Note on the Concept of m-Dependence

✍ Scribed by Peter Schatte

• Publisher: John Wiley and Sons
• Year: 1989
• Tongue: English
• Weight: 106 KB
• Volume: 140
• Category: Article
• ISSN: 0025-584X
• DOI: 10.1002/mana.19891400117

✦ Synopsis

Let XI, Xz, ... be a sequence of r.v.'s and assume that the vectors (XI, ..., X r ) and (X,, ..., X t ) are independent for 1 s r -c s s t , s-rwm. Then the sequence (X,) is called na-dependent. For ?n = 0 we obtain independence. But also for m w 0 many limit theorems hold as in the case of independence, cf. HEINRICH [2] and further papers of the same and other authors.

In the following the concept of m-dependence shall be weakened. 'A sequence XI, &, ... is called weakly m-dependent if X, is independent of (XI, ..., Xr) for swr+m. For m=O one again obtains independence. But if m=-O the central Zimit theorem does not hold for weakly m-dependent r.v.'s. The aim of the present uhort note is to give an example for this fact. This example contradicts Theorem 7.3.1

in CHUNG [I]. The proof of this theorem contains the erroneous conclusion that the independence of X and 2 on the one hand and the independence of Y and Z on the other hand together imply the independence of X + Y and 2.

Let Y , Y1, Y z , ... be a sequence of independent r.v.'s being uniformly distributed on [O, 1). We put Xri-1= Y + Yt (mod 1) and Xz, = Y -Yi (mod I), 0 ~Xzi-1, Xzi-cl, i=1, 2 , ... Since for real y€[O, 1) the r.v. y + Yr (mod 1) is uniformly distributed independent of the actual value of y, the r.v. Xar-1 is independent of Y , cf. also Lemma 1 in SCHATTE [4]. But (Xzt-1, Y ) is independent of (YI, ..., Yi-I), and thus X2i-1 is independent of ( Y , Y l , ..., Yt-1) and consequently independent of XI, ..., Xzi-2. Since we can analogously conclude for S Z i , the sequence XI, Xz, ...