The variance and covariance of fuzzy random variables and their applications

Fuzzy Sets and Systems 120 (2001) 487–497

www.elsevier.com/locate/fss

The variance and covariance of fuzzy random variables and their applications Yuhu Feng ∗ , Liangjian Hu, Huisheng Shu Department of Basic Sciences, China Textile University, Shanghai, 200051, People’s Republic of China Received January 1998; received in revised form December 1998

Abstract The concepts of the variance and covariance of fuzzy random variables and their properties are introduced. Examples show their computation and applications in statistical estimation of parameters when samples or prior information are fuzzy. As their further applications the correlation function and the criterions of mean-square calculus for fuzzy stochastic processes c 2001 Elsevier Science B.V. All rights reserved. are established.
Keywords: Fuzzy number; Fuzzy random variables; Variance and covariance; Analysis; Probability

1. Introduction Motivated by the applications in several areas of applied science, such as mathematical economics, fuzzy optimal, process control and decision theory, etc., the theory of fuzzy random variables and fuzzy stochastic processes was developed in recent years (see [3,4, 6] and their references). Some of the most useful information concerning a real-valued random variable as well as a fuzzy random variable is revealed by its moments, particularly those of the
Corresponding author. Tel.: + 86-21-6237-3331; fax: + 86-216219-4722. E-mail address: [email protected] (Y. Feng).

to study their properties. The variance or covariance of fuzzy random variables is of great importance in statistical analysis, linear theory of fuzzy stochastic proceses and other
c 2001 Elsevier Science B.V. All rights reserved. 0165-0114/01/$ - see front matter  PII: S 0 1 6 5 - 0 1 1 4 ( 9 9 ) 0 0 0 6 0 - 3

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Y. Feng et al. / Fuzzy Sets and Systems 120 (2001) 487–497

new and it inherits many properties in the case of real-valued random variables. The rest of this paper is organized as follows. In Section 2 we brieEy state some results related to the fuzzy number space (E; dp ); 16p6∞, and prove that (E; dp ) is a complete generalized metric space. To be well prepared for discussing the variance and covariance of fuzzy random variables, we de
1 2




1 0

(Cov(X − (r); Y − (r)) + Cov(X + (r); Y + (r))) dr

and the variance of a fuzzy random variable X as DX = Cov(X; X ), where [X − (r); X + (r)] = [X ]r is the r-level set of X for r ∈ (0; 1], and discuss their properties which are similar to the ones of real-valued random variables. Three examples show the computation of DX and the applications in statistical estimation of parameters when samples or prior information are fuzzy. Applied to fuzzy stochastic processes in Section 4, we discuss the correlation function for a fuzzy stochastic process. The importance of the correlation function rests in part on the fact that its properties de
2. Preliminaries De
not make the restrictive assumption that the fuzzy number u has compact support. For u ∈ E; [u]r = {x ∈ R | u(x)¿r}; 0¡r61; is the r-lever set of u. Then (i) [u]r is a closed interval, 0¡r1 6r2 61; r ∈ (0; 1]; (ii) [u]r1 ⊃[u]r2 ; whenever ∞ (iii) for any rn increasing to r; n=1 [u]rn = [u]r : Conversely, if {Mr ; r ∈ (0; 1]} ful
u; v ∈ E and  ∈ R:

Lemma 2.1 (Ma [5, Theorem 3.1]). For u ∈ E; let u+ (r); u− (r) be the upper and lower endpoints of [u]r . Then (i) u− (r) is a left continuous nondecreasing function on (0; 1]; (ii) u+ (r) is a left continuous nonincreasing function on (0; 1]; (iii) u− (r)6u+ (r): Conversely; if a(r); b(r) satisfy the above (i) – (iii), then there exists a unique u ∈ E such that [u]r = [a(r); b(r)]; r ∈ (0; 1]: Let u; v ∈ E, and set 
dp (u; v) =

