Some Connections of Weak Convergence with the Convergence of the Dispersion Functions

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Some Connections of Weak Convergence with the Convergence of the Dispersion Functions

^*

Tran Loc Hung and Nguyen Van Son

Department of Mathematics, College of Sciences Hue University, 77 Nguyen Hue str., Hue city, Vietnam

Received June 11, 2002 Revised October 1, 2002

Abstract. The aim of this note is to establish some connections of weak convergence ofL¹- random variables with the convergence of the dispersion functions.

1. Introduction

In this note, let (Ω,A, P) denote a probability space, and let L¹ denote the set of allL¹- random variables. LetX∈ L¹andF_X be the distribution function of X. We define the dispersion function ofX by

D_X(u) :=E|X−u| for every u∈R= (−∞,+∞),

that is, the absolute moment of order r = 1 of the random variable X with respect to u, for allu∈R.

In recent years some results concerning the dispersion function D_X(u) have been investigated by Munoz-Perez and Sanchez-Gomez in [1, 2], Thu and Turkan in [3], Thu and Hung in [4 - 6], Hung in [7].

It is worth pointing out that the dispersion functionD_X(u) can be considered as a generalization of the mean absolute deviation and the median absolute deviation (see for more details [3 - 6]).

The dispersion function has the following elementary properties (see [1, 2, 4 - 7] for the complete bibliography).

∗This work was supported by MOET Grant B2002-07-03, Vietnam.

a. The functionD_X(u) is almost everywhere differentiable onRand its deriva- tive has at most a countable number of discontinuity points.

b. The functionD_X(u) is convex onR. c. lim

u→+∞D_X (u) = 1 and lim

u→−∞D_X(u) =−1.

d. lim

u→+∞(D_X(u)−u) =−EX and lim

u→−∞(D_X(u) +u) =EX.

e. F_X(u) = ¹₂(D_X (u) + 1), for allu ∈ C_F, where D_X(u) is derivative of the functionD_X(u) andC_F denotes a set of continuity points ofF_X.

f. D_X(u) =

+∞

−∞ |F_X(x)−F_u(x)|dx, whereF_u(x) is the distribution function of the degenerate random variable at u.

g. ^+∞

−∞ |D_X(u)−D_EX(u)|du=σ².

The main purpose of this note is to establish some connections between the weak convergence of the L¹- random variables (denoted by ⇒) and the convergence of the dispersion functions in order to outline the relation between the dispersion functions and the distribution functions. The following theorems are main results of this paper.

Theorem 1. Let X, X₁, X₂, . . . , X_n, . . . ∈ L¹. If there exists a p > 1, such that

supn∈NE|X_n|^p<+∞ (A)

and if

F_Xn ⇒F_X, as n→+∞, then

D_Xn(u)→D_X(u) as n→ ∞, for every u∈R.

Theorem 2. Let {Xn, n≥ 1} be a sequence of L¹- random variables and let {Dn, n≥1}be the corresponding sequence of dispersion functions. Suppose that D_n(u)→D(u)asn→ ∞, for every u∈R. Then

a. D(u)is a convex function in R.

b. Let H be the set of all points u ∈ R, such that D(u), D_n(u), n ∈ N exist.

Then His a dense set inRand

n→∞limD_n(u) =D(u), ∀u∈ H.

c. Let H0 be a set of all points such that D(u) exists. Then for everyu∈ H0,

u→+∞lim D(u) = 1, and lim

u→−∞D(u) =−1.

Theorem 3. Let X, X₁, X₂, . . . X_n, . . . ∈ L¹ and let D, D₁, D₂, . . . D_n, . . . be the corresponding dispersion functions. Assume that D_n(u)→D(u)asn→ ∞, for every u∈R. Then

a. F_n ⇒F_X. b. lim

n→∞sup

u∈R|D_n(u)−D(u)|= 0.

Theorem 4. Let {Xn, n≥ 1} be a sequence of L¹- random variables and let {Dn, n≥1}be the corresponding sequence of dispersion functions. Let assump- tion (A) of Theorem 1 hold, and D_n(u)→ D(u), as n → ∞ for every u∈R.

Then there exists a distribution function F such that D(u) =

R|x−u|dF(x).

