Let {S t H, t ≥ 0） be a linear combination of a Brownian motion and an independent sub-fractional Brownian motion with Hurst index 0 〈 H 〈 1. Its main properties are studied. They suggest that SH lies between the ...Let {S t H, t ≥ 0） be a linear combination of a Brownian motion and an independent sub-fractional Brownian motion with Hurst index 0 〈 H 〈 1. Its main properties are studied. They suggest that SH lies between the sub-fractional Brownian motion and the mixed fractional Brownian motion. We also determine the values of H for which SH is not a semi-martingale.展开更多
In this paper, we introduce the definition of a multi-parameter fractional Lévy process and its local time, and show its decomposition. Using the decomposition, we prove existence and joint continuity of its loca...In this paper, we introduce the definition of a multi-parameter fractional Lévy process and its local time, and show its decomposition. Using the decomposition, we prove existence and joint continuity of its local time.展开更多
Let BH,K = {BH,K(t), t ∈ R+} be a bifractional Brownian motion in Rd. This process is a selfsimilar Gaussian process depending on two parameters H and K and it constitutes a natural generalization of fractional Brown...Let BH,K = {BH,K(t), t ∈ R+} be a bifractional Brownian motion in Rd. This process is a selfsimilar Gaussian process depending on two parameters H and K and it constitutes a natural generalization of fractional Brownian motion (which is obtained for K = 1). The exact Hausdorff measures of the image, graph and the level set of BH,K are investigated. The results extend the corresponding results proved by Talagrand and Xiao for fractional Brownian motion.展开更多
Let XH = {xH（t）,t ∈ R＋} be a subfractional Brownian motion in Rd. We provide asufficient condition for a self-similar Gaussian process to be strongly locally nondeterministic and show that XH has the property of s...Let XH = {xH（t）,t ∈ R＋} be a subfractional Brownian motion in Rd. We provide asufficient condition for a self-similar Gaussian process to be strongly locally nondeterministic and show that XH has the property of strong local nondeterminism. Applying this property and a stochastic integral representation of XH, we establish Chung＇s law of the iterated logarithm for XH.展开更多
Operator self-similar processes, as an extension of self-similar processes, have been studied extensively. In this work, we study limit theorems for functionals of Gaussian vectors. Under some conditions, we determine...Operator self-similar processes, as an extension of self-similar processes, have been studied extensively. In this work, we study limit theorems for functionals of Gaussian vectors. Under some conditions, we determine that the limit of partial sums of functionals of a stationary Gaussian sequence of random vectors is an operator self-similar process.展开更多
For a risk process R_u(t) = u + ct- X(t), t≥0, where u≥0 is the initial capital, c > 0 is the premium rate and X(t), t≥0 is an aggregate claim process, we investigate the probability of the Parisian ruin P_S(u, ...For a risk process R_u(t) = u + ct- X(t), t≥0, where u≥0 is the initial capital, c > 0 is the premium rate and X(t), t≥0 is an aggregate claim process, we investigate the probability of the Parisian ruin P_S(u, T_u) = P{inf (t∈[0,S]_(s∈[t,t+T_u])) sup R_u(s) < 0}, S, T_u > 0.For X being a general Gaussian process we derive approximations of P_S(u, T_u) as u →∞. As a by-product, we obtain the tail asymptotic behaviour of the infimum of a standard Brownian motion with drift over a finite-time interval.展开更多
In Internet environment, traffic flow to a link is typically modeled by superposition of ON/OFF based sources. During each ON-period for a particular source, packets arrive according to a Poisson process and packet si...In Internet environment, traffic flow to a link is typically modeled by superposition of ON/OFF based sources. During each ON-period for a particular source, packets arrive according to a Poisson process and packet sizes （hence service times） can be generally distributed. In this paper, we establish heavy traffic limit theorems to provide suitable approximations for the system under first-in first-out （FIFO） and work-conserving service discipline, which state that, when the lengths of both ON- and OFF-periods are lightly tailed, the sequences of the scaled queue length and workload processes converge weakly to short-range dependent reflecting Gaussian processes, and when the lengths of ON- and/or OFF-periods are heavily tailed with infinite variance, the sequences converge weakly to either reflecting fractional Brownian motions （FBMs） or certain type of long- range dependent reflecting Gaussian processes depending on the choice of scaling as the number of superposed sources tends to infinity. Moreover, the sequences exhibit a state space collapse-like property when the number of sources is large enough, which is a kind of extension of the well-known Little＇s law for M/M/1 queueing system. Theory to justify the approximations is based on appropriate heavy traffic conditions which essentially mean that the service rate closely approaches the arrival rate when the number of input sources tends to infinity.展开更多
文摘Let {S t H, t ≥ 0） be a linear combination of a Brownian motion and an independent sub-fractional Brownian motion with Hurst index 0 〈 H 〈 1. Its main properties are studied. They suggest that SH lies between the sub-fractional Brownian motion and the mixed fractional Brownian motion. We also determine the values of H for which SH is not a semi-martingale.
