Records for time-dependent stationary Gaussian sequences

2020 ◽  
Vol 57 (1) ◽  
pp. 78-96
Author(s):  
Michael Falk ◽  
Amir Khorrami Chokami ◽  
Simone A. Padoan
Keyword(s):  
Distribution Function ◽  
Joint Distribution ◽  
Arrival Time ◽  
Marginal Distribution ◽  
Stationary Process ◽  
Time Dependent ◽  
Time Process ◽  
Expected Number ◽  
Unit Variance ◽  
Strictly Stationary Process

AbstractFor a zero-mean, unit-variance stationary univariate Gaussian process we derive the probability that a record at the time n, say $X_n$ , takes place, and derive its distribution function. We study the joint distribution of the arrival time process of records and the distribution of the increments between records. We compute the expected number of records. We also consider two consecutive and non-consecutive records, one at time j and one at time n, and we derive the probability that the joint records $(X_j,X_n)$ occur, as well as their distribution function. The probability that the records $X_n$ and $(X_j,X_n)$ take place and the arrival time of the nth record are independent of the marginal distribution function, provided that it is continuous. These results actually hold for a strictly stationary process with Gaussian copulas.

scholarly journals A simple integer-valued bilinear time series model

2006 ◽  
Vol 38 (2) ◽  
pp. 559-578 ◽  
Author(s):  
P. Doukhan ◽  
A. Latour ◽  
D. Oraichi
Keyword(s):  
Time Series ◽  
Method Of Moments ◽  
Joint Distribution ◽  
Time Series Data ◽  
Stationary Process ◽  
Existence Condition ◽  
Series Data ◽  
Bilinear Model ◽  
Numerical Examples ◽  
Strictly Stationary Process

In this paper, we extend the integer-valued model class to give a nonnegative integer-valued bilinear process, denoted by INBL(p,q,m,n), similar to the real-valued bilinear model. We demonstrate the existence of this strictly stationary process and give an existence condition for it. The estimation problem is discussed in the context of a particular simple case. The method of moments is applied and the asymptotic joint distribution of the estimators is given: it turns out to be a normal distribution. We present numerical examples and applications of the model to real time series data on meningitis and Escherichia coli infections.

On the expected number of crossings of a level in certain stochastic processes

1970 ◽  
Vol 7 (3) ◽  
pp. 766-770 ◽  
Author(s):  
P. B. M. Roes
Keyword(s):  
Present Method ◽  
Busy Period ◽  
Time Dependent ◽  
Renewal Processes ◽  
Time Process ◽  
Expected Number ◽  
Original Process ◽  
Period Distribution ◽  
Virtual Waiting Time ◽  
Waiting Time Process

We consider a stochastic process which increases and decreases by simple jumps as well as smoothly. The rate of smooth increase and decrease with time is a function of the state of the process. The process is not constant in time except when in the zero state. For such processes a relation is derived between the expected number of true crossings (as opposed to skippings by which we mean vertical crossings due to jumps) of a level x, say, and the time dependent distribution of the process. This result is applied to the virtual waiting time process of the GI/G/1 queue, where it is of particular interest when the zero level is considered, as the underlying crossing process is then a renewal process. It leads to a new derivation of the busy period distribution for this system. This serves as an example for the last brief section, where an indication is given as to how this method may be applied to the GI/G/s queue. Naturally, the present method is most powerful when the original process is a Markov process, so that renewal processes are imbedded at all levels. For an application to the M/G/1 queue, see Roes [3].

On the expected number of crossings of a level in certain stochastic processes

1970 ◽  
Vol 7 (03) ◽  
pp. 766-770 ◽  
Author(s):  
P. B. M. Roes
Keyword(s):  
Present Method ◽  
Busy Period ◽  
Time Dependent ◽  
Renewal Processes ◽  
Time Process ◽  
Expected Number ◽  
Original Process ◽  
Period Distribution ◽  
Virtual Waiting Time ◽  
Waiting Time Process

We consider a stochastic process which increases and decreases by simple jumps as well as smoothly. The rate of smooth increase and decrease with time is a function of the state of the process. The process is not constant in time except when in the zero state. For such processes a relation is derived between the expected number of true crossings (as opposed to skippings by which we mean vertical crossings due to jumps) of a level x, say, and the time dependent distribution of the process. This result is applied to the virtual waiting time process of the GI/G/1 queue, where it is of particular interest when the zero level is considered, as the underlying crossing process is then a renewal process. It leads to a new derivation of the busy period distribution for this system. This serves as an example for the last brief section, where an indication is given as to how this method may be applied to the GI/G/s queue. Naturally, the present method is most powerful when the original process is a Markov process, so that renewal processes are imbedded at all levels. For an application to the M/G/1 queue, see Roes [3].

