Haan, L.F.M. (Laurens) de

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STICHTING MATHEMATISCH CENTRUM 2e BOERHAAVESTRAAT 49 AMSTERDAM AFDELING MATHEMATISCHE STATISTIEK S 410 Quantiles and stabilizing constants by L. de Haan August 1969 -1-O. Introduction and Summarz. A sequence of distribution functions {F} belongs to the domain of attraction of a non-degenerate n distribution function G ( notation {F }G.D(G)) when it is possible n to choose sequences {a} (a > O, n = 1,2,3, ••• ) and {b} such that n n n ( 1 ) F (a X + b ) ➔ G(x) n n n in the weak sense. A well-known theorem of Gnedenko states to which extent we may change the sequences of stabilizing constants. He give the theorem in its extended form (see [1}, p.246). Theorem 1. If (1) holds, we have ( 2) F (ax+ B ) ➔ a*(x) n n n weakly (where a* is non-degenerate) iff (3) a 'v A.a n n -1 (B - b) ➔ B 'an n n for n ➔ 00 and G*(x) = G(Ax + B). In this report we give an explicite expression of the constants a n b n as functions of the given distribution functions F when G is n and one-to-one. As an example we consider the case F n = Fn where·F is a given distribution function; then Gnedenko's expression for stabilizing constants for maxima of independent random variables is seen to be a special case of theorem 2. There is also an application concerning stabilization by moments. Finally we give a connection between quantiles and centering constants used with the weak law of large numbers. -2-1. ~_ice of stabilizing constants. For a sequence of distribution functions {F} satisfying (1) it is not true in general that (1) holds with n bn = µn = r x d F n (x) and -oo 00 00 2 2 I x2 dF (x) - { I 2 a = 0 = xdF(x)}, n n n n -00 _oo even ifµ and 0 exist for every n. Defining n n F (x) = (1 - .:!.) F(x) + _:1_ ~(x - n) n n n where F(x) is an arbitrary distribution function withµ= 0 and o = 1, we have F (x).- F(x) n weakly, so (1) holds with a = 1 but n 0 n a n = / n 1 --➔ oo n He forI!l.ulate a theorem giving the stabilizing constants as functions -of F for a class of limit distributions. The formulation involves n quantiles; the precise definition of a quantile does not matter. However, it is convenient to have a definition which determines the quantile uniquely. For each a (0 <a< 1) we define for the distribution function F (4) Then ( 5) n ~ (n) = inf {x a F (~ (n) - 0) n a < a F (x) > a} n < F (~ (n)). - n a -3-_!~eorem 2. Suppose that the distribution function G is continuous on the whole real line and strictly increasing on {x I O < G(x) < 1}. If {F }£D(G) then n (6) F (a X + b ) ➔ G(ax + b) n n n weakly (hence for all x), with b = E;,a (n) b = G-· 1(a) n (7) a = E;, (3 (n) - E;, (n) a= G- 1 ( f3) -b n a and a and f3 arbitrary (provided O <a< f3 < 1). Proof. Given (G) F (a' x + b') + G(x) n :n n weakly with sequences a' > 0 end b' for every pair of positive n n ' numbers E 1 and E2 there is a positive integer n0 such that for n .:_ n0 (9) Choosing E2 = min {G(b) - G(b - E1), G(b + E1) - G(b)} > o, we have ( 10) F ( a ' ( b - E ) + b ' ) < F ( a ' ( b + E 1 ) + bn' ) • n n 1 n n n Fron ( 5) it follows in view of the continuity of G ( 11 ) F (b - 0) < G(b) < F (b ) n n - - n n n = 1,2,3, ••• -4-Combining (10) and (11) we obtain so ( 12) an'(b - e: 1 ) + b' < b < a'(b + e: 1 ) + b' n n- n n b - b' n n a' n --+-b n + 00 Starting in (9) with a+b ~nstead of b we obtain ( 13) a b b 1 ..E. + n - ~ + a+b n + 00 • a' a' n n n = 1 ,2 ,3, . . . , Application of theorem 1 gives the statement of the theorem. ~~-~• A slight adaptation of the proof shows that the r~uirements on G can be weakened to the following ones: a. ~a< ~S; here ~a= inf < x I G(x) ~a}. b. There exists no e: > 0 such that G is constant on [~ , ~ + e:) a a and [~a, ~ S + . e:). As an application we prove Gnedenko's theorem about choosing n stabilizing constants for sequences F = F attracted by the double-n exponential law ( [2) p.446). -~orollary 1. If for a distribution function F the sequence {Fn} is in the domain of ( 14) then ( 15) with ( 16) and G(x) = exp{-e-x} n F (a X + b) + G(x) n n b = inf {x n a = inf {x n F(x) > 1 F(x) > 1 for x all x, n + 00 - .!.