1 0

1=p p

r

r

h ([u] ; [v] ) dr

;

16p¡∞;

d∞ (u; v) = sup h([u]r ; [v]r ); 0¡r61

where h is the HausdorG metric, i.e. h([u]r ; [v]r ) = max{ |u− (r) − v− (r) |; |u+ (r) − v+ (r) | }: For 16p6∞; dp ful
Y. Feng et al. / Fuzzy Sets and Systems 120 (2001) 487–497

that (E n ; dp ) is not complete for 16p¡∞, where E n denote the set of all fuzzy numbers with compact support in Rn . However, in the present case we have the following theorem. Theorem 2.2. The generalized metric space (E; dp ) is complete for 16p6∞: Proof. p = ∞ follows Theorem 4:1 in [6]. Let 16p¡∞ and (un ; n¿1) be a Cauchy sequence in E, i.e. 
1=p dp (un ; um ) =

1

0

hp ([un ]r ; [um ]r ) dr

→0

(ii) u; v = v; u; (iii) u + v; w = u; w + v; w; (iv) u; v = u;  v; (v) | u; v |6 u; uv; v; where u; v; w ∈ E and  ∈ [0; ∞). For u; v ∈ E, if u; u¡∞; v; v¡∞; from the property (v) we can de
as n; m → ∞ (un− (r);

+ (u+ (r) − v+ (r))2 ] dr

(un+ (r);

then both n¿1) and n¿1) are Cauchy sequences in Lp ((0; 1]), where Lp ((0; 1]) denotes all Lebesgue measurable functions f on (0; 1] 1 for which 0 |f(r)| p dr¡∞: Since Lp ((0; 1]) is complete, it follows that un± (r) → f± (r) ∈ Lp ((0; 1])

489

as n → ∞

the convergence being in the norm of Lp , and there exists a subsequnce (un±k ) such that un±k (r) → f± (r); as k → ∞, for r ∈ (0; 1]\A; where (A) = 0 ( denotes the Lebesgue measure). It is easy matter to verify that f− (r) is nondecreasing and f+ (r) is nonincreasing on (0; 1]\A. Because (0; 1]\A is dense in (0; 1], for each r ∈ (0; 1], there exists an increasing sequences rn ∈ (0; 1]\A; rn ↑ r; de
If the indeterminacy of the form ∞−∞ arises in the Lebesgue integral then we say that u; v does not exist. It is easy to see that the operation ·; · has the following properties. ˆ (i) u; u¿0 and u; u = 0 ⇔ u = 0;

= u; u − 2u; v + v; v = d∗2 (u; v)
62

1 0

h2 ([u]r ; [v]r ) dr = 2d22 (u; v):

(2.2)

3. The variance and covariance of fuzzy random variables Let (; A; P) be a complete probability space. A fuzzy random variable (f.r.v., for short) is a Borel measurable function X : (; A) → (E; d∞ ): If X is an f.r.v. then [X ]r = [X − (r); X + (r)]; r ∈ (0; 1], is a random closed interval set and X − (r); X + (r) are realvalued r.v.’s. [4]. Since for every 16p6∞; |dp (u; v) − dp (u0 ; v0 ) | 6 dp (u; u0 ) + dp (v; v0 ) 6 d∞ (u; u0 ) + d∞ (v; v0 ) and | u; v − u0 ; v0  | 6 | u; v − u0 ; v | + | u0 ; v − u0 ; v0  | 62( v ∞ d1 (u; u0 ) + u0 ∞ d1 (v; v0 )) 62( v ∞ d∞ (u; u0 ) + u0 ∞ d∞ (v; v0 )); so the mappings dp (u; v) : (E; d∞ ) × (E; d∞ ) → R : (u; v) → dp (u; v)