scholarly journals A simple integer-valued bilinear time series model

2006 ◽  
Vol 38 (02) ◽  
pp. 559-578 ◽  
Author(s):  
P. Doukhan ◽  
A. Latour ◽  
D. Oraichi
Keyword(s):  
Time Series ◽  
Method Of Moments ◽  
Joint Distribution ◽  
Time Series Data ◽  
Stationary Process ◽  
Existence Condition ◽  
Series Data ◽  
Bilinear Model ◽  
Numerical Examples ◽  
Strictly Stationary Process

In this paper, we extend the integer-valued model class to give a nonnegative integer-valued bilinear process, denoted by INBL(p,q,m,n), similar to the real-valued bilinear model. We demonstrate the existence of this strictly stationary process and give an existence condition for it. The estimation problem is discussed in the context of a particular simple case. The method of moments is applied and the asymptotic joint distribution of the estimators is given: it turns out to be a normal distribution. We present numerical examples and applications of the model to real time series data on meningitis and Escherichia coli infections.

Nonstationary Narrow-Band Response and First-Passage Probability

1979 ◽  
Vol 46 (4) ◽  
pp. 919-924 ◽  
Author(s):  
S. Krenk
Keyword(s):  
Stochastic Process ◽  
Distribution Function ◽  
Joint Distribution ◽  
Stationary Process ◽  
Narrow Band ◽  
Algebraic Relation ◽  
Joint Distribution Function ◽  
First Passage ◽  
First Passage Probability ◽  
First Passage Problem

The notion of a non-stationary narrow-band stochastic process is introduced without reference to a frequency spectrum, and the joint distribution function of two consecutive maxima is approximated by use of an envelope. Based on these definitions the first passage problem is treated as a Markov point process. The theory is applied to the response of a linear oscillator excited by a stationary process from t = 0, and a simple algebraic relation between the non-stationary and stationary correlation functions of the response is derived.

SPECTRUM OF COSMIC RAYS WITH ENERGY ABOVE 1017 EV

2005 ◽  
Vol 20 (29) ◽  
pp. 6878-6880 ◽  
Author(s):  
V. P. EGOROVA ◽  
A. V. GLUSHKOV ◽  
A. A. IVANOV ◽  
S. P. KNURENKO ◽  
V. A. KOLOSOV ◽  
...  
Keyword(s):  
Distribution Function ◽  
Energy Spectrum ◽  
Cosmic Rays ◽  
Time Distribution ◽  
Arrival Time ◽  
Lateral Distribution ◽  
Arrival Time Distribution ◽  
Array Data ◽  
Lateral Distribution Function ◽  
High Energies

The energy spectrum of primary cosmic rays with ultra-high energies based on the Yakutsk EAS Array data is presented. For the largest events values of S600 and axis coordinates have been obtained using revised lateral distribution function. The effect of the arrival time distribution at several axis distance on estimated density for Yakutsk and AGASA is considered.

scholarly journals One-dimensional distributions of subordinators with upper truncated Lévy measure, and applications

2009 ◽  
Vol 41 (2) ◽  
pp. 367-392 ◽  
Author(s):  
Shai Covo
Keyword(s):  
Distribution Function ◽  
Lévy Process ◽  
Marginal Distribution ◽  
Levy Process ◽  
Gamma Process ◽  
Lévy Measure ◽  
Original Process ◽  
One Dimensional ◽  
Analogous Formula

Given a pure-jump subordinator (i.e. nondecreasing Lévy process with no drift) with continuous Lévy measure ν, we derive a formula for the distribution function Fs (x; t) at time t of the associated subordinator whose Lévy measure is the restriction of ν to (0,s]. It will be expressed in terms of ν and the marginal distribution function F (⋅; t) of the original process. A generalization concerning an arbitrary truncation of ν will follow. Under certain conditions, an analogous formula will be obtained for the nth derivative, ∂nFs (x; t) ∂ xn. The requirement that ν is continuous is shown to have no intrinsic meaning. A number of interesting results involving the size ordered jumps of subordinators will be derived. An appropriate approximation for the small jumps of a gamma process will be considered, leading to a revisiting of the generalized Dickman distribution.

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