Pub Date : 2024-08-14DOI: 10.1137/s0040585x97t991854
A. A. Borovkov, E. I. Prokopenko
Theory of Probability &Its Applications, Volume 69, Issue 2, Page 186-204, August 2024. We study the distribution of the maximal element $overline{xi}_n$ of a sequence of independent random variables $xi_1,dots,xi_n$ and not only for them. The presented approach is more transparent (in our opinion) than the one used before. We consider four classes of distributions with right-unbounded supports and find limit theorems (in an explicit form) of the distribution of $overline{xi}_n$ for them. Earlier, only two classes of right-unbounded distributions were considered, and it was assumed a priori that the normalization of $overline{xi}_n$ is linear; in addition, the components of the normalization (in their explicit form) were unknown. For the two new classes, the required normalization turns our to be nonlinear. Results of this kind are also obtained for four classes of distributions with right-bounded support, which are analogues of the above four right-unbounded distributions (earlier, only the class of distributions with right-bounded support was considered). Some extensions of these results are obtained.
{"title":"On Limit Theorems for the Distribution of the Maximal Element in a Sequence of Random Variables","authors":"A. A. Borovkov, E. I. Prokopenko","doi":"10.1137/s0040585x97t991854","DOIUrl":"https://doi.org/10.1137/s0040585x97t991854","url":null,"abstract":"Theory of Probability &Its Applications, Volume 69, Issue 2, Page 186-204, August 2024. <br/> We study the distribution of the maximal element $overline{xi}_n$ of a sequence of independent random variables $xi_1,dots,xi_n$ and not only for them. The presented approach is more transparent (in our opinion) than the one used before. We consider four classes of distributions with right-unbounded supports and find limit theorems (in an explicit form) of the distribution of $overline{xi}_n$ for them. Earlier, only two classes of right-unbounded distributions were considered, and it was assumed a priori that the normalization of $overline{xi}_n$ is linear; in addition, the components of the normalization (in their explicit form) were unknown. For the two new classes, the required normalization turns our to be nonlinear. Results of this kind are also obtained for four classes of distributions with right-bounded support, which are analogues of the above four right-unbounded distributions (earlier, only the class of distributions with right-bounded support was considered). Some extensions of these results are obtained.","PeriodicalId":51193,"journal":{"name":"Theory of Probability and its Applications","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2024-08-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142198635","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-08-14DOI: 10.1137/s0040585x97t991866
M. V. Zhitlukhin
Theory of Probability &Its Applications, Volume 69, Issue 2, Page 205-216, August 2024. We consider a stochastic multiagent market model with endogenous asset prices and find a market strategy which cannot be asymptotically outperformed by a single agent. Such a strategy should distribute its capital among the assets proportionally to the conditional expectations of their discounted relative dividend intensities. The main assumption, under which the results are obtained, is that all agents should be small in the sense that actions of an individual agent do not affect the asset prices. The optimal strategy is found as a solution of a linear backward stochastic differential equation.
概率论及其应用》（Theory of Probability &Its Applications），第 69 卷第 2 期，第 205-216 页，2024 年 8 月。 我们考虑了一个具有内生资产价格的随机多代理市场模型，并找到了一种单个代理无法渐进地超越其表现的市场策略。这种策略应根据资产贴现相对红利强度的条件预期，按比例在资产间分配资本。得出结果的主要假设是，所有代理都是小代理，即单个代理的行为不会影响资产价格。最优策略是线性反向随机微分方程的解。
{"title":"Optimal Growth Strategies in a Stochastic Market Model with Endogenous Prices","authors":"M. V. Zhitlukhin","doi":"10.1137/s0040585x97t991866","DOIUrl":"https://doi.org/10.1137/s0040585x97t991866","url":null,"abstract":"Theory of Probability &Its Applications, Volume 69, Issue 2, Page 205-216, August 2024. <br/> We consider a stochastic multiagent market model with endogenous asset prices and find a market strategy which cannot be asymptotically outperformed by a single agent. Such a strategy should distribute its capital among the assets proportionally to the conditional expectations of their discounted relative dividend intensities. The main assumption, under which the results are obtained, is that all agents should be small in the sense that actions of an individual agent do not affect the asset prices. The optimal strategy is found as a solution of a linear backward stochastic differential equation.","PeriodicalId":51193,"journal":{"name":"Theory of Probability and its Applications","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2024-08-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142198634","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-08-14DOI: 10.1137/s0040585x97t991933
A. V. Bulinski
Theory of Probability &Its Applications, Volume 69, Issue 2, Page 313-321, August 2024. We study the distribution of the maximal element $overline{xi}_n$ of a sequence of (possibly) independent random variables $xi_1,dots,xi_n$. A formula for evaluation of a random variable expectation based on a quantile function is considered. This formula is applied to evaluation of the expectation for a nondecreasing function of a random variable transformed via its distribution function. The case of a discontinuous distribution function is the most interesting. As a corollary, we refine an example proposed in the author's previous article [Theory Probab. Appl., 68 (2023), pp. 392--410].