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Y. Feng et al. / Fuzzy Sets and Systems 120 (2001) 487–497

and

is an integrably real-valued r.v. Set

u; v : (E; d∞ ) × (E; d∞ ) → R : (u; v) → u; v

D∗ (X; Y ) = (Ed∗2 (X; Y ))1=2 ;

are continuous, hence if X and Y are f.r.v.’s then dp (X; Y ); X; Y  and X p are real-valued r.v.’s. An f.r.v. X is called integrably bounded if E X ∞ ¡∞ and the expected value EX is de
X; Y ∈ Lp :

where d∗ is de
n→∞

n→∞

Theorem 3.1. Let (Xn ; n¿1) be a sequence in Lp ; 16p¡∞. The following conditions are equivalent. (i) X ∈ Lp and Dp (Xn ; X ) → 0; (ii) (Xn ; n¿1) is a Cauchy sequence in Lp ; i.e. Dp (Xm ; Xn ) → 0; as m; n → ∞; (iii) ( Xn pp ; n¿1) is uniformly integrable and dp (Xn ; X ) converges in probability to zero; as n → ∞. Corollary 3.2. (Lp ; Dp ) is a complete metric space. In the next discussion we
(3.3)

Lemma 3.3. (i) The mapping E·; ·: (L2 ; D2 ) × (L2 ; D2 ) → R; (X; Y ) → EX; Y ; is continuous. (ii) (Xn ; n¿1) is a Cauchy sequence in (L2 ; D2 ) if and only if as n; m → ∞; the limit EXn ; Xm  exists. Proof. (i) From the Schwarz inequality we have the following inequality. |EX; Y  − EX0 ; Y  |
6E

1 0

max( |Y − (r) |; |Y + (r) |)

×( |X − (r) − X0− (r) | + |X + (r) − X0+ (r) |) dr 
1 ˆ ( |X − (r) − X − (r) | 6D2 (Y; 0)E 0

0

1=2

(3.1)

Then (Lp ; Dp ) is a metric space. Using Theorem 2.2 similar to the proof of Theorem 2.1 in [4], we have the following theorem and corollary.

(3.2)

+ |X + (r) − X0+ (r) |)2 dr ˆ 62D2 (X; X0 )D2 (Y; 0):

(3.4)

So for any !¿0, choose a " = max(1; !(2(1 + ˆ + D2 (Y0 ; 0))) ˆ −1 ); as D2 (X; X0 )¡" and D2 (X0 ; 0) D2 (Y; Y0 )¡"; since ˆ 6 D2 (Y; Y0 ) + D2 (Y0 ; 0) ˆ D2 (Y; 0) ˆ ˆ 6 " + D2 (Y0 ; 0)61 + D2 (Y0 ; 0); thus we have |EX; Y  − EX0 ; Y0  | 6 |EX; Y  − EX0 ; Y  | + |EX0 ; Y  − EX0 ; Y0  | ˆ + D2 (Y; Y0 )D2 (X0 ; 0))6!: ˆ 62(D2 (X; X0 )D2 (Y; 0)

Y. Feng et al. / Fuzzy Sets and Systems 120 (2001) 487–497

(ii) If limn; m→∞ D2 (Xn ; Xm ) = 0, by Corollary 3.2, there exists an X ∈ L2 such that limn→∞ D2 (Xn ; X ) = 0, so limn; m→∞ EXm ; Xn  = EX; X  from (i). Conversely, let limn; m→∞ EXm ; Xn  = a, then D∗ (Xm ; Xn ) = EXm ; Xm −2EXm ; Xn +EXn ; Xn  → a − 2a + a = 0, as n; m → ∞. Thus (Xn ; n¿1) is a Cauchy sequence from (3.4). Now we discuss the concepts of variance and covariance of f.r.v.’s. The expected value of an f.r.v. is de
(iii) D(X + u) = 2 DX ; (iv) D(X + Y ) = DX + DY + 2Cov(X; Y ); (v) |#(X; Y ) |61 and #(X; Y ) = 1 if and only if Y + EX = EY + X; a:s:; #(X; Y ) = −1 if and only if Y + X = EY + EX; a:s:;  where  = DY=DX ; (vi) (Chebyshev inequality) P(d2 (X; EX )¿!)6 (2DX )=!2 ; for any !¿0. Proof. The proof of (i)–(iv) is completely straightforward from the de
The variance of X is de
(3.6)

