### Abstract

We study limiting distributions of exponential sums S_{N}(t) = ∑^{N}
_{i=1}e^{1xi} 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 e^{λH0(t)} (0 < λ <) where the rate function H _{0}(t) is a certain asymptotic version of the function H (t)= log E[e^{tXi}](case B) or H (t)= -log E[e^{tXi}] (case A). We have found two critical points, λ_{1}<λ_{2}, below which the Law of Large Numbers and the Central Limit Theorem, respectively, break down. For 0 < λ < λ_{2}, 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) |
---|---|

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 |

### Fingerprint

### 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

- Mathematics(all)
- Analysis
- Statistics and Probability

### Cite this

*Probability Theory and Related Fields*,

*132*(4), 579-612. https://doi.org/10.1007/s00440-004-0406-3

**Limit theorems for sums of random exponentials.** / Ben Arous, Gérard; Bogachev, Leonid V.; Molchanov, Stanislav A.

Research output: Contribution to journal › Article

*Probability Theory and Related Fields*, vol. 132, no. 4, pp. 579-612. https://doi.org/10.1007/s00440-004-0406-3

}

TY - JOUR

T1 - Limit theorems for sums of random exponentials

AU - Ben Arous, Gérard

AU - Bogachev, Leonid V.

AU - Molchanov, Stanislav A.

PY - 2005/7

Y1 - 2005/7

N2 - We study limiting distributions of exponential sums SN(t) = ∑N i=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.

AB - We study limiting distributions of exponential sums SN(t) = ∑N i=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.

KW - Central limit theorem

KW - Exponential Tauberian theorems

KW - Infinitely divisible distributions

KW - Random exponentials

KW - Regular variation

KW - Stable laws

KW - Sums of independent random variables

KW - Weak limit theorems

UR - http://www.scopus.com/inward/record.url?scp=23744456992&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=23744456992&partnerID=8YFLogxK

U2 - 10.1007/s00440-004-0406-3

DO - 10.1007/s00440-004-0406-3

M3 - Article

AN - SCOPUS:23744456992

VL - 132

SP - 579

EP - 612

JO - Probability Theory and Related Fields

JF - Probability Theory and Related Fields

SN - 0178-8051

IS - 4

ER -