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Authors:I. A. Alexeev Pages: 499 - 515 Abstract: Theory of Probability & Its Applications, Volume 67, Issue 4, Page 499-515, February 2023. This paper, which is a continuation of [I. A. Alexeev, Theory Probab. Appl., 67 (2022), pp. 335--351], is concerned with $\alpha$-stable distributions with complex stability index $\alpha$. Sufficient conditions for membership in the domain of attraction of $\alpha$-stable random variables (r.v.'s) are given, and $\alpha$-stable Lévy processes and the corresponding semigroups of operators are constructed. Necessary and sufficient conditions are given for membership in the class of limit laws for sums of independent and identically distributed (i.i.d.) complex r.v.'s with complex normalization and centering. Citation: Theory of Probability & Its Applications PubDate: 2023-02-08T08:00:00Z DOI: 10.1137/S0040585X97T991118 Issue No:Vol. 67, No. 4 (2023)

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Authors:V. A. Vatutin, E. E. D'yakonova Pages: 516 - 534 Abstract: Theory of Probability & Its Applications, Volume 67, Issue 4, Page 516-534, February 2023. We consider a Galton--Watson branching process with particles of two types in which particles of type one produce both particles of types one and two, and particles of type two generate offsprings of only type two. It is known that if both types are critical, then, for a process that is initiated at time $0$ by a single type-one particle, the number of particles of type two at time $n$ (provided that the process is not degenerate by this time) is proportional to $n$. We find the asymptotics of the probability that the number of type-two particles at time $n$ is of the order $o(n) $ (provided that the process is not degenerate by this time). Citation: Theory of Probability & Its Applications PubDate: 2023-02-08T08:00:00Z DOI: 10.1137/S0040585X97T99112X Issue No:Vol. 67, No. 4 (2023)

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Authors:E. S. Palamarchuk Pages: 535 - 547 Abstract: Theory of Probability & Its Applications, Volume 67, Issue 4, Page 535-547, February 2023. A linear control system over an infinite time-horizon is considered, where external excitations are defined as polynomials based on a time-varying Ornstein--Uhlenbeck process. An optimal control law with respect to long-run average type criteria is established. It is shown that the optimal control has the form of a linear feedback law, where the affine term satisfies a backward linear stochastic differential equation. The normalizing functions in the optimality criteria depend on the stability rate of the dynamic equation for the Ornstein--Uhlenbeck process. Citation: Theory of Probability & Its Applications PubDate: 2023-02-08T08:00:00Z DOI: 10.1137/S0040585X97T991131 Issue No:Vol. 67, No. 4 (2023)

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Authors:S. N. Smirnov Pages: 548 - 569 Abstract: Theory of Probability & Its Applications, Volume 67, Issue 4, Page 548-569, February 2023. We consider a guaranteed deterministic approach to the discrete-time superreplication problem in which it is required to cover a contingent liability on a written option in all feasible scenarios. These scenarios are described by a priori given compact sets depending on the price history: at each time instant, the price increments must lie in the corresponding compact sets. We assume no transaction costs. The problem statement is game-theoretic and leads to the Bellman--Isaacs equations in pure and mixed “market” strategies. In the case of no trading constraints, we study the relationship between the Bellman functions in the “deterministic” and “probabilistic” statements of the superhedging problem. As established under very general conditions, the “probabilistic” Bellman function does not exceed the “deterministic” counterpart. Sufficient conditions for their coincidence are found. Citation: Theory of Probability & Its Applications PubDate: 2023-02-08T08:00:00Z DOI: 10.1137/S0040585X97T991143 Issue No:Vol. 67, No. 4 (2023)

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Authors:A. Klump, M. Kolb Pages: 570 - 592 Abstract: Theory of Probability & Its Applications, Volume 67, Issue 4, Page 570-592, February 2023. Given a distribution on the positive extended real line, the two-sided inverse first-passage time problem for Brownian motion asks for a function such that the first passage time of this function by a reflected Brownian motion has the given distribution. We combine the ideas of Ekström and Janson, which were developed within the scope of the one-sided inverse first-passage time problem, with the methods of De Masi et al., which were used in the context of free boundary problems, in order to give a different proof for the uniqueness for the two-sided inverse first-passage time problem by using a stochastic order relation. We provide criteria for qualitative properties of solutions of the inverse first-passage problem, which apply to the boundary corresponding to the exponential distribution. Citation: Theory of Probability & Its Applications PubDate: 2023-02-08T08:00:00Z DOI: 10.1137/S0040585X97T991155 Issue No:Vol. 67, No. 4 (2023)

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Authors:R. Maller, S. Shemehsavar Pages: 593 - 612 Abstract: Theory of Probability & Its Applications, Volume 67, Issue 4, Page 593-612, February 2023. We study exchangeable random partitions based on an underlying Dickman subordinator and the corresponding family of Poisson--Dirichlet distributions. The large sample distribution of the vector representing the block sizes and the number of blocks in a partition of $\{1,2,\dots,n\}$ is shown to be, after norming and centering, a product of independent Poissons and a normal distribution. In a species or gene sampling situation, these quantities represent the abundances and the numbers of species or genes observed in a sample of size $n$ from the corresponding Poisson--Dirichlet distribution. We include a summary of known convergence results concerning the Dickman subordinator in this context. Citation: Theory of Probability & Its Applications PubDate: 2023-02-08T08:00:00Z DOI: 10.1137/S0040585X97T991167 Issue No:Vol. 67, No. 4 (2023)

