General branching processes: Convergence to irzhina processes

Translated from Problemy Ustoichivosti Stokhasticheskikh Modelei, Trudy Seminara, pp. 74–82, 1990.     

Limit behavior of additive functionals of semistable processes and processes attracted to semistable processes

The limit behavior of distributions of functionals of the form $$v_T = \frac{1}{T}\int\limits_o^T {h\left( {x_t } \right)dt,} $$ is studied in the paper; here x_t is a process with values in R_m which is semistable or is attracted to a semistable process, and h, h:R_m→-R₁, is bounded and measurable. Sufficient conditions are obtained for the existence of a limit distribution. It is shown that in the one-dimensional case these conditions are actually also necessary. Results are also presented on the joint behavior of several functionals of similar type.     

Sample-path analysis of processes with imbedded point processes

The purpose of this paper is to review, unify, and extend previous work on sample-path analysis of queues. Our main interest is in the asymptotic behavior of a discrete-state, continuous-time process with an imbedded point process. We present a sample-path analogue of the renewal-reward theorem, which we callY=X. We then applyY=X to derive several relations involving the transition rates and the asymptotic (long-run) state frequencies at an arbitrary point in time and at the points of the imbedded point process. Included are sample-path versions of the rate-conservation principle, the global-balance conditions, and the insensitivity of the asymptotic frequency distribution to the distribution of processing time in a LCFS-PR service facility. We also provide a natural sample-path characterization of the PASTA property.The research of this author was partially supported by the U.S. Army Research Office, Contract DAAG29-82-K-0151 at N.C. State University, and by the National Science Foundation, Grant No. ECS-8719825, at the University of North Carolina, Chapel Hill.The research of this author was partially supported by the U.S. Army Research Office, Contract DAAG29-82-K-0151 at N.C. State University.     

Fuzzy processes

In this paper, contributions to fuzzy probability and to differential equations with fuzzy parameters are made.After an introductory section, a review of fuzzy sets and fuzzy algebra is given in Section 2. The main new results of the investigation are contained in Section 3.In Section 3, Zadeh's definition of the probability of a ‘fuzzy event’ the average value of a fuzzy function are extended into the time domain. It is then shown that not only grades of membership, but also probabilistic processes with notions of fuzziness contained, can be defined which obey ordinary, matric, or integro-differential equations. Applications are also given in Section 3.     

Branching processes

Author's abstract of a thesis for the attainment of the academic degree of Doctor of Physics and Mathematics. The thesis was defended on April 25, 1968 at a meeting of the Scientific Council of the V. A. Steklov Mathematical Institute, Academy of Sciences of the USSR. Official opponents: Academician A. N. Kolmogorov, Yu. V. Prokhorov, Corresponding Member of the Academy of Sciences of the USSR, and Prof. Yaglom, Doctor of Physics and Mathematics.Translated from Matematicheskie Zametki, Vol. 4, No. 2, pp. 239–251, August, 1968.     

Wishart processes

We propose some matrix generalizations of square Bessel processes and we indicate their first properties: hitting time of 0 of the smallest eigenvalue, additivity property, associated Martingales, distributions, which mainly extend the real-valued classical results. We explain why these processes are indecomposable and therefore differ from the real-valued ones. We conclude with some formulae concerning matrix quadratic functionals analogous to the Cameron Martin formula.     

Interval-dividing processes

Summary If and then P(n ^–1·[(Y ₁)++(Y _n )] converges to cnts. law on R ¹) = P(n ^–1·[(Y ₁)++(Y _n )] converges to a cnts. law on R ¹). Thus if ,n then n ^–1[(X ₁)+...+(X _n )] converges a.s. The main result here generalizes this: Let X ₍₁₎ ⁿ , X ₍₂₎ ⁿ ,..., X _(n) ⁿ be the order statistics associated with X ₁, X ₂,,X _n. Define random variables Z ₁,Z ₂, by {Z _n =i}={X _n =X _(i) ⁿ }. Then if Z ₁,Z ₂,Z ₃, are independent and P(Z_ni)i/n, and {X _i} is bounded, n ^–1·[(X ₁)++(X _n)] converges a.s.     

Macdonald processes

Macdonald processes are probability measures on sequences of partitions defined in terms of nonnegative specializations of the Macdonald symmetric functions and two Macdonald parameters $q,t \in [0,1)$ . We prove several results about these processes, which include the following. (1) We explicitly evaluate expectations of a rich family of observables for these processes. (2) In the case $t=0$ , we find a Fredholm determinant formula for a $q$ -Laplace transform of the distribution of the last part of the Macdonald-random partition. (3) We introduce Markov dynamics that preserve the class of Macdonald processes and lead to new “integrable” 2d and 1d interacting particle systems. (4) In a large time limit transition, and as $q$ goes to 1, the particles of these systems crystallize on a lattice, and fluctuations around the lattice converge to O’Connell’s Whittaker process that describe semi-discrete Brownian directed polymers. (5) This yields a Fredholm determinant for the Laplace transform of the polymer partition function, and taking its asymptotics we prove KPZ universality for the polymer (free energy fluctuation exponent $1/3$ and Tracy-Widom GUE limit law). (6) Under intermediate disorder scaling, we recover the Laplace transform of the solution of the KPZ equation with narrow wedge initial data. (7) We provide contour integral formulas for a wide array of polymer moments. (8) This results in a new ansatz for solving quantum many body systems such as the delta Bose gas.     

Convergence of processes of step sums on mixing processes

We prove a limit theorem on the convergence of nonhomogeneous centered processes of step sums, constructed on random mixing sequences, to a process with independent increments. We also prove a invariance-principle type theorem for schemes of summation of functionals on random mixing sequences.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 42, No. 5, pp. 716–721, May, 1990.     

Projection of processes with orthogonal increments and subordinated processes

Second-order random processes are considered as curves in a Hilbert space ? of random variables ξ with Eξ=0, E∣ξ∣²<∞. Processes which are a projection of a given process x(t) with orthogonal increments on some subspaces of ? are considered. Processes subordinate to x(t) are also considered.     

On coupling of stochastic processes with embedded point processes

Criteria for semi-, wide-sense-, traditional regeneration and a coupling construction of stochastic processes with embedded point processes are presented.     

Necessary conditions for renewal processes approximating to Wiener processes

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Constructions of coupling processes for Lévy processes

We construct optimal Markov couplings of Lévy processes, whose Lévy (jump) measure has an absolutely continuous component. The construction is based on properties of subordinate Brownian motions and the coupling of Brownian motions by reflection.