WHEN DOES A RANDOMLY WEIGHTED SELF{NORMALIZED SUM CONVERGE IN DISTRIBUTION?

We determine exactly when a certain randomly weighted self{normalized sum converges in distribution, partially verifying a 1965 conjecture of Leo Breiman, and then apply our results to characterize the asymptotic distribution of relative sums and to provide a short proof of a 1973 conjecture of Logan, Mallows, Rice and Shepp on the asymptotic distribution of self{ normalized sums in the case of symmetry.

Throughout this paper {Y i } i≥1 will denote a sequence of i.i.d. Y random variables, where Y is non-negative with distribution function G. Let Y ∈ D (α), with 0 < α ≤ 2, denote that Y is in the domain of attraction of a stable law of index α. We shall use the notation Y ∈ D (0) to mean that 1 − G is a slowly varying function at infinity. Now let {X i } i≥1 be a sequence of i.i.d. X random variables independent of {Y i } i≥1 , where X satisfies E|X| < ∞ and EX = 0. (1) Consider the randomly weighted self-normalized sum (Here and elsewhere we define 0/0 = 0.) In a beautiful paper, Breiman (1965) proved the following result characterizing when R n converges in distribution to a non-degenerate law.
Theorem 1 Suppose for each such sequence {X i } i≥1 of i.i.d X random variables, independent of {Y i } i≥1 , the ratio R n converges in distribution, and the limit law of R n is non-degenerate for at least one such sequence {X i } i≥1 . Then Y ∈ D (α), with 0 ≤ α < 1.
Theorem 1 is a restatement of his Theorem 4. At the end of his 1965 paper Breiman conjectured that the conclusion of Theorem 1 remains true as long as there exist one i.i.d. X sequence {X i } i≥1 , satisfying (1), such that R n converges in distribution to a non-degenerate law. We shall provide a partial solution to his conjecture (we assume E |X| p < ∞ for some p > 2) and at the same time give a new characterization for a non-negative random variable Y ∈ D (α), with 0 ≤ α < 1. Before we do this, let us briefly describe and comment upon Breiman's proof of Theorem 1. Let . . , n. Clearly along subsequences {n } of {n}, the ordered random variables D (i) n , i = 1, . . . , n, converge in distribution to random sequences From this one readily concludes that the limit laws of R n are of the form This implies that along the full sequence {n} , where D 1 is either non-degenerate or D 1 = 1. Breiman proves that when D 1 = 1, Y ∈ D(0), and when D 1 is non-degenerate necessarily Y ∈ D (α) , with 0 < α < 1. At first glance it may seem reasonable that it would be enough for (3) However, Jim Fill has shown that there exist two non-identically distributed random sequences This indicates that one must look for another way to try to establish Breiman's conjecture, than merely to refine his original proof. Our partial solution to Breiman's conjecture is contained in the following theorem.
, where X satisfies E|X| p < ∞ for some p > 2 and EX = 0, then the ratio R n converges in distribution to a non-degenerate random variable R if and only if Y ∈ D (α), with 0 ≤ α < 1.
The proof of Theorem 1 will follow readily from the following characterization of when Y ∈ D (α), with 0 ≤ α < 1. We shall soon see that whether Y ∈ D (α), with 0 ≤ α < 1, or not depends on the limit of ET 2 n , where with {s i } i≥1 being a sequence of independent Rademacher random variables independent of Remark 1 It can be inferred from Theorems 1, 2 and Proposition 1 of Breiman (1965) that the limit in (7) is equal to zero if and only if if and only if there exist constants B n such that Remark 2 Proposition 1 should be compared to a result of Bingham and Teugels (1981), which says where Proof. First assume that Y ∈ D (α), where 0 < α < 1. Notice that with T n defined as in (6), By Corollary 1 of Le Page, Woodroofe and Zinn (1981), where Γ i = i j=1 ξ j , with {ξ j } j≥1 being a sequence of i.i.d. exponential random variables with mean 1 independent of {s i } i≥1 . Since clearly |T n | ≤ 1, we can infer by (11) that for any We shall prove that where From this one gets from (13) after a little calculus that We get Now for any fixed 0 < α < 1 the limit in (11) remains the same for any Y ∈ D (α). Therefore for convenience we can and shall choose Y = U −1/α , where U is Uniform (0, 1). Therefore we can write the expession in (14) as which by the change of variables t = s/n 1/α ,

ds.
A routine limit argument now shows that this last expression converges to (14) we get that Arguing as in the proof of Theorem 3 of Breiman (1965) this implies that For y ≥ 0, let q (y) denote the inverse of − log ϕ (v). Changing variables to t = q(y) we get from (15) that By Karamata's Tauberian theorem, see Theorem 1.7.1 on page 37 of Bingham et al (1987), we conclude that which, in turn, by the change of variables y = q (x) gives t 0 yϕ (y) dy − log ϕ (t) → 1 − α, as t 0.
Since − log(1 − s)/s → 1 as s 0, this implies that or in other words Set f (x) = −x −2 ϕ (1/x) = (ϕ (1/x)) , for x > 0. With this notation we can rewrite (16) as By Theorem 1.6.1 on page 30 of Bingham et al (1987) this implies that f (y) is regularly varying at infinity with index ρ = −α − 1, which, in turn, by their Theorem 1.5.11 implies that 1 − ϕ (1/x) is regularly varying at infinity with index −α, which says that 1 − ϕ (s) is regularly varying at 0 with index α. Set for x ≥ 0, We see that for any s > 0, To finish the proof we must show that holds if and only if Y ∈ D (0). It is well-known going back to Darling (1952 (Refer to Haeusler and Mason (1991) and the references therein.) Thus clearly whenever Y ∈ D (0) we have T n → d s 1 and therefore we have (18). To go the other way, assume that (18) holds. This implies that Proof of Theorem 2. First assume that for some non-degenerate random variable R, By Jensen's inequality for any r ≥ 1, This implies that whenever R n → d R, where R is non-degenerate, then where necessarily 0 ≤ α < 1. Thus by Proposition 1, Y ∈ D (α) , with 0 ≤ α < 1. Breiman (1965) shows that whenever Y ∈ D (α) , with 0 ≤ α < 1, then (20) holds for some non-degenerate random variable R. To be specific, when α = 0, R = d X and when 0 < α < 1, it can be shown by using the methods of Le Page et al (1981) that This completes the proof of Theorem 2.
The proof just given is highly dependent on the assumption that E|X| p < ∞ for some p > 2.
To replace it by the weaker assumption E|X| < ∞ would require an entirely different approach. Therefore the complete Breiman conjecture remains open. In the next section we provide some applications of our results to the study of the asymptotic distribution of relative ratio and selfnormalized sums.