The results obtained are intended to show that a connection between the weak convergence ofL¹- random variables and the convergence of the dispersion measures based on the dispersion functions can be used in studying weak limit theorems.

2. Proofs.

Proof of Theorem 1. By assumption (A) of Theorem 1 and sinceX_n ∈ L¹,for alln∈N, we see that (X_n)_n∈Nare uniformly integrable. We thus get

n→∞lim

R|x|dF_Xn(x) =

R|x|dF_X(x).

It follows easily that the function g_u(x) =|x−u| − |x|, x∈R, is continuous and bounded onR,so we have

n→∞lim

R(|x−u| − |x|)dF_Xn(x) =

R(|x−u| − |x|)dF_X(x) or

n→∞lim

R|x−u|dF_n(x) =

R|x−u|dF(x).

The proof is complete.

Remark to Theorem 1. It should be noted that Theorem 1 does not hold without the assumption (A).

Let {X_n, n ≥ 1} be a sequence of random variables with the distribution functions

F_n(x) =

⎧⎪

⎪⎪

⎪⎪

⎨

⎪⎪

⎪⎪

⎪⎩

0, if x≤0,

1−exp(−x), if 0< x≤n,

1

exp(2n)−nexp(n)(x−n) + 1−exp(−n), if n < x≤exp(n),

1, if exp(n)< x.

and letX be a random variable with the distribution function F(x) =

0, if x≤0, 1−exp(−x), if 0< x.

Then, although

n∈NsupE|X_n |<+∞, and

n→∞limF_n(x) =F(x), ∀x∈R but D_Xn(u)D_X(u) as n→+∞.

Lemma 1. The family of (D_X)_X∈L¹ is equally continuous on R.

Proof. LetX ∈ L¹.For everyu∈R,we have

−1≤D⁻_X(u)≤D_X⁺(u)≤1, ∀u∈R. Consequently, withh >0

−1≤D⁺_X(u)≤ D_X(u+h)−D_X(u)

h ≤D⁻_X(u+h)≤1, it follows that

|D_X(u+h)−D_X(u)|≤h.

The proof of Lemma 1 is complete.

Proof of Theorem 2.

a. For everyα, u, v∈R,0≤α≤1, we have

D_n(αu+ (1−α)v)≤αD_n(u) + (1−α)D_n(v), ∀n∈N.

It follows that

n→∞limD_n(αu+ (1−α)v)≤ lim

n→∞(αD_n(u) + (1−α)D_n(v)).

Consequently,

D(αu+ (1−α)v)≤αD(u) + (1−α)D(v).

ThereforeD(u) is convex onR.

b. Since a convex functionRhas at most a countable number of undifferentiable points, it follows that R\ H has at most a countaible number, too. Thus, the set His dense inR.

To complete the proof it remains to show that

n→∞limD_Xn(u) =D_X (u), ∀u∈ H. (2.1) Leta, b∈R, a < b.Since the sequence (D_n)_n∈Nis equally continuous on [a, b]

(see Lemma 1), it follows that the sequence (D_n)_n∈N converges to D at every point in [a, b].Thus we get

n→∞lim sup

u∈[a,b]|D_n(u)−D(u)|= 0.

Assume thatδ >0,then there exists an₀ such that

D(u)−δ < D_n(u)< D(u) +δ, ∀n > n₀.

Then, we can show that for everyu∈R, D⁻(u)≤lim inf

n→∞D⁻_n (u)≤lim sup

n→∞ D⁺_n (u)≤D⁺(u), and consequently the proof follows.

c. Since H ⊂ H₀ and H is a dense set in R, and D(u) is not a decreasing function on H₀,from (2.1), we deduce that −1≤D(u)≤1.

On the other hand

|E(X_n)|≤

R|x|dF_n(x), and

n→∞lim

R|x|dF_n(x) = lim

n→∞D_n(0) =D(0).

Thus (E(X_n))_n∈Nare bounded, i.e. there existsM >0 such that 0≤|E(X_n)|<

M, for alln∈N.

SinceD_n(u)≥|E(X_n)−u|, for allu∈R, n∈N,it follows that D_n(u)−u≥ −E(X_n)>−M, ∀u∈R, n∈N, and

D_n(u) +u≥E(X_n)>−M, ∀u∈R, n∈N.