} n - _1 } - b ne n n = 1 ,2 ,3, ••• -5-Proof. It is not difficult to see that theorem 2 holds with ( 17) and b = ~(n) n a n. a = ~(n) - b n Sn n when an~ a and Sn:....-+ S (n + 00 ). Applying the adapted theorem 2 with a = (1 - .!.)n and S = (1 - -1 )n n n n ne we obtain ( 16). Remark. Obviously (15) and (16) are also true for the two other possible types of limit laws G. Another application concerns stabilization by moments. Coro_llary 2. If ( 10) F (ax+ b) -r G(x) n n n for all x, where G is one-to-one and b arbitrary, then there exists n a B such that ( 19) ~ (a X + µ ) - G(x + B). n n n Proof. Let {X} be random variables with distribution functions --- n {F} and n (20) Then ( 21) X - ~(n) y =-n __ a__ n ~(n) ~(n) S a (n = 1,2,3, •• ). on and o ( Y ) = --,---.----.--.-n ~(n) ~(n) a a -6-From (10) and theorem 2 it follows that for some positive A (22) The inequality cr(Y) - A. n (see [3J p.244) gives that the sequence {µ (Y )} is bounded so that n ( from (22)) the sequence {E Y2} is bounded; hence the sequence n µ(Y) n = ~(n) _ ~(n) Ci. s converges. Application of theorem 1 gives (19), 2, Weak law of large numbers, A sequence of distribution functions {F} is said to satisfy the n weak law of large numbers if there is a sequence of real numbers {b} such that r. (23) F (x + b ) n n ---\,(X) for x f 0. Theorem 3, For a sequence of distribution functions {F} the n following propositions are equivalent: a. The sequence {F} satisfies the weak law of large numbers. n b. For each a (0 <a< 1) (24) F (x + ~(n)) n a \( X) for x f O. -7-c. For each a and S (0 <a< S < 1) (25) Proof. b ~ a: Trivial. lim n-+<><> a ~b: Choose a (0 <a< 1) and E (E > 0) ; from (26) lim n-+<><> F (x + b) n n it follows that when n ~ n0 r 0 =1 for x < 0 for x > 0 F ( - E + b) < c < F (E + b ). n n n n Applying (5) we have (27) b - E < n E,;(n) < b + E, a n hence for each x F (x - E + b) < F (x + E,;(n)) < F (x + E + b ). n n-n a -n n From this and (26) we obtain (24). a,b ~c: relation (27) gives As a is arbitrary we have (25). b} = 0, n c~b: Choose a (0 <a< 1), x > 0 and E > 0 arbitrary and E1 such that O < E 1 < X • Relation (25) implies for n ~ n0 E,;(n) - E < 1-E 1 E,; (n) a -8-hence by (5) 1 - e: < F (E,;(n)) < F (x - e: 1 + E,;(n)) < F (x + E,;(n)) < 1, n 1-e: - n 1-e: - n a so for x > 0 lim F (x + E,;(n)) = 1. n a n+oo Analogously one proves (24) for x < 0. By a simple transformation we can restate the results of theorem 3 as conditions for a sequence of distribution functions {F} concen-n trated on the non-negative half-axis which is relatively stable i.e. for which F (ax)- l(x-1) n n weakly for suitably chosen positive constants {a}. n Theorem 4. For a sequence of distribution functions {F} with n F (0-) = 0 for n = 1,2,3, ... the following propositions are equi-n valent: a. The sequence {F} is relatively stable. n b. For each a (0 <a< 1) for x + 1. c. For each a and S (0<a<S<1) lim n+oo Proof. A sequence {F} satisfies the conditions of theorem 4 iff the n sequence {G} defined by n G (x) = F ( ex) n n satisfies the conditions of theorem 3, As in section 1 the results of Gnedenko ( [2] p.426) concerning the law of large numbers and the relative stability of the sequence of maxima of independent identically distributed random variables can be seen as corollaries to the theorems 3 and 4. -9-References [1] Feller, W. (1966). An introduction to probability theory and its applic~tions _g_. Wiley, New-York. [2} Gnedenko, B.V. (1943), Sur la distribution limite du terme maximum d'une serie aleatoire. Annals of Math. 44 423-453, [3] Loeve, M. (1963), Probability theory. van Nostrand, New-York. 

Quantiles and stabilizing constants

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