ELECTRONICCOMMUNICATIONSinPROBABILITY ASYMPTOTICS FOR PRODUCTS OF SUMS AND U -STATISTICS

The product of subsequent partial sums of independent, identically distributed, square integrable, positive random variables is asymptotically lognormal. The result extends in a rather routine way to non-degenerate U -statistics.


Introduction
It is well known that the products of independent, identically distributed (iid), positive, square integrable random variables (rv's) are asymptotically lognormal. This fact is an immediate consequence of the classical central limit theorem (clt). In the present paper, we are interested in the limiting law of the products of sums of iid rv's. It appears that their asymptotic behavior is rather similar. We also derive a parallel result for the products of non-degenerate U -statistics. While considering limiting properties of sums of records, Arnold and Villaseñor (1998) obtained the following version of the clt for a sequence (X n ) of iid exponential rv's with the mean equal one n k=1 log(S k ) − n log(n) + n √ 2n as n → ∞, where S k = X 1 + . . . + X k , k = 1, 2, . . ., and N is a standard normal rv. Their proof is heavily based on a very special property of exponential(gamma) distributions: namely that there is independence of ratios of subsequent partial sums and the last sum. It uses also Resnick's (1973) result on weak limits for records.

Electronic Communications in Probability
Observe that, via the Stirling formula, the relation (1) can be equivalently stated as The main purpose of this note is to show that this limit behavior of a product of partial sums has a universal character and holds true for any sequence of square integrable positive iid rv's. This is done in Section 2. Section 3 extends the result to a U -statistics setting.

Main Result
Our main result is a general version of (1), without assuming any particular distribution for the X i 's.
where N is a standard normal rv.
Before proving the above result we will establish a version of the classical central limit theorem, essentially, for scaled iid rv's. To this end we will use the clt for triangular arrays, so the basic step in the proof will rely on verifying the Lindeberg condition.
Observe that Define now

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and observe that E(Z i,n ) = 0 and as n → ∞. Observe that as a by-product of the above computation we have obtained the identity In order to complete the proof we need to check that the Lindeberg condition is satisfied for the triangular array [Z i,n ]. Take any ε > 0. Then, using (4), we get does not depend on i. Since a n → 0 as n → ∞ then the Lindeberg condition holds. 2 Proof of Theorem 1. The proof relies on the delta-method expansion. In what follows we use only elementary considerations to justify its validity. Denote C k = S k /(µk), k = 1, 2, . . .. By the strong law of large numbers it follows that for any Consequently, there exist two sequences (δ m ) ↓ 0 (δ 1 = 1/2) and (R m ) ↑ ∞ such that Take now any real x and any m. Then To compute B m,n we will expand the logarithm: where θ k , k = 1, . . . , n are (0, 1)-valued rv's and F m,n < δ m . Rewrite now D m,n as Observe now that for any fixed m as n → ∞ (as a matter of fact this sequence converges to zero a.s.). Note that for |x| < 1/2 and any θ ∈ (0, 1) it follows that x 2 /(1 + θx) 2 ≤ 4x 2 . Then for any m as n → ∞. The above is a consequence of the Markov inequality, since for any ε > 0

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as n → ∞ for any fixed m. Since by Lemma 1 it follows that as n → ∞ then by (5) and (6) we conclude that for any fixed m where Φ is the standard normal distribution function. Finally, observe that Remark 1. It is perhaps worth to notice that by the strong law of large numbers and the property of the geometric mean it follows directly that if only existence of the first moment is assumed.
Remark 2. In Arnold and Villaseñor (1998) the following identity was proved: where (X n ) and (X n ) are independent sequences of iid standard exponential rv's and R n is nth record from an independent sequence of iid Gumbel rv's (R 1 is the first observation). Consequently Now, in view of the result above, we can apply the argument used in Arnold and Villaseñor (1998) in the reverse order. From Theorem 1 it follows that the left hand side converges in distribution to the standard normal law and the same holds true by the standard clt for the first element at the rhs. Now since the first and the second element at the rhs are independent it follows that √ n(R n − log(n)) is asymptotically standard normal, which proves Resnick's result for records from the Gumbel distribution.

Extension to U -statistics
A useful notion of a "U -statistic" has been introduced by Hoeffding (1948) and pertains to any estimator based on the statistic of the form where h is a symmetric real function of m arguments, the X i 's are iid rv's, and the summation is carried over all possible choices of m indices out of the set {1, 2, . . . , n}. Let us note that if m = 1 and h(x) = x then the above definition gives simply S n . If we assume that E h(X 1 , . . . , X m ) 2 < ∞ and define h 1 (x) = E h(x, X 2 , . . . , X m ) as well aŝ then we may write where It is well known (cf. e.g., Serfling 1980) that, The result of Theorem 1 can be extended to U -statistics as follows.
In order to prove the theorem we shall first consider a more general version of (3).

Lemma 2. Under the assumptions of Theorem 2
Proof of Lemma 2. Using the decomposition (8) we have

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By Lemma 1, applied to the rv's m h 1 (X i ) for i = 1, 2, . . . we have since the second expression converges to zero a.s. as n → ∞. Therefore, in order to prove the lemma it suffices to showR To argue the above, it is, in turn, enough to argue that ER 2 n → 0 as n → ∞.
To this end let us note that, due to symmetries involved, we have if E |h| < ∞.