Fourier series associated with the sample functions of a stochastic process

1970 ◽  
Vol 67 (1) ◽  
pp. 101-106
Author(s):  
G. Samal
Keyword(s):  
Stochastic Process ◽  
Continuous Process ◽  
Finite Measure ◽  
Unit Interval ◽  
Finite Mass ◽  
Process With Independent Increments ◽  
Independent Increments ◽  
Image Position ◽  
Almost All ◽  
Essential Restriction

We consider a stochastically continuous process ω(t, α) with independent increments, whose sample functions are bounded in the unit interval 0 ≤ t ≤ 1 for almost all α. If ω(t, α) is a process with independent increments, the characteristic function of ω(t, α) is of the form exp {tψ(u)} where where F is a σ-finite measure with finite mass outside every neighbourhood of o, and a and σ are constants. There is no essential restriction in supposing ω(0, α) = 0.

Representation of the characteristic function of a stochastic integral

1980 ◽  
Vol 17 (2) ◽  
pp. 448-455 ◽  
Author(s):  
M. Riedel
Keyword(s):  
Stochastic Process ◽  
Continuous Function ◽  
Characteristic Function ◽  
Stochastic Integral ◽  
Representation Formula ◽  
Canonical Representation ◽  
Convergence In Probability ◽  
Process With Independent Increments ◽  
Independent Increments ◽  
Image Position

Let X(t) be a continuous, homogeneous stochastic process with independent increments characterized by a, σ, M, N in the Lévy representation formula. In this note we obtain the Lévy canonical representation of the characteristic function of a stochastic integral (in the sense of convergence in probability) of the form (where υ(t) is a non-decreasing, non-negative and left-continuous function) in terms of υ(t), a, σ, M, N.

Representation of the characteristic function of a stochastic integral

1980 ◽  
Vol 17 (02) ◽  
pp. 448-455 ◽  
Author(s):  
M. Riedel
Keyword(s):  
Stochastic Process ◽  
Continuous Function ◽  
Characteristic Function ◽  
Stochastic Integral ◽  
Representation Formula ◽  
Canonical Representation ◽  
Convergence In Probability ◽  
Process With Independent Increments ◽  
Independent Increments ◽  
Image Position

Let X(t) be a continuous, homogeneous stochastic process with independent increments characterized by a, σ, M, N in the Lévy representation formula. In this note we obtain the Lévy canonical representation of the characteristic function of a stochastic integral (in the sense of convergence in probability) of the form (where υ(t) is a non-decreasing, non-negative and left-continuous function) in terms of υ(t), a, σ, M, N.

Characterization of stable processes by identically distributed stochastic integrals

1980 ◽  
Vol 12 (3) ◽  
pp. 689-709 ◽  
Author(s):  
M. Riedel
Keyword(s):  
Stochastic Process ◽  
Stable Process ◽  
Convergence In Probability ◽  
Stochastic Integrals ◽  
Stable Processes ◽  
Process With Independent Increments ◽  
Independent Increments ◽  
The Subject

Let X(t) be a homogeneous and continuous stochastic process with independent increments. The subject of this paper is to characterize the stable process by two identically distributed stochastic integrals formed by means of X(t) (in the sense of convergence in probability). The proof of the main results is based on a modern extension of the Phragmén-Lindelöf theory.

A characterization of stable processes

1969 ◽  
Vol 6 (02) ◽  
pp. 409-418 ◽  
Author(s):  
Eugene Lukacs
Keyword(s):  
Stochastic Process ◽  
Distribution Function ◽  
Interior Point ◽  
Random Variable ◽  
Stable Processes ◽  
Process With Independent Increments ◽  
Independent Increments

Let X(t) be a stochastic process whose parameter t runs over a finite or infinite n terval T. Let t 1 , t 2 ɛ T, t 1 〈 t2; the random variable X(t 2) – X(t 1) is called the increment of the process X(t) over the interval [t 1, t 2]. A process X(t) is said to be homogeneous if the distribution function of the increment X(t + τ) — X(t) depends only on the length τ of the interval but is independent of the endpoint t. Two intervals are said to be non-overlapping if they have no interior point in common. A process X(t) is called a process with independent increments if the increments over non-overlapping intervals are stochastically independent. A process X(t) is said to be continuous at the point t if plimτ→0 [X(t + τ) — X(t)] = 0, that is if for any ε > 0, limτ→0 P(| X(t + τ) — X(t) | > ε) = 0. A process is continuous in an interval [A, B] if it is continuous in every point of [A, B].

On the asymptotic behaviour of sample spacings

1981 ◽  
Vol 90 (2) ◽  
pp. 293-303 ◽  
Author(s):  
John Hawkes
Keyword(s):  
Growth Rate ◽  
Asymptotic Behaviour ◽  
Growth Rates ◽  
Maximum Growth ◽  
Maximum Growth Rate ◽  
Unit Interval ◽  
Sample Spacing ◽  
Image Position ◽  
Almost All ◽  
Sample Spacings

SummarySuppose that we are given a random sample of size n chosen according to the uniform distribution on the unit interval. Let Zn(x) = Zn(x, ω) be the length of the unique left-closed and right-open sample spacing that contains x. The purpose of this paper is to examine the almost sure, and exceptional, growth rates of the process {Zn}. The typical maximum growth rate and the growth rate of the maximum can be of quite different orders of magnitude as is shown by the following two results.Theorem 2. With probability one we havefor almost all x.Theorem 3. With probability one we have

Characterization of stochastic processes by stochastic integrals

1983 ◽  
Vol 15 (1) ◽  
pp. 81-98 ◽  
Author(s):  
B. L. S. Prakasa Rao
Keyword(s):  
Stochastic Process ◽  
Stochastic Processes ◽  
Recent Work ◽  
Stochastic Integrals ◽  
Stable Processes ◽  
Process With Independent Increments ◽  
Independent Increments

Let be a continuous homogeneous stochastic process with independent increments. A review of the recent work on the characterization of Wiener and stable processes and connected results through stochastic integrals is presented. No proofs are given but appropriate references are mentioned.

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