A Generic Central Limit Theorem
In this section I will explain McLeish’s (1974) central limit theorems for dependent random variables with an emphasis on stationary martingale difference processes.
The following approximation of exp(i ■ x) plays a key role in proving central limit theorems for dependent random variables.
Lemma 7.1: For x є К with x < 1, exp(i ■ x) = (1 + i ■ x)exp(x2/2 + r(x)), where r(x) < x3.
Proof: It follows from the definition of the complex logarithm and the series expansion of log(1 + i ■ x) for x  < 1 (see Appendix III) that
TO
log(1 + i ■ x) = i ■ x + x1 /2 + ^2(1)k1ikxk/k + i ■ m ■ n
k=3
= i ■ x + x2/2 — r(x) + i ■ m ■ n,
where r(x) = — Eь=з(— 1)k— 1ikxk/k. Taking the exp of both sides of the equation for log(1 + i ■ x) yields exp(i ■ x) = (1 + i ■ x) exp(—x2/2 + r(x)). To prove the inequality r(x) < x 3, observe that
TO TO
r (x) = – (—1)k—1ikxk / k = x 3J2(— 1)kik+1 xk/(k + 3)
k=3 k=0
TO
= x3 2(— 1)2ki2k+1 x2k/(2k + 3)
k=0
TO
+ x3 ^( 1)2k+1i2k+2x2k+1/(2k + 4)
k=0
= x3£(1)kx2k+1/(2k + 4) + i ■ x3£(1)kx2k/(2k + 3)
k=0 k=0
TOTO
= J2(1)kx2k+4/(2k + 4) + i J2(1)kx2k+3/(2k + 3)
k=0
where the last equality in (7.37) follows from
d TO TO
— J2 (1)kx 2k+4/(2k + 4) = J2 (1)kx 2k+3
k0
The theorem now follows from (7.37) and the easy inequalities
x 3 x 1
/ 1 + y2 dy < f y3—y = 4 x 4 < x 3h/2
x 2 x l
f J+y2 йУ < f y2dy = 3 lx I3 < lx 3Д/2,
which hold for x  < 1. Q. E.D.
The result of Lemma 7.1 plays a key role in the proof of the following generic central limit theorem:
Lemma 7.2: Let Xt, t = 1, 2,…, n,… be a sequence of random variables satisfying the following four conditions:
plim max XtI/Vn = 0, (7.38)
п^ж 1<t <n
plim(1/n)^2X2t = о2 є (0, ж),
t=1
and
Then
Proof: Without loss of generality we may assume that о2 = 1 because, if not, we may replace Xt by Xt/о. It follows from the first part of Lemma 7.1 that
Condition (7.39) implies that






















n
Zn(f) = exp(f2/2) – exp (f2/2)(1/n)J^ X) j
x ex^^ r(f Xt/Vn^ ^p 0.
Because  Zn (f ) < 2 with probability 1 given that  exp( x2/2 + r(x)) < 1,
it follows from (7.49) and the dominatedconvergence theorem that lim E [Zn(f )2] = 0.
Moreover, condition (7.41) implies (using zw = z ■ w and z = Vzz) that
2
ҐІ(1 + І f Xt/Vn)
t = 1
Therefore, it follows from the CauchySchwarz inequality and (7.51) and (7.52) that
Finally, it follows now from (7.40), (7.48), and (7.53) that
Because the righthand side of (7.54) is the characteristic function of the N(0, 1) distribution, the theorem follows for the case a2 = 1 Q. E.D.
Lemma 7.2 is the basis for various central limit theorems for dependent processes. See, for example, Davidson’s (1994) textbook. In the next section, I will specialize Lemma 7.2 to martingale difference processes.
7.5.3. Martingale Difference Central Limit Theorems
Note that Lemma 7.2 carries over if we replace the Xt’s by a double array Xn, t, t = 1, 2,…,n, n = 1, 2, 3, In particular, let
Yn, i = Xi,
Yn, t = XtI (1/n) YX2 < a2 + ij for t > 2. (7.55)
Then, by condition (7.39),
n
P[Yn, t = Xt for some t < n] < P [(1/n) ^ Xf > a2 + 1] ^ 0;
t=1
(7.56)
hence, (7.42) holds if
1n
^Y Yn, t ^d N(0, a2). (7.57)
Therefore, it suffices to verify the conditions of Lemma 7.2 for (7.55).
First, it follows straightforwardly from (7.56) that condition (7.39) implies
n
plim(1/n)Y Ylt = a2. (7.58)
t=1
Moreover, if Xt is strictly stationary with an a mixing base and E [X2] = a2 є (0, то), then it follows from Theorem 7.7 that (7.39) holds and so does (7.58).
Next, let us have a closer look at condition (7.38). It is not hard to verify that, for arbitrary є > 0,
(7.59)
Hence, (7.38) is equivalent to the condition that, for arbitrary є > 0,
n
(1/n)Y X21 (I Xt  > є JR) ^ p 0. (7.60)
t=1
Note that (7.60) is true if Xt is strictly stationary because then
where
Hence,
Thus, (7.63) is finite if supn>1 (1 /n)J2’t=1 E[Xj] < to, which in its turn is true if Xt is covariance stationary.
Finally, it follows from the law of iterated expectations that, for a martingale difference process Xt, E [ПП= 1 (1 + i*Xt/^n)] = E [П”= 1 (1 +
ifE[Xt.1ІД/Й)] = 1, Vf є К, and therefore also E[П"=і(1 +
if Y„,t Д/й)] = E[ПП=і(1 + if E[Y„,t.tі]А/и)] = 1, Vf є К.