The normalized covariance is given by Cov(X; Y ) #(X; Y ) = √ √ DX DY

(3.7)

and is called the correlation coeAcient of X and Y . If #(X; Y ) = 0 then the f.r.v.’s X and Y are said to be uncorrelated. Note that if X ∈ L2 , then E(X ± (r))2 ¡∞ for all r ∈ (0; 1]\A, where (A) = 0;  is the Lebesgue measure. On the other hand, if X; Y ∈ L2 , then X; Y  is an integrably real-valued r.v. and EX; EY  exists. So (3.5) makes sense. The variance and covariance of f.r.v.’s have many properties which are similar to the ones of real-valued r.v.’s. Theorem 3.5. Let f.r.v.’s X and Y be in L2 . Then (i) Cov(X; Y ) = 12 (EX; Y  − EX; EY ); DX = 12 D∗2 (X; EX ); (ii) Cov(X + u; kY + v) = k Cov(X; Y ); where u; v ∈ E and ; k ∈ R; k¿0;

491

1 2 2 D∗ (Y

+ tEX; EY + tX )

1 2 2 D∗ (Y

+ |t |X; EY + |t |EX ) if t¡0:

if t¿0; (3.8)

It is easy to see that f(0) = DY = 12 D∗2 (Y; EY ). If t¿0, then using the fact EX; u = EX; u, for each X ∈ L2 and u ∈ E with u; u¡∞, and the property (i), we have D∗2 (Y + tEX; EY + tX ) = E(Y + tEX; Y + tEX  − 2Y + tEX; EY + tX  + EY + tX; EY + tX ) = EY; Y  − EY; EY  − 2tEX; Y  + 2tEX; EY  + t 2 EX; X  − t 2 EX; EX  = 2(DY − 2t Cov(X; Y ) + t 2 DX ) = 2f(t): If t¡0, then using properties (i)–(iv), 1 2 2 D∗ (Y

+ |t |X; EY + |t |EX )

= D(Y + |t |X )

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Y. Feng et al. / Fuzzy Sets and Systems 120 (2001) 487–497

= DY + 2|t |Cov(X; Y ) + t 2 DX = f(t):

(vi) For any !¿0, by (2.2) and the property (i), P(d2 (X; EX )¿!)6

Hence we complete the proof of (3.8). Since f(t)¿0, it is well known that (Cov(X; Y ))2 − (DX )(DY )60, i.e. |#(X; Y )|61. If |#(X; Y ) | = 1, then there exists a t0 such by (2.2), (3.1) and (3.2), if that f(t0 ) = 0. Thus  #(X; Y ) = 1, then t0 = DY=DX ¿0 and so D2 (Y + t0 EX; EY + t0 X ) = 0; Y + t0 EX = EY + t0 X; a:s:;  if #(X; Y ) = −1, then t0 = − DY=DX ¡0 and so D2 (Y + |t0 |X; EY + |t0 |EX ) = 0; Y + |t0 |X = EY + |t0 |EX; a:s: Hence the necessary condition holds. The suAciency is due to the fact that if Y + EX = EY + X , a.s. then by the property (ii), 1 Cov(X; Y ) = Cov(X + EY; Y + EX )  1 = Cov(Y + EX; Y + EX )  √ √ 1 = Cov(Y; Y ) = DX DY  i.e. #(X; Y ) = 1, if Y + X = EY + EX , a.s. then by the property (i),

6

= EX; Y  − EX; EY  = EY + X; X  − EX; X  − EX; EY  = EEY + EX; X  − EX; X  − EX; EY  = EX; EX  − EX; X  √ √ = − 2DX = − 2 DX DY i.e. #(X; Y ) = −1.