基金supported by the National Natural Science Foundation of China (No. 10871177)the Ph. D.Programs Foundation of Ministry of Education of China (No. 20060335032)the Natural Science Foundation of Zhejiang Province of China (No. Y7080044)
文摘In this paper, we introduce the definition of a multi-parameter fractional Lévy process and its local time, and show its decomposition. Using the decomposition, we prove existence and joint continuity of its local time.
基金supported by National Natural Science Foundation of China (Grant No.10721091)
文摘Let BH,K = {BH,K(t), t ∈ R+} be a bifractional Brownian motion in Rd. This process is a selfsimilar Gaussian process depending on two parameters H and K and it constitutes a natural generalization of fractional Brownian motion (which is obtained for K = 1). The exact Hausdorff measures of the image, graph and the level set of BH,K are investigated. The results extend the corresponding results proved by Talagrand and Xiao for fractional Brownian motion.
基金Supported by NSFC(Grant Nos.11201068,11671041)“the Fundamental Research Funds for the Central Universities”in UIBE(Grant No.14YQ07)
文摘Let XH = {xH（t）,t ∈ R＋} be a subfractional Brownian motion in Rd. We provide asufficient condition for a self-similar Gaussian process to be strongly locally nondeterministic and show that XH has the property of strong local nondeterminism. Applying this property and a stochastic integral representation of XH, we establish Chung＇s law of the iterated logarithm for XH.
文摘Operator self-similar processes, as an extension of self-similar processes, have been studied extensively. In this work, we study limit theorems for functionals of Gaussian vectors. Under some conditions, we determine that the limit of partial sums of functionals of a stationary Gaussian sequence of random vectors is an operator self-similar process.
基金the National Natural Science Foundation of China(Grant Nos.12002089 and 11902081)Science and Technology Projects in Guangzhou(Grant Nos.202201010326 and 2023A04J1323)Guangdong Basic and Applied Basic Research Foundation(Grant No.2023A1515010833).
基金the Swiss National Science Foundation (Grant No. 200021140633/1)the project Risk Analysis, Ruin and Extremes (an FP7 Marie Curie International Research Staff Exchange Scheme Fellowship) (Grant No. 318984)Narodowe Centrum Nauki (Grant No. 2013/09/B/ST1/01778 (2014-2016))
文摘For a risk process R_u(t) = u + ct- X(t), t≥0, where u≥0 is the initial capital, c > 0 is the premium rate and X(t), t≥0 is an aggregate claim process, we investigate the probability of the Parisian ruin P_S(u, T_u) = P{inf (t∈[0,S]_(s∈[t,t+T_u])) sup R_u(s) < 0}, S, T_u > 0.For X being a general Gaussian process we derive approximations of P_S(u, T_u) as u →∞. As a by-product, we obtain the tail asymptotic behaviour of the infimum of a standard Brownian motion with drift over a finite-time interval.
基金Supported by the National Natural Science Foundation of China (No.10371053,10971249)
文摘In Internet environment, traffic flow to a link is typically modeled by superposition of ON/OFF based sources. During each ON-period for a particular source, packets arrive according to a Poisson process and packet sizes （hence service times） can be generally distributed. In this paper, we establish heavy traffic limit theorems to provide suitable approximations for the system under first-in first-out （FIFO） and work-conserving service discipline, which state that, when the lengths of both ON- and OFF-periods are lightly tailed, the sequences of the scaled queue length and workload processes converge weakly to short-range dependent reflecting Gaussian processes, and when the lengths of ON- and/or OFF-periods are heavily tailed with infinite variance, the sequences converge weakly to either reflecting fractional Brownian motions （FBMs） or certain type of long- range dependent reflecting Gaussian processes depending on the choice of scaling as the number of superposed sources tends to infinity. Moreover, the sequences exhibit a state space collapse-like property when the number of sources is large enough, which is a kind of extension of the well-known Little＇s law for M/M/1 queueing system. Theory to justify the approximations is based on appropriate heavy traffic conditions which essentially mean that the service rate closely approaches the arrival rate when the number of input sources tends to infinity.
