geometric equivalences to the asymptotic normality condition
GEOMETRIC
EQUIVALENCES TO THE ASYMPTOTIC NORMALITY CONDITION
by Galen
TECHNICAL
R.
Shorack
REPORT
November
Department Box University Seattle,
of
No.
336
1998
Statistics
354322 of
Washington
Washington
98195
USA
GEOMETRIC EQUIVALENCES TO THE ASYMPTOTIC NORMALITY CONDITION G. R. SHORACK
Department of Statsiiics, University of Washington, Seattle, WA 98195, USA. Abstract: The asymptotic normality of the sample mean of iid rv 's is equivalent to the well known conditions of Levy or Feller. More recently, additional equivalences have been developed in terms of the quantile function (qf). And other useful probabilistic equivalences could be cited. The emphasis here is on equivalences, and many other useful and informative equivalences will be developed. Many depend only on simple comparisons of areas, and those are the ones that will be developed herein. [Because the asymptotic normality above is equivalent to appropriately phrased consistency of the sample second moment, many additional equivalences can be developed in the simpler context of the weak law of large numbers. But this will be done elsewhere.] Roughly, one can learn all about asymptotic normality by studying the conditions in simpler settings. Here, we do the geometric part.
O.
Introduction
, ... , ... be iid with distribution function (df) F(·) and quantile function 1 K (.) == F- (-). The classical central limit theorem (CLT) states that when this distribution has finite variance, then == y'n[.¥n - 1tJ/(1 -7d N(O, 1). When (12 is finite, it holds that
(0.1)
t [K;(ct) V K 2 (1
ct)] /
(12 -7
0
as t
-7
0,
for each fixed
c>
O.
But what if (12 is infinite? Let a> 0 be tiny. We agree that dom(a, a) denotes [0,1 a), (a, 1] or (a, 1 - a) according as X 2:: 0, X ~ 0 or general X, and that I 0, always) it becomes the well known equivalence (1.29) again. Maller(1979) also obtained the result of Feller's problem, and discussed it for X 2 in Maller(1980). See also Raikov(1938). [See O'Brien(1980), S. Csorgo and Mason(1989), and Gotze and Gine(l995) for equivalences phrased in terms of the negligibility of the maximal summand.] Thus the ANC can be studied in the context of the simpler WLLN problem, and this is therefore a preferable context in which to study and develop equivalences. Then, the most convenient equivalence is available for the easiest possible proof, while all equivalences are available for applications and understanding. The full list of informative equivalences obtained is large. Related work: The reader is also referred to Shorack(l998b) where the consistency of sample moments is studied in the possibly infinite moment case for non iid rv's (conditions reduce to (0.4) in the iid case). While in this simpler context, many other equivalences are developed (probabilistic conditions, not geometric). And the reader is referred to Shorack(1998c) where quite general necessary and sufficient conditions for normality are given (the non iid case is covered with necessary conditions and with sufficient conditions, and these both reduce to (0.4) in the iid case). Both papers put emphasis on the fact that many equivalences are possible. Having them available offers insight and makes other proofs simpler. point of view and notation of Shorack{1998b,c) are specifically geared to that of the current paper, and so papers seem natural to mention next theorem is a teaser {given in a
J
's In
row mdependent.,
F.
that L = log(I/t) and lex) = log(x) are slowly varying. They are the prototypes. Note also that when 0- 2 is finite, the Winsorized variance function ij2(t) is always slowly varying, -t 0- 2/0- 2 = 1 for each c > O. since Infinite variance facts. Whenever the variance in infinite, the Winsorized variance ij2 (a) completely dominates the square j12(a) of the Winsorized mean. Let ka,a' denote I< Winsorized outside (a, I-a l ) . Gnedenko-Kolmogorov(I954, p.I83) showed (0.7) below, while (0.8) and (0.9) are then trivial. For every non degenerate qf I< having EI
x)j f[o.x] yr dF(y) ,
lim sup tIC'(lt-+O
X-t-OO
'
for each r > O. The same is true for the lim inf , and for the lim (if it exists). By change of variable
Proof.
