# Limit theorems for sums of random exponentials

Gérard Ben Arous, Leonid V. Bogachev, Stanislav A. Molchanov

Research output: Contribution to journalArticle

### Abstract

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, λ12, 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) 579-612 34 Probability Theory and Related Fields 132 4 https://doi.org/10.1007/s00440-004-0406-3 Published - Jul 2005

### Fingerprint

Limit Theorems
Regular Variation
Characteristic Exponents
H-function
Limit Laws
I.i.d. Random Variables
Exponential Sums
Rate Function
Law of large numbers
Skewness
Limiting Distribution
Central limit theorem
Breakdown
Critical point

### 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

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

In: Probability Theory and Related Fields, Vol. 132, No. 4, 07.2005, p. 579-612.

Research output: Contribution to journalArticle

Ben Arous, Gérard ; Bogachev, Leonid V. ; Molchanov, Stanislav A. / Limit theorems for sums of random exponentials. In: Probability Theory and Related Fields. 2005 ; Vol. 132, No. 4. pp. 579-612.
@article{b2974027a1344d5ba8af05df78349a58,
title = "Limit theorems for sums of random exponentials",
abstract = "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.",
keywords = "Central limit theorem, Exponential Tauberian theorems, Infinitely divisible distributions, Random exponentials, Regular variation, Stable laws, Sums of independent random variables, Weak limit theorems",
author = "{Ben Arous}, G{\'e}rard and Bogachev, {Leonid V.} and Molchanov, {Stanislav A.}",
year = "2005",
month = "7",
doi = "10.1007/s00440-004-0406-3",
language = "English (US)",
volume = "132",
pages = "579--612",
journal = "Probability Theory and Related Fields",
issn = "0178-8051",
publisher = "Springer New York",
number = "4",

}

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 -