On crossing of a level by processes defined by sums of a random number of terms

Gusak, D. V.

doi:10.1007/bf01084899

Cited by 3 publications

(3 citation statements)

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“…The integral transforms of this distribution [9] differ from our results (the formula (24) for x = 0 ). However, to solve the two-boundary problem, we require a more general one-boundary functional of the process {D x (t)} t≥0 , i.e.…”

Section: Introduce the Following Notation For The Mathematical Expectcontrasting

confidence: 54%

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Intersections of an Interval By a Difference of a Compound Poisson Process and a Compound Renewal Process

2009

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In this article we determine the Laplace transforms of the one-boundary characteristics and of the distribution of the number of intersections of a fixed interval by a difference of a compound Poisson process and a compound renewal process. The results obtained are applied for a particular case of this process, namely, for the difference of the compound Poisson process and the renewal process whose jumps are geometrically distributed. The advantage is that these results are in a closed form, in terms of resolvent sequences of the process. In this case, under certain assumptions, we find the limit distributions of the one-boundary and two-boundary characteristics of the process. In addition, we prove the weak convergence of these distributions to the corresponding distributions of a symmetric Wiener process.

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Section: Introduce the Following Notation For The Mathematical Expectcontrasting

confidence: 54%

“…The joint distribution of {τ k (0), T k (0)} was studied by Gusak [9]. The integral transforms of this distribution [9] differ from our results (the formula (24) for x = 0 ).…”

Section: Introduce the Following Notation For The Mathematical Expectcontrasting

confidence: 51%

Intersections of an Interval By a Difference of a Compound Poisson Process and a Compound Renewal Process

2009

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“…The joint distribution of {τ k (0), T k (0)} was studied by D. V. Gusak [14]. The integral transforms of this distribution were obtained in terms of the projectors without explicit form and they do not coincide with our results (formulae (3.4), (3.5) for x = 0).…”

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confidence: 66%

Intersections of the interval and reflections for a semi-Markov walk with linear drift

Kadankov¹

2008

Random Operators and Stochastic Equations

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Several two-boundary problems are solved for a semi-Markov random walk with a linear drift. The Laplace transforms of the joint distribution of the number of upward and downward intersection of the interval by the process. We also find the Laplace transforms of the boundary functionals of the processes reflected at their boundaries. The results obtained are applied for a special case when the jumps of the process are exponentially distributed.Key words. The semi-Markov walk with a linear drift; exit from an interval; upward and downward intersection of the interval; reflected processes.

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Two-boundary problems for semi-Markov walk with a linear drift

Kadankov¹,

Kadankova²

2007

Random Operators and Stochastic Equations

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On crossing of a level by processes defined by sums of a random number of terms

Cited by 3 publications

References 3 publications

Intersections of an Interval By a Difference of a Compound Poisson Process and a Compound Renewal Process

Intersections of an Interval By a Difference of a Compound Poisson Process and a Compound Renewal Process

Intersections of the interval and reflections for a semi-Markov walk with linear drift

Two-boundary problems for semi-Markov walk with a linear drift

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