We can now specialize Lemma 7.2 to martingale difference processes:
Theorem 7.10: Let Xt є К be a martingale difference process satisfying the following three conditions:
(a) (1/n)J2n=1 Xf ^p a2 є (0, то);
(b) For arbitrary є > 0, (1/n)YTt=1 X21 (Xt > ejn) ^ p 0;
(c) suPn>l(1/n)JTt=l E[Xf] < то.
Then, (1/Vn) m=1 Xt ^d N(0, a2).
Moreover, it is not hard to verify that the conditions of Theorem 7.10 hold if the martingale difference process Xt is strictly stationary with an аmixing base and E[Xf] = a2 є (0, то):
Theorem 7.11: Let Xt є К be a strictly stationary martingale difference process with an аmixing base satisfying E[X2] = a2 є (0, то). Then (1/Vn)En=1 Xt ^dN(0, a2).
1. Let U and V be independent standard normal random variables, and let Xt = U ■ cos(Xt) + V ■ sin(Xt) for all integers t and some nonrandom number X є (0, n). Prove that Xt is covariance stationary and deterministic.
2. Show that the process Xt in problem 1 does not have a vanishing memory but that nevertheless plim^^ (1 fn)^f П= 1 Xt = 0.
3. Let Xt be a time series process satisfying E[ Xt ] < то, and suppose that the events in the remote a – algebra.то = ПТО0a (Xt, Xt1, Xt2,…) have either probability 0 or 1. Show that P(E[Xt .то] = E[Xt]) = 1.
4. Prove (7.30).
5. Prove (7.31) by verifying the conditions on Theorem 7.8(b) for gt(в) = (Yt – ft (в))2 with Yt defined by (7.18) and ft (в) by (7.26).
6. Verify the conditions of Theorem 7.9 for gt (в) = (Yt – ft (в ))2 with Yt defined by (7.18) and ft(в) by (7.26).
7. Prove (7.50).
8. Prove (7.59).
APPENDIX
7.A.1. Introduction
In general terms, a Hilbert space is a space of elements for which properties similar to those of Euclidean spaces hold. We have seen in Appendix I that the Euclidean space Rn is a special case of a vector space, that is, a space of elements endowed with two arithmetic operations: addition, denoted by “+,” and scalar multiplication, denoted by a dot. In particular, a space V is a vector space if for all x, y, and z in Vand all scalars c, ci, and c2,
(a) x + y = y + x;
(b) x + (y + z) = (x + y) + z;
(c) There is a unique zero vector 0 in V such that x + 0 = x;
(d) For each x there exists a unique vector – x in V such that x + (—x) = 0;
(e) 1 ■ x = x;
(f) (C1C2) ■ x = C1 ■ (c2 ■ x);
(g) c ■ (x + y) = c ■ x + c ■ y;
(h) (c1 + c2) ■ x = c1 ■ x + c2 ■ x.
Scalars are real or complex numbers. If the scalar multiplication rules are confined to real numbers, the vector space Vis a real vector space. In the sequel I will only consider real vector spaces.
The inner product of two vectors x and y in Rn is defined by xTy. If we
denote (x, y) = xTy, it is trivial that this inner product obeys the rules in the
more general definition of the term:
Definition 7.A.1: An inner product on a real vector space V is a real function (x, y) on V x Vsuch that for all x, y, z in V and all c in R,
(1) (x, y) = (y, x);
(2) (cx, y) = c(x, y);
(3) (x + y, z) = (x, z) + (y, z);
(4) (x, x) > 0 when x = 0.
A vector space endowed with an inner product is called an innerproduct space. Thus, Rn is an innerproduct space. In Rn the norm of a vector x is defined by x II = VxTx. Therefore, the norm on a real innerproduct space is defined similarly as x  = s/(x, x). Moreover, in Rn the distance between two vectors x and y is defined by x – y  = у/(x — y)T(x – y). Therefore, the distance between two vectors x and y in a real innerproduct space is defined similarly as x — y\ = V(x — y, x — y). The latter is called a metric.
An innerproduct space with associated norm and metric is called a pre – Hilbert space. The reason for the “pre” is that still one crucial property of Rn is missing, namely, that every Cauchy sequence in Rn has a limit in Rn.
Definition 7.A.2: A sequence of elements Xn of an innerproduct space with associated norm and metric is called a Cauchy sequence if, for every є > 0, there exists an n0 such that for all k, m > n0, \xk — xm  < є.
Theorem 7.A.1: Every Cauchy sequence in Шг,£ < то has a limit in the space involved.
Proof: Consider first the case R. Let x = limsup„^TOxn, where xn is a Cauchy sequence. I will show first that x < то.
There exists a subsequence nk such that x = limk^TOx„t. Note that xnk is also a Cauchy sequence. For arbitrary є > 0 there exists an index k0 such that x„k — x„m  < є if k, m > k0. If we keep k fixed and let m ^то, it follows that x„k — x  < є; hence, x < то, Similarly, x = liminf„^TOxn > —то. Now we can find an index k0 and subsequences nk and nm suchthatfor k, m > k0, x„k — x  < є, lx„m — x  < є, and x„k — x„m  < є; hence, x — x  < 3є. Because є is arbitrary, we must have x = Jc = limй^тоxn. If we apply this argument to each component of a vectorvalued Cauchy sequence, the result for the case Re follows. Q. E.D.
For an innerproduct space to be a Hilbert space, we have to require that the result in Theorem 7.A1 carry over to the innerproduct space involved:
Definition 7.A.3: A Hilbert space H is a vector space endowed with an inner product and associated norm and metric such that every Cauchy sequence in H has a limit in H.
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