On the distribution of duration of stay in an interval of the semi-continuous process with independent increments

2004 ◽  
Vol 12 (4) ◽  
Author(s):  
V. F. Kadankov ◽  
T. V. Kadankova
Keyword(s):  
Continuous Process ◽  
Duration Of Stay ◽  
Process With Independent Increments ◽  
Independent Increments

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.

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].

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