{"title":"On an Example of Expectation Evaluation","authors":"A. V. Bulinski","doi":"10.1137/s0040585x97t991933","DOIUrl":"https://doi.org/10.1137/s0040585x97t991933","url":null,"abstract":"Theory of Probability &Its Applications, Volume 69, Issue 2, Page 313-321, August 2024. <br/> We study the distribution of the maximal element $overline{xi}_n$ of a sequence of (possibly) independent random variables $xi_1,dots,xi_n$. A formula for evaluation of a random variable expectation based on a quantile function is considered. This formula is applied to evaluation of the expectation for a nondecreasing function of a random variable transformed via its distribution function. The case of a discontinuous distribution function is the most interesting. As a corollary, we refine an example proposed in the author's previous article [Theory Probab. Appl., 68 (2023), pp. 392--410].","PeriodicalId":51193,"journal":{"name":"Theory of Probability and its Applications","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2024-08-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142198627","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-08-14DOI: 10.1137/s0040585x97t991908
V. L. Kulikov, E. F. Olekhova, V. I. Oseledets
Theory of Probability &Its Applications, Volume 69, Issue 2, Page 265-280, August 2024. We consider a power series at a fixed point $rho in (0.5,1)$, where random coefficients assume a value $0$ or $1$ and form a stationary ergodic aperiodic process. The Erdös measure is the distribution law of such a series. The problem of absolute continuity of the Erdös measure is reduced to the problem of determining when the corresponding hidden Markov chain is a Parry--Markov chain. For the golden ratio and a 1-Markov chains, we give necessary and sufficient conditions for absolute continuity of the Erdös measure and, using Blackwell--Markov chains, provide a new proof that the necessary conditions obtained earlier by Bezhaeva and Oseledets [Theory Probab. Appl., 51 (2007), pp. 28--41] are also sufficient. For tribonacci numbers and 1-Markov chains, we give a new proof of the theorem on singularity of the Erdös measure. For tribonacci numbers and 2-Markov chains, we find only two cases with absolute continuity.
{"title":"On Absolute Continuity of the Erdös Measure for the Golden Ratio, Tribonacci Numbers, and Second-Order Markov Chains","authors":"V. L. Kulikov, E. F. Olekhova, V. I. Oseledets","doi":"10.1137/s0040585x97t991908","DOIUrl":"https://doi.org/10.1137/s0040585x97t991908","url":null,"abstract":"Theory of Probability &Its Applications, Volume 69, Issue 2, Page 265-280, August 2024. <br/> We consider a power series at a fixed point $rho in (0.5,1)$, where random coefficients assume a value $0$ or $1$ and form a stationary ergodic aperiodic process. The Erdös measure is the distribution law of such a series. The problem of absolute continuity of the Erdös measure is reduced to the problem of determining when the corresponding hidden Markov chain is a Parry--Markov chain. For the golden ratio and a 1-Markov chains, we give necessary and sufficient conditions for absolute continuity of the Erdös measure and, using Blackwell--Markov chains, provide a new proof that the necessary conditions obtained earlier by Bezhaeva and Oseledets [Theory Probab. Appl., 51 (2007), pp. 28--41] are also sufficient. For tribonacci numbers and 1-Markov chains, we give a new proof of the theorem on singularity of the Erdös measure. For tribonacci numbers and 2-Markov chains, we find only two cases with absolute continuity.","PeriodicalId":51193,"journal":{"name":"Theory of Probability and its Applications","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2024-08-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142198629","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-08-14DOI: 10.1137/s0040585x97t991921
V. I. Piterbarg
Theory of Probability &Its Applications, Volume 69, Issue 2, Page 294-312, August 2024. Gaussian random fields on finite-dimensional smooth manifolds, whose variance functions reach their maximum values at smooth submanifolds, are considered, and the exact asymptotic behavior of large excursion probabilities is established. It is shown that our conditions on the behavior of the covariation and variance are best possible in the context of the classical Pickands double sum method. Applications of our asymptotic formulas to large deviations of Gaussian vector processes are considered, and some examples are given. This paper continues the previous study of the author with Kobelkov, Rodionov, and Hashorva [J. Math. Sci., 262 (2022), pp. 504--513] which was concerned with Gaussian processes and fields on manifolds with a single point of maximum of the variance.
