Large deviations for renewal processes

Jiang Tiefeng

doi:10.1016/0304-4149(94)90147-3

Large deviations for renewal processes

Jiang Tiefeng

Statistics (Twin Cities)

Research output: Contribution to journal › Article › peer-review

19 Scopus citations

Abstract

Let {X_n, n ≥1} be a sequence of identically distributed real random variables with EX₁ = μ > 0. Define S_n=Σⁿ_i=1X_i and N_α(t) = inf{n≥1;S_n>n^αt}, where t>0, α∈[0, 1) and the infimum o f the empty set is defined to be +∞. Let P_α,t be the distribution of N_α(t)/t^1/(1-α), t > 0. In this paper, we establish the large deviation principle for {P_{α, t}; t > 0} when {X_n; n≥ 1 } is a sequence of i.i.d. random variables or, more generally, an exchangeable sequence.

Original language	English (US)
Pages (from-to)	57-71
Number of pages	15
Journal	Stochastic Processes and their Applications
Volume	50
Issue number	1
DOIs	https://doi.org/10.1016/0304-4149(94)90147-3
State	Published - 1994

Bibliographical note

Funding Information:
Corres[>or&nce to: Dr. Jiang Tiefeng, Department of Mathematics, Jilin University, Changchun Province. People’s Republic of China. Research supported by the National Natural Science Foundation of China.

Keywords

exchangeable random variables
large deviations
rate function
renewal processes

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10.1016/0304-4149(94)90147-3

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title = "Large deviations for renewal processes",

abstract = "Let {Xn, n ≥1} be a sequence of identically distributed real random variables with EX1 = μ > 0. Define Sn=Σni=1Xi and Nα(t) = inf{n≥1;Sn>nαt}, where t>0, α∈[0, 1) and the infimum o f the empty set is defined to be +∞. Let Pα,t be the distribution of Nα(t)/t1/(1-α), t > 0. In this paper, we establish the large deviation principle for {Pα, t; t > 0} when {Xn; n≥ 1 } is a sequence of i.i.d. random variables or, more generally, an exchangeable sequence.",

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note = "Funding Information: Corres[>or&nce to: Dr. Jiang Tiefeng, Department of Mathematics, Jilin University, Changchun Province. People{\textquoteright}s Republic of China. Research supported by the National Natural Science Foundation of China.",

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T1 - Large deviations for renewal processes

AU - Tiefeng, Jiang

N1 - Funding Information: Corres[>or&nce to: Dr. Jiang Tiefeng, Department of Mathematics, Jilin University, Changchun Province. People’s Republic of China. Research supported by the National Natural Science Foundation of China.

PY - 1994

Y1 - 1994

N2 - Let {Xn, n ≥1} be a sequence of identically distributed real random variables with EX1 = μ > 0. Define Sn=Σni=1Xi and Nα(t) = inf{n≥1;Sn>nαt}, where t>0, α∈[0, 1) and the infimum o f the empty set is defined to be +∞. Let Pα,t be the distribution of Nα(t)/t1/(1-α), t > 0. In this paper, we establish the large deviation principle for {Pα, t; t > 0} when {Xn; n≥ 1 } is a sequence of i.i.d. random variables or, more generally, an exchangeable sequence.

AB - Let {Xn, n ≥1} be a sequence of identically distributed real random variables with EX1 = μ > 0. Define Sn=Σni=1Xi and Nα(t) = inf{n≥1;Sn>nαt}, where t>0, α∈[0, 1) and the infimum o f the empty set is defined to be +∞. Let Pα,t be the distribution of Nα(t)/t1/(1-α), t > 0. In this paper, we establish the large deviation principle for {Pα, t; t > 0} when {Xn; n≥ 1 } is a sequence of i.i.d. random variables or, more generally, an exchangeable sequence.

KW - exchangeable random variables

KW - large deviations

KW - rate function

KW - renewal processes

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DO - 10.1016/0304-4149(94)90147-3

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EP - 71

JO - Stochastic Processes and their Applications

JF - Stochastic Processes and their Applications

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IS - 1

ER -

Large deviations for renewal processes

Abstract

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