence in probability and allied results, Math. Japon. 15 (1970), 111–117.

[6] Zięba, W., On the L¹_F convergence for conditional amarts, J. Multivariate Anal. 26 (1988), 104–110.

Wioletta Nowak Wiesław Zięba

Department of Mathematics Institute of Mathematics

Lublin University of Technology Maria Curie-Skłodowska University ul. Nadbystrzycka 38 pl. Marii Curie-Skłodowskiej 1 20-618 Lublin, Poland 20-031 Lublin, Poland

e-mail: wnowak@antenor.pol.lublin.pl e-mail: zieba@golem.umcs.lublin.pl Received May 11, 2005