D∗2 (X; EX ) 2DX = 2 : !2 !

It is clearly seen that if X and Y are independent, i.e. &(X ) and &(Y ) are independent where &(X ) is the smallest &-
 vk (x) = sup inf (vk (yk )) ; k¿1

k=1

2Cov(X; Y )

Ed22 (X; EX ) !2

where the supremum is taken over all sequences ∞ {y1 ; y2 ; : : :} such that x = j=1 yj . Let X ∈ L2 , then ∞ EX = k=1 pk uk by Example ∞ 3 of [6]. It is easy to check that EX; X = k=1 pk uk ; uk . Then by using Theorem 3.5(i) and the fact that EX; EX  = ∞ ∞ k=1 j=1 pk pj uk ; uj , we have DX = 12 (EX; X  − EX; EX )

  ∞ ∞ ∞ 1  pk uk ; uk  − pk pj uk ; uj  : = 2 k=1

k=1 j=1

Y. Feng et al. / Fuzzy Sets and Systems 120 (2001) 487–497

Example 3.7. Let  x−'     (     1 X (!) =  '+(+,−x     ,     o

if '¡x¡' + (;

(iii) S 2 is an unbiased estimator of the variance &2 of X . In fact, by Theorem 3.5(iii) and (iv),

if x = ' + (;

E XP =

if ' + (¡x¡' + ( + ,; elsewhere;

where '; ( and , are real-valued r.v.’s with (¿0 and ,¿0, a.s. Fix ! ∈ ; X (!) is a continuous triangle fuzzy number. It is easy to show that [X ]r = [' + r(; ' + ( + (1 − r),], for r ∈ (0; 1]. Therefore, D(X − (r)) = D' + r 2 D( + 2r Cov('; (); D(X + (r)) = D' + D( + (1 − r)2 D, + 2Cov('; () + 2(1 − r) Cov('; ,) + 2(1 − r) Cov((; ,): Hence from (3.5) we have that DX = D' + 23 D( + 16 D, + 32 Cov('; () + 12 Cov('; ,) + 12 Cov((; ,): If ( = ,, a.s., i.e. X (!) is a symmetric triangle fuzzy number, then DX = D' + 43 D( + 2Cov('; (): Example 3.8. A simple random sample X1 ; X2 ; : : : ; Xn is taken from an f.r.v. X with EX = u and DX = &2 , i.e. X1 ; X2 ; : : : ; Xn are independent and identically distributed with X . Let 1 XP = n S2 =

n

n

Xi ;

M2 =

i=1

493

1 2 d∗ (Xi ; XP ); 2n i=1

2

1 d2∗ (Xi ; XP ) 2(n − 1) i=1

denote the sample average, the sample second central moment and the sample variance, respectively. Then (i) XP converges in probability to u in d2 , as n → ∞. (ii) XP is the minimum variance linear unbiased estimator of the expected value u of X .

n

1 EXi = u; n i=1

n &2 1 DXi = DXP = 2 n n i=1

thus the Chebyshev inequality (Theorem 3.5(vi)) P yields that nin probability d2 (X ; u) → 0, as n → ∞. If Y = i=1 ai Xi is a linear unbiased n estimator of the expected value u of X , since i=1 ai = 1 and n DY = i=1 a2i &2 ¿(1=n)&2 = DX , hence (ii) holds. In order to show (iii), note that EXi ; Xi  = 2&2 + u; u and EXP ; XP  = (2&2 =n) + u; u from Theorem 3.5(i), then by (2.1), n

