A NOTE ON A DISTRIBUTION OF WEIGHTED SUMS OF I.I.D. RAYLEIGH RANDOM VARIABLES
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1 Sankhyā : The Indian Journal of Statistics 1998, Volume 6, Series A, Pt. 2, pp A NOTE ON A DISTRIBUTION OF WEIGHTED SUMS OF I.I.D. RAYLEIGH RANDOM VARIABLES By P. HITCZENKO North Carolina State University, Raleigh SUMMARY. In this note we show that if X, X 1, X 2,... are independent and identically distributed random variables with density f X t) = 2t exp t 2 ), t > and a 1,..., a n) are n nonnegative numbers such that a2 k = 1, then for every t > the following inequality is true P a k X k t) P 1 X k t). n We will actually prove a comparison inequality in a more general context. 1. Introduction Let X, X 1,... be independent identically distributed iid) random variables and consider the weighted sum n a kx k, where a 1,..., a n are real numbers. It is of interest in probability theory to establish upper bounds on the tail probability of a k X k, i.e. the quantity P n a kx k t). One possibility is to dominate this probability by tail probability of weighted sum with all coefficients equal. The reason for this is that the latter quantity, being a tail probability of a sum of iid random variables, is often easier to estimate from above. Perhaps the most precise method of estimating tail probabilities for sums of iid random variables is given in Hahn and Klass, 1997), see also Hitczenko et. al., 1997)). In this note we will be concerned with standard Rayleigh random variables i.e. X has a probability density function f X t) = 2te t2, t, see Johnson et. al., 1994), Chapter 18, for more information on Rayleigh distribution). We will assume that a k s are nonnegative. We will show that if n a2 k is constant, say, equal to 1, then P a k X k t) P X k nt).... 1) As it frequently happens, it is easier to prove a more general result. Following Marshall and Olkin, 1979), Definition A.1 let us introduce Paper received. January AMS 1991) subject classification. 6E15, 6G5. Key words and phrases. tail probability, weighted sum, Rayleigh distribution.
2 172 p. hitczenko Definition. Let a = a 1,..., a n ), b = b 1,..., b n ) be two sequences of real numbers. Suppose that a 1 a 2... a n, and b 1 b 2... b n. We say that b is majorized by a if n a k = n b k, and j b k j a k, for j = 1,..., n. Theorem. Let X 1,..., X n ) be a sequence of standard Rayleigh random variables and let a and b be two sequences of nonnegative numbers such that b 2 1,..., b 2 n) is majorized by a 2 1,..., a 2 n). Then, for every t > we have P a k X k t) P b k X k t).... 2) Remark. For a as above let b = b 1,..., b n ), where b i = 1/ n) n a2 k )1/2, i = 1,..., n. Then b 2 1,..., b 2 n) is majorized by a 2 1,..., a 2 n). In particular, the above result implies that if n a2 k = 1, then P a k X k t) P X k nt), so that the conclusion of Theorem is stronger than 1). However, as we shall see below, 2) is easier to prove than 1), since it can be reduced to the case n = Proof of Theorem We first show that it suffices to prove the result when n = 2. Pick i < j such that a i is strictly larger than a j and let f n 2 t) be density of S n 2 = k i,j a kx k. Then P n a kx k t) = P a ix i + a j X j t s)f n 2 s)ds = t P a ix i + a j X j t s)f n 2 s)ds + P S n 2 t) = t P a ix i + a j X j t s)f n 2 s)ds + P S n 2 t).... 3) Since f n 2 is nonnegative, it follows that if the result is true for n = 2, then one can decrease a i and increase a j keeping a 2 i + a2 j, and each of the remaining a k s constant). The squares of the new sequence are majorized by the squares of the old sequence and therefore the last term in 3) above will increase. It is now easy to see, that if b 2 1,..., b 2 n) is majorized by a 2 1,..., a 2 n) then the former can be obtained from the latter by a finite number of such operations on pairs of coefficients and each time the tail probability will increase. This implies 2). So it remains to prove 2) for n = 2. In that case, since a a 2 2 = 1, we have:
3 distribution of weighted sums of i.i.d. 173 P a 1 X 1 + a 2 X 2 t) = P X 1 a 1 t, X 2 a 2 t) + P a 1 X 1 + a 2 X 2 t, X 1 < a 1 t) +P a 1 X 1 + a 2 X 2 t, X 2 < a 2 t) = exp a 2 1t 2 ) exp a 2 2t 2 ) + a 1t P a 2 X 2 t a 1 u)2ue u2 du + a 2 t P a 1 X 1 t a 2 u)2ue u2 du = exp t 2 ) + 2 a 1 t exp t a 1u) 2 u 2 )udu a a 2 t exp t a 2u) 2 u 2 )udu. a 2 1 Substituting v = u/a 1 t) and v = u/a 2 t) in the first and second integral, respectively, we obtain that a1t exp t a1u)2 a 2 2 = t 2 u 2 )udu + a 2t a 2 1 exp t 2 1 a2 1 v)2 a 2 2 Writing x = a 2 1, 1 x = a 2 2 and using exp t a2u)2 a 2 1 u 2 )udu + a 2 1)) + a 2 2 exp t 2 1 a2 2 v)2 a a 2 2)) ) vdv. 1 xv) 2 1 x + xv 2 = 1 + x 1 x 1 v)2 and we see that it suffices to show that 1 1 x)v) x)v 2, x Hx) = xe t 2 x 1 x 1 v)2 + 1 x)e 1 x t2 1 v)2) x vdv is increasing in x on, 1/2). This is because Hx) is symmetric about 1/2, i.e. Hx) = H1 x), < x < 1.) The derivative of Hx) is e t2 x 1 x v 1)2 1 xt2 v 1) 2 1 x) 2 ) ) 2 1 x e t x v 1)2 1 1 x)t2 v 1) 2 x 2 )vdv,... 4) so we need to show that the latter expression is nonnegative for < x < 1/2. To this end we will compute e t2 x Letting a 2 = t 2 x/1 x) we get 1 x v 1)2 1 xt2 v 1) 2 1 x) 2 )vdv. v 1) 2 e a2 1 a2 1 x v 1)2 )v 1)dv + v 1) 2 e a2 v 1) 2 )dv. a2 1 x e a2 v 1) 2 dv
4 174 p. hitczenko By straightforward, but a bit tedious computation we see that the last expression is equal to π1 2x) e xt /1 x) ) + 2t2 t x1 x) P N1, 1 x ), 1)), 2xt2 where Nµ, σ 2 ) is a normal random variable with mean µ and variance σ 2. Interchanging the role of x and 1 x we get a similar formula for the second term in 4) and it follows that 4) is equal to Since π1 2x) t x1 x) P N, 1), t 2x/1 x))) + P N, 1), t 21 x)/x))) 1 2t 2 e t2 x/1 x) e t2 1 x)/x ). P N, 1), t 2x/1 x))) = 1 t 2x/1 x) e u2 /2 du t 2x/1 x) = 1 1 x t 2x 1 1 x t 2x we get that 4) is no less than t 2x /2 1 x e u2 du t 2x/1 x) ue u2 /2 du = 1 x 2 πt x 1 e t2 x/1 x) ), 1 2x 2xt 1 e t2x/1 x) ) + 1 2x 2 21 x)t 1 e t2 1 x)/x ) t 1 e t2x/1 x) ) 1 2 2t 1 e t2 1 x)/x 2 = 1 1 x 2t 2 x 1 x/1 x) e t2 ) x 1 x 1 1 x)/x e t2 ) ) = 1 1 e a 2 a 1 e b b ), where a = t 2 x/1 x), b = t 2 1 x)/x. The last expression is nonnegative for < x < 1/2, since a b and the function hu) = 1 e u )/u is nonincreasing on R +. This completes the proof. 3. Remarks i) Rayleigh distribution is a specific example of a larger class of Weibull distributions i.e. distributions with density ft) = pt p 1 exp t p ), t >, < p, where the case p = is interpreted as a degenerate random variable equal to 1 with probability 1.) For 1 p let q be its conjugate exponent, i.e. p 1 + q 1 = 1, and for a = a 1,..., a n ) let a q = n a k q ) 1/q, if q < and a = sup a k. We suspect that the following is true: if X 1,..., X n are iid
5 distribution of weighted sums of i.i.d. 175 Weibull random variables with parameter p, 1 p and a, b are sequences of nonnegative numbers such that b q 1,..., bq n) is majorized by a q 1,..., aq n), then P a k X k t) P b k X k t). Note that this is trivially true if p = 1 or, and we just have shown that it is true for p = 2. For other values of p, by the argument given above, the problem reduces to showing that the function Gx) = xe tp hx,u) + 1 x)e tp h1 x,u) )u p 1 du, 1 ux)p is increasing in x in, 1/2), where hx, u) = 1 x) p 1 + xup, < x, u < 1. ii) Since for every sequence a such that a 2 k = 1, a 2 1,... a 2 n) is majorized by 1,..., ), 2) implies that P a k X k t) P X t) = exp t 2 ). Analogous inequality is true for other values of p, 1 p. Indeed, let X be a Weibull random variable with parameter p. If p = 1 or, this is obvious, otherwise, assuming 1/p + 1/q = 1 and a q = 1, we can write P n a kx k t) P X k a q 1 since pq 1) = q and a q = 1. k t, k = 1..., n) = n P X aq 1 k t) = exp t p n apq 1) k ) = exp t p ), References Hahn, M.G., Klass, M.J. 1997). Approximation of partial sums of arbitrary i.i.d. random variables and the precision of the usual exponential bound, Ann. Probab. 25, Hitczenko, P., Montgomery - Smith, S.J., Oleszkiewicz, K. 1997). Moment inequalities for linear combinations of certain i.i.d. symmetric random variables, Studia Math. 123, Johnson, N.L., Kotz, S., Balakrishnan, N. 1994). Continuous Univariate Distributions,vol.1, Wiley, New York. Marshall, A.W., Olkin, I. 1979). Inequalities: Theory of Majorization and Its Applications, Academic Press, New York. Department of Mathematics North Carolina State University Raleigh, NC 27695, USA pawel@math.ncsu.edu
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