Application to relative ratio sums
Let {Y i } i≥1 be a sequence of i.i.d. Y non-negative random variables and for any n ≥ 0 let S n = n i=1 Y i , where S 0 := 0. For any n ≥ 1 and 0 ≤ t ≤ 1, consider the relative ratio sum Our first corollary characterizes when such relative ratio sums converge in distribution to a non-degenerate law.

Corollary 4 For any
where V (t) is non-degenerate if and only if Y ∈ D (α) , with 0 ≤ α < 1.
The proof Corollary 1 will be an easy consequence of the following proposition. Independent , with 0 < t < 1. For any n ≥ 1 and 0 < t < 1 let [nt] denote the integer part of nt and set Proposition 5 For all 0 < t < 1, Proof of Proposition 2. We have S n and, clearly, where M m (t) = |N n (t) − [nt]|. Now (recalling that we define 0/0 = 0), we have Thus since E |N n (t) − [nt]| /n → 0, we get (24).
Proof of Corollary 1. Note that Therefore by Proposition 2 and (25), we readily conclude that (23) holds with a non-degenerate converges in distribution to a non-degenerate random variable. Thus Corollary 1 follows from Theorem 2.
When Y ∈ D (0), it is easy to apply Proposition 2, (25) and (19) to get that V n (t) → d 1 (t) , and when Y ∈ D (α) , with 0 < α < 1, one gets from Proposition 2, (25) and by arguing as in Le Page et al (1981), that Also, one can show using Theorem 1, Theorem 2 and Proposition 1 of Breiman (1965 if and only if there exists a sequence of positive constants B n such that (9) holds. Furthermore, by Proposition 2 and (25), we see that this happens if and only if V n (t) → p t. An easy variation of Corollary 1, says that if S n = d S n , with S n and S n independent, then where K is non-degenerate if and only if Y ∈ D (α), with 0 < α < 1. Again by using the techniques of Le Page et al (1981) one can show that where W α = d W α , W α and W α are independent and Curiously, it can be shown that L α := log W α and log K provide examples of random variables that have finite positive moments of any order, yet have distributions that are not uniquely determined by their moments. To see this, let h α denote the density of L α . Using known results about densities of stable laws that can be found in Ibragimov and Linnik (1971) and Zolotarev (1986) it can be proved that L α has all positive moments and its density h α is in C ∞ . Moreover, it is readily checked that This implies that the distribution of L α is not uniquely determined by its moments. Refer to Lin (1997). Furthermore, by a result of Devinatz (1959), this in turn implies that the distribution of log K = d L α − L α , where L α is an independent copy of L α , is also not uniquely determined by its moments.

Application to self-normalized sums
Let {X i } i≥1 be a sequence of i.i.d. X random variables and consider the self-normalized sums Logan, Mallows, Rice and Shepp (1973) conjectured that S n (2) converges in distribution to a standard normal random variable if and only if EX = 0 and X ∈ D (2) , and more generally that S n (2) converges in distribution to a non-degenerate random variable not concentrated on two points if and only if X ∈ D (α), with 0 < α ≤ 2, where EX = 0 if 0 < α < 1 and X is in the domain of attraction of a Cauchy law in the case α = 1. The first conjecture was proved by Giné, Götze and Mason (1997) and the more general conjecture has been recently established by Chistyakov and Götze (2004). Griffin and Mason (1991) where S(2) is a non-degenerate if and only if sY ∈ D (α) , where 0 ≤ α ≤ 2.
In the proof of Corollary 2 we describe the possible limit laws and when they occur.
Proof of Corollary 2. When sY ∈ D (0), then by using (19) one readily gets that Whenever sY ∈ D (α), with 0 < α < 2, we apply Corollary 1 of Le Page et al (1981) to get that and when sY ∈ D (2), Raikov's theorem (see Lemma 3.2 in Giné et al (1997)), implies that for any non-decreasing positive sequence {a i } i≥1 such that n i=1 s i Y i /a n → d Z, where Z is a standard normal random variable, one has n i=1 Y 2 i /a 2 n → p 1 , which gives S n (2) → d Z.

A conjecture
As in Breiman (1965) we shall end our paper with a conjecture. For a sequence of i.i.d. positive random variables {Y i } i≥1 , a sequence of independent Rademacher random variables {s i } i≥1 independent of {Y i } i≥1 and 1 ≤ p < 2, we conjecture that where S(p) is a non-degenerate random variable if and only if Y ∈ D (α), where 0 ≤ α < p. At present we can only verify it for case p = 1 and the limit case p = 2.