EM2 =

1 E (Xi ; Xi  − 2Xi ; XP  + XP ; XP ) 2n i=1

1 E = 2n



n

 Xi ; Xi  − XP ; XP 

i=1

=

n−1 2 & n

2

hence ES = E((n=n − 1)M2 ) = &2 . 4. Application in fuzzy stochastic processes In this section we discuss the linear theory of fuzzy stochastic processes. The results of Section 3 will play the leading role. Let T be a
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Y. Feng et al. / Fuzzy Sets and Systems 120 (2001) 487–497

correlation function of the f.s.p. X (t). If EX (t + 2); X (t) = B(2); t; t + 2 ∈ T , is independent of t, then X (t) is called wide-sense (or weakly) stationary f.s.p. Let (X (t); t ∈ T ) be a Gaussian f.s.p., from Theorem 5:3 of [4], for every t ∈ T , X (t) = EX (t) +  where ('(t); t ∈ T ) is a real-valued Gaussian '(t), [!) is the fuzzy number s.p. with mean zero and '(t; in E whose membership function equals 1 at '(t; !) and zero elsewhere. Denote (t; s) = E('(t)'(s)), the correlation function of '(t), then B(t; s) = 2(t; s) + EX (t); EX (s):

(4.1)

As in the case of real-valued s.p., a function B(t; s) de
B(ti ; tj )i j =

n

|B(t + 2) − B(t) | = |EX (t + 2); X (0) − EX (t); X (0)| 62D2 (X (t + 2); X (t)) X (0) 2 : Example 4.3 (Autoregressive sequence). Let N be the set of all integers. A fuzzy stochastic sequence (f.s.s., for short) (Xn ; n ∈ N ) ⊂ L2 satis
EX (ti ); X (tj )i j



i; j=1


=E

X (t + h ) → EX (t); X (t) = B(t; t), as h; h → 0. This condition, that is B(t + h; t + h ) → B(t; t), as h; h → 0 in any manner whatever, is just the requirement that B(t; s) is continuous at (t; t). Conversely, by EX (t + h); X (t + h ) → EX (t); X (t) and EX (t + h); X (t) → EX (t); X (t), as h; h → 0; we have D∗ (X (t + h); X (t)) → 0, as h → 0, so X (t) is m.s. continuous at t from (3.3). (ii) (a) ⇔ (b) ⇔ (c) follow (i). (d) ⇒ (b) is obvious. (a) ⇒ (d) follows (3.4), i.e.

1

 n

0

+E

EXn ; Xm  =

2 −

i X (ti ) (r)

dr

i=1


1  n 0

2 +

i X (ti ) (r)

dr¿0

i=1

and the suAciency is due to that we can construct a Gaussian f.s.p. such that its correlation function equals to B(t; s) by (4.1). Theorem 4.2 (Continuity in mean square criterion). (i) A second-order f.s.p. (X (t); t ∈ T ) is m.s. continuous at t if and only if B(t; s) is continuous at (t; t). (ii) Let (X (t); t ∈ T ) be a wide-sense stationary f.s.p. with correlation function B(2). The following conditions are equivalent. (a) X (t) is m.s. continuous ; (b) B(2) is continuous at 2 = 0; (c) X (t) is m.s. continuous at t = 0; (d) B(2) is continuous . Proof. (i) From Lemma 3.3(i), if D2 (X (t + h); X (t)) → 0, as h → 0, then B(t + h; t + h ) = EX (t + h);

1

if n = m;

0

if n = m:

Then (Xn ; n ∈ N ) is wide-sense stationary, we call it standardized f.s.s. sliding sum of an f.s.s.: De
(4.2)

for some real numbers 41 ; : : : ; 4s ; h0 ; h1 ; : : : ; hs . The mathematical model as (4.2) exists in many practical problems. It is easy to see that if we know the values of Y (0); : : : ; Y (n − 1); then from Eq. (4.2) we can express the values of Y (s); Y (s − 1); : : : one by one by the initial values Y (0); : : : ; Y (n − 1) and the given values X (0); X (1); : : : . Next we consider the existence of the wide-sense stationary solution Y (n) of Eq. (4.2) which is expressed by X (m); m6n.

