Journal of Statistical Research ISSN 0256 - 422 X 2010, Vol. 44, No. 2, pp. 289-292
Bangladesh
ON NORMAL CONVERGENCE CRITERIA FOR SUMS OF ROW-WISE INDEPENDENT RANDOM VARIABLES
Mark Inlow
Rose-Hulman Institute of Technology, CM 143, 5500 Wabash Avenue Terre Haute, IN 47803, USA
Email: [email protected]
summary
In this article we present new criteria for the asymptotic normality of sums of row- wise independent random variables. Besides providing an alternate approach for demonstrating a sum of independent random variables is asymptotically normal, our criteria provide new insight into the nature of asymptotic normality and the Lindeberg condition.
Keywords and phrases: Central limit theorem, infinitesimal array, Lindeberg con- dition
AMS Classification: 60F05
1 Introduction
Classic criteria ensuring the asymptotic normality of sums of row-wise independent random variables (RV’s) were determined early in the 20th century; see Le Cam (1986) for historical details. More recently, alternative criteria have been developed in terms of the first four moments, Kruglov (2004), and Lyapunov’s condition, Kruglov (2008). Here we present new criteria for asymptotic normality based on L´evy’s splitting of a RVX according to the size of its absolute value, Le Cam (1986): for any >0,
X =XI(|X|< ) +XI(|X| ≥) (1.1) whereI(·) is the indicator functionI(A) = 1,Atrue; 0 otherwise. In addition to providing a new means of establishing the asymptotic normality of a sum of row-wise independent RV’s, our criteria provide an alternate interpretation of asymptotic normality and, in particular, the Lindeberg condition.
c Institute of Statistical Research and Training (ISRT), University of Dhaka, Dhaka 1000, Bangladesh.
290 Inlow
2 Normal Convergence Criteria
Let Xnj, j = 1, . . . , kn, limnkn = ∞, be an array of row-wise independent RV’s which is infinitesimal, i.e., for any >0 limnmax1≤j≤knP(|Xnj| ≥) = 0. (Here and in the sequel all limits are for n→ ∞.) Classic normal convergence criteria for Yn = Pkn
j=1Xnj are provided by Lo`eve (1977):
Theorem 1(Classic Normal Convergence Criteria). LetXnj be an infinitesimal array of row-wise independent RV’s. Yn =Pkn
j=1Xnj converges in distribution to a normal RV with finite meanαand varianceσ2 (Yn→dN(α, σ2))if and only if for every >0and a τ >0, I. limnPkn
j=1P(|Xnj| ≥) = 0,
II. limnαn(τ) =αwhereαn(τ) =Pkn
j=1αnj(τ)andαnj(τ) = E[XnjI(|Xnj|< τ)], III. limnσ2n(τ) =σ2whereσn2(τ) =Pkn
j=1σ2nj(τ)andσ2nj(τ) = E[Xnj2 I(|Xnj|< τ)]−α2nj(τ).
Further, under I, II and III hold for anyτ0 in(0, τ].
The following theorem provides our new criteria based on 1.1.
Theorem 2(Alternate Normal Convergence Criteria). LetXnj be an infinitesimal array of row-wise independent RV’s. Yn=Pkn
j=1Xnj →dN(α, σ2)if and only if there is a τ >0 such that for all ∈(0, τ],
A. Wn() =Pkn
j=1XnjI(|Xnj| ≥)converges in probability to zero (Wn→p0), B. limnαn() =α, and
C. limnσ2n() =σ2.
Proof. SupposeYn →d N(α, σ2). By Theorem 1, there is aτ >0 such thatB and C hold for all∈(0, τ]. To show thatAholds, we note that for anyδ >0
P(|Wn()| ≤δ) ≥ P(Wn() = 0)
≥ P kn
\
j=1
{|Xnj|< }
= 1−P kn
[
j=1
{|Xnj| ≥}
≥ 1−
kn
X
j=1
P(|Xnj| ≥).
Thus by I of Theorem 1, limnP(|Wn()| ≤δ)≥1−limnPkn
j=1P(|Xnj| ≥) = 1 for any >0.
