Abstract
We study limiting distributions of exponential sums SN(t) = ∑Ni=1e1xi as t→∞, N→∞, where (X i ) are i.i.d. random variables. Two cases are considered: (A) ess sup X i = 0 and (B) ess sup X i = ∞. We assume that the function h(x)= -log P{X i >x} (case B) or h(x) = -log P {X i >-1/x} (case A) is regularly varying at ∞ with index 1 < script Q sign <∞ (case B) or 0 < script Q sign < ∞ (case A). The appropriate growth scale of N relative to t is of the form eλH0(t) (0 < λ <) where the rate function H 0(t) is a certain asymptotic version of the function H (t)= log E[etXi](case B) or H (t)= -log E[etXi] (case A). We have found two critical points, λ1<λ2, below which the Law of Large Numbers and the Central Limit Theorem, respectively, break down. For 0 < λ < λ2, under the slightly stronger condition of normalized regular variation of h we prove that the limit laws are stable, with characteristic exponent α = α (script Q sign λ) ∈(0,2) and skewness parameter β ≡ 1.
Original language | English (US) |
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Pages (from-to) | 579-612 |
Number of pages | 34 |
Journal | Probability Theory and Related Fields |
Volume | 132 |
Issue number | 4 |
DOIs | |
State | Published - Jul 2005 |
Keywords
- Central limit theorem
- Exponential Tauberian theorems
- Infinitely divisible distributions
- Random exponentials
- Regular variation
- Stable laws
- Sums of independent random variables
- Weak limit theorems
ASJC Scopus subject areas
- Analysis
- Statistics and Probability
- Statistics, Probability and Uncertainty