Therefore

n→+∞lim D_n(u)−u=D(u)−u≥ −M, ∀u∈R,

and lim

n→+∞D_n(u) +u=D(u) +u≥ −M, ∀u∈R.

Finally, the functions f(u) =D(u)−uandg(u) =D(u) +uare convex onR, where f(u)≤0,andg(u)≥0. We thus conclude that

u→+∞lim f(u) = 0

and lim

u→−∞g(u) = 0, thus, foru∈ H₀,

u→+∞lim D(u) = 1, lim

u→−∞D(u) =−1.

Proof of Theorem 3.

a. The proof is immediate from the part b) of Theorem 2 and the properties of the dispersion function from Sec. 1.

b. We have

D_n(u)−u≥ −E(X_n), D_n(u) +u≥E(X_n), ∀n∈N, and D(u)−u≥ −E(X), D(u) +u≥E(X).

By virtue of the properties of dispersion functions and remark above, it follows that for all >0,there existu₀ andn₀,such that

−E(X)≤D(u)−u≤ −E(X) +

2, ∀u≥u₀,

and D(u₀)−u₀+

2 ≥D_n(u₀)−u₀, ∀n > n0. In short, we have

−E(X) +≥D(u₀)−u₀+

2 ≥D_n(u₀)−u₀≥ −E(Xn), ∀n > n0, u≥u₀, or for all >0,there exists n₀such that

E(X)≤E(X_n) +, ∀n > n₀.

In the same manner we can see that for all >0, there existsn₁ such that E(X)≤E(X_n)−, ∀n > n1.

Consequently,

n→∞limE(X_n) =E(X).

According to properties of the dispersion function, one finds that supu∈R|D_n(u)−D(u)|<+∞.

It remains to prove that

n→∞limsup

u∈R|D_n(u)−D(u)|= 0. (2.2) Let (2.2) be not true, i.e. there exists > 0 such that for all n₀, there exists n > n₀ such that

supu∈R|D_n(u)−D(u)|> . For everyk∈N,let us put

n_k = inf{n∈N|n > n₁+k,sup

u∈R|D_n(u)−D(u)|> }, (2.3) where n₁is a positive integer number satisfying sup

u∈R|D_n1(u)−D(u)|> . It follows that, for every k∈N,there areu_nk such that

|D_nk(u_nk)−D(u_nk)|> . (2.4) According to Theorem 2, the sequence (u_nk)_k∈N is not bounded. Thus we can extract from (u_nk)_k∈Na subsequence convergent to +∞or to−∞.Without loss of generality we can assume

k→∞limu_nk = +∞.

Then we have

m→∞lim lim

k→∞(D_m(u_nk)−u_nk) = lim

m→∞(−EX_m) =−E(X),

k→∞lim lim

m→∞(D_m(u_nk)−u_nk) = lim

k→∞(D(u_nk)−u_nk) =−E(X).

Thus lim

k→∞(D_nk(u_nk)−u_nk) =−E(X). (2.5) On the other hand, it is obvious that

k→∞lim(D(u_nk)−u_nk) =−E(X). (2.6) But this shows that (2.3) - (2.6) are in contradictions, and we have the desired

proof.

Proof of Theorem 4. According to the Theorem 3, if we put F(x) =

⎧⎨

⎩ 1

2(D(x) + 1), ifx∈ H,

xlimn→xF(x_n) ifx /∈ H(x_n< x, x_n ∈ H), thenF(x) is a distribution function andF_n⇒F.

On the other hand, for alla, b∈ H, a < b,we have

n→∞lim _b

a |x|dF_n(x) = _b

a |x|dF(x).

Since

R|x|dF_n(x)< M,it shows that

R|x|dF(x)< M,thus we conclude that F is a distribution function of a random variable with finite mean.

LetD(u) be a corresponding dispersion function of the functionF.By virtue of the Theorem 1 we have

n→∞limD_n(u) =D(u), ∀u∈R.

It follows that D(u) =D(u), for all u∈ R, and this completes the proof of

the theorem.

Acknowledgement. The authors would like to express their gratitude to the referee for valuable remarks and comments which improve the presentation of this paper.

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