Laws of large numbers and moderate deviations for stochastic processes with stationary and independent increments

Jiang Tiefeng; M. Bhaskara Rao; Wang Xiangchen; Li Deli

doi:10.1016/0304-4149(93)90025-Y

Laws of large numbers and moderate deviations for stochastic processes with stationary and independent increments

Jiang Tiefeng, M. Bhaskara Rao, Wang Xiangchen, Li Deli

Statistics (Twin Cities)

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

5 Scopus citations

Abstract

Let {X(t); t ≥ 0} be a stochastic process with stationary and independent increments which has no Gaussian component. Assume that X(1) has a finite moment generating function. Let P_λ be the probability measure of the process {Z_λ(t); 0 ≤ t ≤ 1}, where Z_λ(t) = (1/λ^q)X(λα[0, t]), α is a probability measure on [0, 1] and 1< q < 2. We may regard P_λ as a probability measure on BV[0, 1], the space of functions of bounded variation on [0, 1]. In this paper, we establish some results on moderate deviations for {P_λ;λ>0}. We also present the Marcinkiewicz-Zygmund type Strong Law of Large Numbers for {X(t); t ≥ 0}.

Original language	English (US)
Pages (from-to)	205-219
Number of pages	15
Journal	Stochastic Processes and their Applications
Volume	44
Issue number	2
DOIs	https://doi.org/10.1016/0304-4149(93)90025-Y
State	Published - 1993

Bibliographical note

Funding Information:
Correspondence to: Prof. M.B. Rao, Department of Statistics, North Minard Hall, SU Station, P.O. Box 5075, Fargo, ND 581055075, USA. * Supported by the Youth Science Foundation of China. ** Supported by the National Natural Science Foundation of China.

Keywords

large deviations
rate function
stationary and independent increments
strong law of large numbers

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10.1016/0304-4149(93)90025-Y

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@article{53ca9a8d7fac4d4aa21a8d8e9856e53b,

title = "Laws of large numbers and moderate deviations for stochastic processes with stationary and independent increments",

abstract = "Let {X(t); t ≥ 0} be a stochastic process with stationary and independent increments which has no Gaussian component. Assume that X(1) has a finite moment generating function. Let Pλ be the probability measure of the process {Zλ(t); 0 ≤ t ≤ 1}, where Zλ(t) = (1/λq)X(λα[0, t]), α is a probability measure on [0, 1] and 1< q < 2. We may regard Pλ as a probability measure on BV[0, 1], the space of functions of bounded variation on [0, 1]. In this paper, we establish some results on moderate deviations for {Pλ;λ>0}. We also present the Marcinkiewicz-Zygmund type Strong Law of Large Numbers for {X(t); t ≥ 0}.",

keywords = "large deviations, rate function, stationary and independent increments, strong law of large numbers",

author = "Jiang Tiefeng and {Bhaskara Rao}, M. and Wang Xiangchen and Li Deli",

note = "Funding Information: Correspondence to: Prof. M.B. Rao, Department of Statistics, North Minard Hall, SU Station, P.O. Box 5075, Fargo, ND 581055075, USA. * Supported by the Youth Science Foundation of China. ** Supported by the National Natural Science Foundation of China.",

year = "1993",

doi = "10.1016/0304-4149(93)90025-Y",

language = "English (US)",

volume = "44",

pages = "205--219",

journal = "Stochastic Processes and their Applications",

issn = "0304-4149",

publisher = "Elsevier B.V.",

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TY - JOUR

T1 - Laws of large numbers and moderate deviations for stochastic processes with stationary and independent increments

AU - Tiefeng, Jiang

AU - Bhaskara Rao, M.

AU - Xiangchen, Wang

AU - Deli, Li

N1 - Funding Information: Correspondence to: Prof. M.B. Rao, Department of Statistics, North Minard Hall, SU Station, P.O. Box 5075, Fargo, ND 581055075, USA. * Supported by the Youth Science Foundation of China. ** Supported by the National Natural Science Foundation of China.

PY - 1993

Y1 - 1993

N2 - Let {X(t); t ≥ 0} be a stochastic process with stationary and independent increments which has no Gaussian component. Assume that X(1) has a finite moment generating function. Let Pλ be the probability measure of the process {Zλ(t); 0 ≤ t ≤ 1}, where Zλ(t) = (1/λq)X(λα[0, t]), α is a probability measure on [0, 1] and 1< q < 2. We may regard Pλ as a probability measure on BV[0, 1], the space of functions of bounded variation on [0, 1]. In this paper, we establish some results on moderate deviations for {Pλ;λ>0}. We also present the Marcinkiewicz-Zygmund type Strong Law of Large Numbers for {X(t); t ≥ 0}.

AB - Let {X(t); t ≥ 0} be a stochastic process with stationary and independent increments which has no Gaussian component. Assume that X(1) has a finite moment generating function. Let Pλ be the probability measure of the process {Zλ(t); 0 ≤ t ≤ 1}, where Zλ(t) = (1/λq)X(λα[0, t]), α is a probability measure on [0, 1] and 1< q < 2. We may regard Pλ as a probability measure on BV[0, 1], the space of functions of bounded variation on [0, 1]. In this paper, we establish some results on moderate deviations for {Pλ;λ>0}. We also present the Marcinkiewicz-Zygmund type Strong Law of Large Numbers for {X(t); t ≥ 0}.

KW - large deviations

KW - rate function

KW - stationary and independent increments

KW - strong law of large numbers

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U2 - 10.1016/0304-4149(93)90025-Y

DO - 10.1016/0304-4149(93)90025-Y

M3 - Article

AN - SCOPUS:0742266458

SN - 0304-4149

VL - 44

SP - 205

EP - 219

JO - Stochastic Processes and their Applications

JF - Stochastic Processes and their Applications

IS - 2

ER -

Laws of large numbers and moderate deviations for stochastic processes with stationary and independent increments

Abstract

Bibliographical note

Keywords

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