On Normal Convergence Criteria . . . 291
Conversely, suppose that for some τ > 0, A, B, and C hold for ∈ (0, τ]. By 1.1 Xnj =Vnj() +Wnj(), whereVnj() =XnjI(|Xnj|< ) and Wnj() =XnjI(|Xnj| ≥), so that Yn=Vn() +Wn() where Vn() =Pkn
j=1Vnj() and Wn() =Pkn
j=1Wnj(). Hence- forth we suppress in our notation. By A, Wn →p 0 for ∈(0, τ]. Thus, by Slutsky’s Theorem,Yn →dN(α, σ2) ifVn→dN(α, σ2). LetUnj= (Vnj−E[Vnj])/σn so thatUnj is a row-wise independent array, E[Unj] = 0, andPkn
j=1Var(Unj) = 1. By Lyapunov’s Theorem, Pkn
j=1Unj =Un→dN(0,1) if limnPkn
j=1E[|Unj|3] = 0. Since |Vnj|< , |Vnj−E[Vnj]| ≤2 so that
|Vnj−E[Vnj]|3 = |Vnj−E[Vnj]||Vnj−E[Vnj]|2
≤ 2|Vnj−E[Vnj]|2. Thus
kn
X
j=1
E[|Unj|3] = 1 σn3
kn
X
j=1
E[|Vnj−E[Vnj]|3]
≤ 2
σn3
kn
X
j=1
E[(Vnj−E[Vnj])2]
= 2
σn3
kn
X
j=1
σnj2 .
Therefore limnPkn
j=1E[|Unj|3] ≤ 2/σ by C. Since this inequality holds for all in (0, τ], limnPkn
j=1E[|Unj|3] = 0.
3 Remarks
If we interpret XnjI(|Xnj|< ) and XnjI(|Xnj| ≥) as constituting the central and tail contributions, respectively, ofXnj then our normality criteria admit the following intuitive interpretation: A sum of asymptotically negligible, independent RV’sYn=PXnjis asymp- totically normal if and only if theXnj are such that the sum of their tail contributions,Wn, is asymptotically negligible and the mean and variance of the sum of their central contribu- tions,Vn, converge. Further, our criteria provide new insight into the Lindeberg condition, a construct admitting various characterizations; see, for example, Goldstein (2009). Suppose thatX1, . . . , Xn is a sequence of RV’s with finite variancesσ2j. Lets2n=Pn
j=1σj2. If theXj
satisfy the uniformly asymptotically negligible (UAN) condition limnmaxjσj2/s2n= 0 then the normalized sumYn=Pn
j=1Xj/snis asymptotically normal if and only if the Lindeberg condition is satisfied: for any >0
limn (1/s2n)
n
X
j=1
E[Xj2I(|Xj| ≥sn)] = lim
n n
X
j=1
E[(Xj/sn)2I(|Xj/sn| ≥)]
= 0.
292 Inlow
Since the Xj are UAN, the normalized summands ˜Xnj =Xj/sn are infinitesimal. Thus, by Theorem 2 the Lindeberg condition is equivalent to the requirement that for every >0 Wn=Pn
j=1X˜njI(|X˜nj| ≥)→p0.
Acknowledgements
We thank Emanuel Parzen and an anonymous referee for their helpful comments.
References
[1] Goldstein, L. (2009). A probabilistic proof of the Lindeberg-Feller central limit theorem.
The American Mathematical Monthly, 116, 45-60.
[2] Le Cam, L. (1986). The central limit theorem around 1935.Statistical Science,1, 78-96.
[3] Lo`eve, M. (1977).Probability Theory, 4th ed., vol. 1. Springer-Verlag, New York.
[4] Kruglov, V. M. (2004). Normal and Poisson convergences. Theory of Probability and Its Applications,48, 347-355.
[5] Kruglov, V. M. (2008). On the necessity of the Lyapunov condition for normal conver- gence.Theory of Probability and Its Applications,52, 164-166.