(2.2)
f[o,F(xlJ [(I' (s)
Define t = 1 (a)
r(t)
ds
f[o,x]
yr dF(y)
for all x.
F(x). Then
tI are equal at the associated pairs of end points re any flat spot of F or any flat spot of K; and are monotone across flat spots. At all non flat spot pairs of points, they are equal. That at all of the key values (i.e., local extremes) the quantities in question are equal. 0
References
[1] Csorgo, S., Haeusler, E. and Mason, D. (1988). The asymptotic distribution of trimmed sums. Ann. Probability 16, 672-699. [2] Csorgo, S., Haeusler, E. and Mason, D. (1989). A probabilistic approach to the asymptotic distribution of sums of independent identically distributed random variables. Adv. in Appl. Math. 9, 259-333. [3] Csorgo.S. and Mason, D. (1989). Bootstrapping empirical functions. Ann. Statist. 17 1447-1471. [4] Feller, W. (1966). An Introduction to Probability Theory and It's Applications. Vol. 2. John Wiley and Sons, NY.
[.5] Gine, E. and Gotze, F. (1995). On the domain of attraction to the normal law and convergence of selfnormalized sums. Universitiit of Bielefeld Mathematics Technical Report 9.5-99. [6] Gine, E., Gotze, F. and Mason, D. (1998). When is the student t-statistic asymptotically standard normal? Ann. Probability 25 1.514-1531.
[7] Gnedenko, B. and Kolmogorov, A. (19.54). Limit Distributions for Sums of Independent Random Variables. Addison-Wesley, Cambridge, Mass. [8] Maller, R. (1979). Relative stability, characteristic functions and stochastic compactness. J. Austral. Math. Soc. Ser. A 28,499-509. [9] Maller, R. (1980). On one-sided boundedness of normal partial sums. Bull. Austral. Math. Soc. 21, 37:3-391. O'Brien, G. (1980). A limit theorem for sample maxima and heavy branches in Galtonwatson trees. J. Appl. Probab. 17 539-54.5.
EQUIVALENCES TO THE ASYMPTOTIC NORMALITY CONDITION
by Galen
TECHNICAL
R.
Shorack
REPORT
November
Department Box University Seattle,
of
No.
336
1998
Statistics
354322 of
Washington
Washington
98195
USA
GEOMETRIC EQUIVALENCES TO THE ASYMPTOTIC NORMALITY CONDITION G. R. SHORACK
Department of Statsiiics, University of Washington, Seattle, WA 98195, USA. Abstract: The asymptotic normality of the sample mean of iid rv 's is equivalent to the well known conditions of Levy or Feller. More recently, additional equivalences have been developed in terms of the quantile function (qf). And other useful probabilistic equivalences could be cited. The emphasis here is on equivalences, and many other useful and informative equivalences will be developed. Many depend only on simple comparisons of areas, and those are the ones that will be developed herein. [Because the asymptotic normality above is equivalent to appropriately phrased consistency of the sample second moment, many additional equivalences can be developed in the simpler context of the weak law of large numbers. But this will be done elsewhere.] Roughly, one can learn all about asymptotic normality by studying the conditions in simpler settings. Here, we do the geometric part.
O.
Introduction
, ... , ... be iid with distribution function (df) F(·) and quantile function 1 K (.) == F- (-). The classical central limit theorem (CLT) states that when this distribution has finite variance, then == y'n[.¥n - 1tJ/(1 -7d N(O, 1). When (12 is finite, it holds that
(0.1)
t [K;(ct) V K 2 (1
ct)] /
(12 -7
0
as t
-7
0,
for each fixed
c>
O.