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Authors:S. O'Rourke, N. Williams Pages: 613 - 632 Abstract: Theory of Probability & Its Applications, Volume 67, Issue 4, Page 613-632, February 2023. For an $n \times n$ independent-entry random matrix $X_n$ with eigenvalues $\lambda_1, \dots, \lambda_n$, the seminal work of Rider and Silverstein [Ann. Probab., 34 (2006), pp. 2118--2143] asserts that the fluctuations of the linear eigenvalue statistics $\sum_{i=1}^n f(\lambda_i)$ converge to a Gaussian distribution for sufficiently nice test functions $f$. We study the fluctuations of $\sum_{i=1}^{n-K} f(\lambda_i)$, where $K$ randomly chosen eigenvalues have been removed from the sum. In this case, we identify the limiting distribution and show that it need not be Gaussian. Our results hold for the case when $K$ is fixed as well as for the case when $K$ tends to infinity with $n$. The proof utilizes the predicted locations of the eigenvalues introduced by E. Meckes and M. Meckes, [Ann. Fac. Sci. Toulouse Math. (6), 24 (2015), pp. 93--117]. As a consequence of our methods, we obtain a rate of convergence for the empirical spectral distribution of $X_n$ to the circular law in Wasserstein distance, which may be of independent interest. Citation: Theory of Probability & Its Applications PubDate: 2023-02-08T08:00:00Z DOI: 10.1137/S0040585X97T991179 Issue No:Vol. 67, No. 4 (2023)

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Authors:E. O. Lenena Pages: 633 - 639 Abstract: Theory of Probability & Its Applications, Volume 67, Issue 4, Page 633-639, February 2023. The accuracy of approximation of the vector of queue lengths for an open Jackson network with regenerative input flow and unreliable servers is estimated. A theorem on the accuracy of approximation of the vector of queue lengths in open Jackson networks is put forward, i.e., an estimate for the probability of deviations of the norm of the difference between the process of queue lengths and the constructed reflected Brownian motion is obtained. As a corollary, an estimate of the Wasserstein distance is given. Citation: Theory of Probability & Its Applications PubDate: 2023-02-08T08:00:00Z DOI: 10.1137/S0040585X97T991180 Issue No:Vol. 67, No. 4 (2023)

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Authors:M. A. Lifshits, A. A. Tadevosian Pages: 640 - 644 Abstract: Theory of Probability & Its Applications, Volume 67, Issue 4, Page 640-644, February 2023. We consider the asymptotic behavior of the expectation of the maximum for a special assignment process with constant or i.i.d. coefficients. We show how this expectation depends on the coefficients' distribution. Citation: Theory of Probability & Its Applications PubDate: 2023-02-08T08:00:00Z DOI: 10.1137/S0040585X97T991192 Issue No:Vol. 67, No. 4 (2023)

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Authors:R. Aliyev, V. Bayramov Pages: 645 - 651 Abstract: Theory of Probability & Its Applications, Volume 67, Issue 4, Page 645-651, February 2023. A renewal--reward process with dependent components and heavy-tailed interarrival times is investigated, and an asymptotic expansion as $t\to\infty$ for the expectation is derived. Citation: Theory of Probability & Its Applications PubDate: 2023-02-08T08:00:00Z DOI: 10.1137/S0040585X97T991209 Issue No:Vol. 67, No. 4 (2023)

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Authors:A. N. Shiryaev, I. V. Pavlov Pages: 652 - 667 Abstract: Theory of Probability & Its Applications, Volume 67, Issue 4, Page 652-667, February 2023. This paper presents abstracts of talks given at the 7th International Conference on Stochastic Methods (ICSM-7), held June 2--9, 2022 at Divnomorskoe (near the town of Gelendzhik) at the Raduga sports and fitness center of the Don State Technical University. The conference was chaired by A. N. Shiryaev. Participants included leading scientists from Russia, France, Portugal, and Tadjikistan. Citation: Theory of Probability & Its Applications PubDate: 2023-02-08T08:00:00Z DOI: 10.1137/S0040585X97T991210 Issue No:Vol. 67, No. 4 (2023)

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Authors: Editorial Board of TPA Pages: 668 - 670 Abstract: Theory of Probability & Its Applications, Volume 67, Issue 4, Page 668-670, February 2023. A remembrance of the life and accomplishments of Larisa Grigor'evna Afanasyeva, who passed away on August 26, 2022. Citation: Theory of Probability & Its Applications PubDate: 2023-02-08T08:00:00Z DOI: 10.1137/S0040585X97T991222 Issue No:Vol. 67, No. 4 (2023)