Y. Feng et al. / Fuzzy Sets and Systems 120 (2001) 487–497

Set the sliding sum Y (n) =

∞

k X (n − k):

(4.3)

k=0

From (4.2) and (4.3), we can obtain the following simultaneous linear equations: 0 = h0 ; k =

(4.4)

k

4i k−i + hk

if 16k6s;

(4.5)

i=1

k =

s

4i k−i

if k¿s + 1:

(4.6)

i=1

(4.4) and (4.5) have the unique solution (0 ; 1 ; : : : ; s ) = (h0 ; h1 ; : : : ; hs )A;

(4.7)

where A = (ai; j ) is an upper triangular matrix which the diagonal elements of A all equal 1, when i¡j; ai; j is a polynomial of 41 ; : : : ; 4s and the coeAcient of its every term is 1. Using (4.6), we have   ∞ s s s 2 1−s 4i k2 6s 4i2 k2 ¡∞: i=1

s

k=s+1

The m.s. integral of X on an in
Next we discuss the criterions of mean-square integrability and mean-square diGerentiability for f.s.p.’s. In the setting of this paper, we introduce the concepts of mean-square integral and diGerential of f.s.p.’s slightly diGerent than those of [4]. Denition 4.4. Let X (t) be a second-order f.s.p. de
b

a




b

a

B(t; s) dt ds

(4.8)

exists and is 9nite, then the m.s. integral exists, where −∞6a¡b6 + ∞.

b a

X (t) dt

Proof. If [a; b] is a
 n

5ti X (ti );

=

m

 5tj X (tj )

j=i

i=1

k=1 k=s+i

Hence if i=1 4i2 ¡1=s and (4.7) is nonnegative, then the autoregressive equation (4.2) has the wide-sense stationary solution (4.3).

495

n m

B(ti ; tj )5ti 5tj

i=1 j=1

exists. By the de
b1

a1

=

=

X (t) dt;

lim E

|51 | →0 |52 | →0

a2

X (t) dt

 n

|51 | →0 |52 | →0 i=1 j=1 b2

a1




5ti X (ti );

i=1 m n

lim


=

b2

b2 a2

m

 5tj X (tj )

j=1

EX (ti ); X (tj )5ti 5tj

B(t; s) dt ds:

(4.9)

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Y. Feng et al. / Fuzzy Sets and Systems 120 (2001) 487–497

Using Lemma 3.3(ii) again we know that the conclusion also holds for an in
a

(ii) (Law of large numbers) If EX (t) = u ∈ E; for any T ∈ R, and B(t; s) is Riemann integrable, then
1 T X (t) dt → u; as T → ∞; T 0 the convergence being in D2 , if and only if T T (1=T ) 0 0 B(t; s) dt ds → u; u; as T → ∞:



1 T




T 0

 X (t) dt; u 

= lim E |5| →0

n 1 5ti X (ti ); u T

 = u; u;

i=1

and recall the de

T  1 D∗ X (t) dt; u T 0  =E




1 T

− 2E

1 T







1 T2

X (t) dt;

0



=

T

T 0




T 0

T 0

1 T




u1  v1 ; u2  v2  = u1 ; u2  − u1 ; v2  − v1 ; u2  + v1 ; v2 : (4.10) Because u  v : (E; d∞ ) × (E; d∞ ) → (E; d∞ ) is continuous, if X and Y are f.r.v.’s and the H-diGerence of X and Y exists a.s., then X  Y is an f.r.v. and X  Y ∈ L2 provided X; Y ∈ L2 . Denition 4.8. A second-order f.s.p. X (t); t ∈ T , is m.s. diGerentiable at t0 ∈ T if there exists an X  (t0 ) ∈ L2 such that the limits in D2 lim

h → 0+

X (t0 + h)  X (t0 ) h

and lim

h → 0+

X (t0 )  X (t0 − h) h

exist and equal X  (t0 ). At the end points of T we consider only the one-sided derivatives. If X (t) is m.s. diGerentiable at every t ∈ T then we call X (t) m.s. diGerentiable on T .