But what if (12 is infinite? Let a> 0 be tiny. We agree that dom(a, a) denotes [0,1 a), (a, 1] or (a, 1 - a) according as X 2:: 0, X ~ 0 or general X, and that I 0, always) it becomes the well known equivalence (1.29) again. Maller(1979) also obtained the result of Feller's problem, and discussed it for X 2 in Maller(1980). See also Raikov(1938). [See O'Brien(1980), S. Csorgo and Mason(1989), and Gotze and Gine(l995) for equivalences phrased in terms of the negligibility of the maximal summand.] Thus the ANC can be studied in the context of the simpler WLLN problem, and this is therefore a preferable context in which to study and develop equivalences. Then, the most convenient equivalence is available for the easiest possible proof, while all equivalences are available for applications and understanding. The full list of informative equivalences obtained is large. Related work: The reader is also referred to Shorack(l998b) where the consistency of sample moments is studied in the possibly infinite moment case for non iid rv's (conditions reduce to (0.4) in the iid case). While in this simpler context, many other equivalences are developed (probabilistic conditions, not geometric). And the reader is referred to Shorack(1998c) where quite general necessary and sufficient conditions for normality are given (the non iid case is covered with necessary conditions and with sufficient conditions, and these both reduce to (0.4) in the iid case). Both papers put emphasis on the fact that many equivalences are possible. Having them available offers insight and makes other proofs simpler. point of view and notation of Shorack{1998b,c) are specifically geared to that of the current paper, and so papers seem natural to mention next theorem is a teaser {given in a
J
's In
row mdependent.,
F.
that L = log(I/t) and lex) = log(x) are slowly varying. They are the prototypes. Note also that when 0- 2 is finite, the Winsorized variance function ij2(t) is always slowly varying, -t 0- 2/0- 2 = 1 for each c > O. since Infinite variance facts. Whenever the variance in infinite, the Winsorized variance ij2 (a) completely dominates the square j12(a) of the Winsorized mean. Let ka,a' denote I< Winsorized outside (a, I-a l ) . Gnedenko-Kolmogorov(I954, p.I83) showed (0.7) below, while (0.8) and (0.9) are then trivial. For every non degenerate qf I< having EI
x)j f[o.x] yr dF(y) ,
lim sup tIC'(lt-+O
X-t-OO
'
for each r > O. The same is true for the lim inf , and for the lim (if it exists). By change of variable
Proof.
(2.2)
f[o,F(xlJ [(I' (s)
Define t = 1 (a)
r(t)
ds
f[o,x]
yr dF(y)
for all x.
F(x). Then
tI are equal at the associated pairs of end points re any flat spot of F or any flat spot of K; and are monotone across flat spots. At all non flat spot pairs of points, they are equal. That at all of the key values (i.e., local extremes) the quantities in question are equal. 0
References
[1] Csorgo, S., Haeusler, E. and Mason, D. (1988). The asymptotic distribution of trimmed sums. Ann. Probability 16, 672-699. [2] Csorgo, S., Haeusler, E. and Mason, D. (1989). A probabilistic approach to the asymptotic distribution of sums of independent identically distributed random variables. Adv. in Appl. Math. 9, 259-333. [3] Csorgo.S. and Mason, D. (1989). Bootstrapping empirical functions. Ann. Statist. 17 1447-1471. [4] Feller, W. (1966). An Introduction to Probability Theory and It's Applications. Vol. 2. John Wiley and Sons, NY.
[.5] Gine, E. and Gotze, F. (1995). On the domain of attraction to the normal law and convergence of selfnormalized sums. Universitiit of Bielefeld Mathematics Technical Report 9.5-99. [6] Gine, E., Gotze, F. and Mason, D. (1998). When is the student t-statistic asymptotically standard normal? Ann. Probability 25 1.514-1531.
[7] Gnedenko, B. and Kolmogorov, A. (19.54). Limit Distributions for Sums of Independent Random Variables. Addison-Wesley, Cambridge, Mass. [8] Maller, R. (1979). Relative stability, characteristic functions and stochastic compactness. J. Austral. Math. Soc. Ser. A 28,499-509. [9] Maller, R. (1980). On one-sided boundedness of normal partial sums. Bull. Austral. Math. Soc. 21, 37:3-391. O'Brien, G. (1980). A limit theorem for sample maxima and heavy branches in Galtonwatson trees. J. Appl. Probab. 17 539-54.5.