Proof. (i) Follows (4.9). (ii) Since by Lemma 3.3(i) E

Let u; v ∈ E. If there exists a w ∈ E such that u = v + w then we call w the H-diGerence of u and v, denoted by u  v. It is easy to verify that if u1  v1 and u2  v2 exist, then

T 0

 X (t) dt

 X (t) dt; u + u; u

B(t; s) dt ds − u; u:

This and (3.3) prove the corollary.

Theorem 4.9 (DiGerentiation in mean square criterion). Let (X (t); t ∈ R) be a second-order f.s.p. with correlation function B(t; s) and the H-di;erence X (t + h)  X (t) exists, a.s., for all t ∈ R and h¿0. (i) If the second generalized derivative @2 B(t; s)= @t@s |t=s=t0 exists, where  @2 B(t; s)  @t@s t=s=t0 = lim


h; h →0

1 (B(t0 + h; t0 + h ) hh

− B(t0 ; t0 + h ) − B(t0 + h; t0 ) + B(t0 ; t0 )); then X (t) is m.s. di;erentiable at t0 . (ii) If @2 B(t; s)=@t@s |t=s=t0 exists for all t0 ∈ R; then X (t) is m.s. di;erentiable and the correlation function of (X  (t); t ∈ R) equals @2 B(t; s)=@t@s.

Y. Feng et al. / Fuzzy Sets and Systems 120 (2001) 487–497

Proof. (i) Let h¿0; h ¿0; from (4.10) we have   1 1 E (X (t0 + h)  X (t0 ));  (X (t0 + h )  X (t0 )) h h =

1 (B(t0 + h; t0 + h ) − B(t0 ; t0 + h ) hh − B(t0 + h; t0 ) + B(t0 ; t0 ));

 E

 1 1  (X (t0 )  X (t0 − h));  (X (t0 )  X (t0 − h )) h h

=

1 (B(t0 − h; t0 − h ) − B(t0 ; t0 − h ) hh − B(t0 − h; t0 ) + B(t0 ; t0 )):

Hence using the condition of (i) and Lemma 3.3(ii), we have that X (t) is m.s. diGerentiable at t0 . (ii) By Lemma 3.3(i) and similar to the proof of (i), we obtain that BX
(t; s) = EX  (t); X  (S)  1 = lim E (X (t + h)  X (t)); h; h
→0 h  1  (X (s + h )  X (s)) h =

@2 B(t; s) : @t@s

497

Remark 4.10. Similar to [4] we can also discuss many properties of m.s. integral and m.s. diGerential for a second-order f.s.p. For example, the Newton–Leibniz formula, the m.s. calculus properties of a Gaussian f.s.p., etc. Except slightly diGerence, the argument is same as in [4]. Acknowledgements The author would like to thank the referees for valuable comments and suggestions. References [1] P. Diamond, P. Kloeden, Metric space of fuzzy sets, Fuzzy Sets and Systems 35 (1990) 241–249. [2] P. Diamond, P. Kloeden, Metric space of fuzzy sets: corrigendum, Fuzzy Sets and Systems 45 (1992) 123. [3] Y. Feng, Convergence theorems for fuzzy random variables and fuzzy martingales, Fuzzy Sets and Systems 103 (1999) 435–441. [4] Y. Feng, Mean-square integral and diGerential of fuzzy stochastic processes, Fuzzy Sets and Systems 102 (1999) 271–280. [5] M. Ma, On embedding problems of number spaces: part 4, Fuzzy Sets and Systems 58 (1993) 185–193. [6] M.L. Puri, D.A. Ralescu, Fuzzy random variables, J. Math. Anal. Appl. 114 (1986) 409– 422. [7] Z. Wang, A remark on the condition of integrability in quadratic mean for their second-order random processes, Chinese Ann. Math. 3 (1982) 349–352 (in Chinese).