概率论及其应用》（Theory of Probability &Its Applications），第 69 卷第 2 期，第 294-312 页，2024 年 8 月。 考虑了有限维光滑流形上的高斯随机场，其方差函数在光滑子流形上达到最大值，并建立了大偏移概率的精确渐近行为。结果表明，我们关于协方差和方差行为的条件在经典皮康兹双和法中是最可行的。本文考虑了我们的渐近公式在高斯向量过程大偏离中的应用，并给出了一些示例。本文是作者与科贝尔科夫、罗迪奥诺夫和哈肖尔瓦先前研究的继续[《数学科学》，262 (2022)，第 504-513 页]，该研究涉及流形上的高斯过程和场，其方差有单点最大值。
{"title":"High Excursion Probabilities for Gaussian Fields on Smooth Manifolds","authors":"V. I. Piterbarg","doi":"10.1137/s0040585x97t991921","DOIUrl":"https://doi.org/10.1137/s0040585x97t991921","url":null,"abstract":"Theory of Probability &Its Applications, Volume 69, Issue 2, Page 294-312, August 2024. <br/> Gaussian random fields on finite-dimensional smooth manifolds, whose variance functions reach their maximum values at smooth submanifolds, are considered, and the exact asymptotic behavior of large excursion probabilities is established. It is shown that our conditions on the behavior of the covariation and variance are best possible in the context of the classical Pickands double sum method. Applications of our asymptotic formulas to large deviations of Gaussian vector processes are considered, and some examples are given. This paper continues the previous study of the author with Kobelkov, Rodionov, and Hashorva [J. Math. Sci., 262 (2022), pp. 504--513] which was concerned with Gaussian processes and fields on manifolds with a single point of maximum of the variance.","PeriodicalId":51193,"journal":{"name":"Theory of Probability and its Applications","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2024-08-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142198628","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-08-14DOI: 10.1137/s0040585x97t991891
O. E. Kudryavtsev, A. S. Grechko, I. E. Mamedov
Theory of Probability &Its Applications, Volume 69, Issue 2, Page 243-264, August 2024. We construct a universal Monte Carlo method for pricing the options whose payout function depends on the final position of the extremum of the Lévy process. The proposed method is capable of evaluating the prices of floating and fixed strike lookback options not only at the initial time but also during the entire period when the current position of the Lévy process may be different from its extremum. Our algorithm involves three stages: approximation of the cumulative distribution function (c.d.f.) of the extremum process, evaluation of its inversion, and simulation of the final position of the extremum of the Lévy process. We obtain new approximate formulas for the c.d.f.'s of the supremum and infimum processes for Lévy models via Wiener--Hopf factorization. We also describe the principles of developing a hybrid Monte Carlo method, which combines classical numerical methods for construction of the c.d.f. of the final position of the extremum process and machine learning methods for inverting the c.d.f. with the help of tensor neural networks. The efficiency of the universal Monte Carlo method for lookback option pricing is supported by numerical experiments.
{"title":"Monte Carlo Method for Pricing Lookback Type Options in Lévy Models","authors":"O. E. Kudryavtsev, A. S. Grechko, I. E. Mamedov","doi":"10.1137/s0040585x97t991891","DOIUrl":"https://doi.org/10.1137/s0040585x97t991891","url":null,"abstract":"Theory of Probability &Its Applications, Volume 69, Issue 2, Page 243-264, August 2024. <br/> We construct a universal Monte Carlo method for pricing the options whose payout function depends on the final position of the extremum of the Lévy process. The proposed method is capable of evaluating the prices of floating and fixed strike lookback options not only at the initial time but also during the entire period when the current position of the Lévy process may be different from its extremum. Our algorithm involves three stages: approximation of the cumulative distribution function (c.d.f.) of the extremum process, evaluation of its inversion, and simulation of the final position of the extremum of the Lévy process. We obtain new approximate formulas for the c.d.f.'s of the supremum and infimum processes for Lévy models via Wiener--Hopf factorization. We also describe the principles of developing a hybrid Monte Carlo method, which combines classical numerical methods for construction of the c.d.f. of the final position of the extremum process and machine learning methods for inverting the c.d.f. with the help of tensor neural networks. The efficiency of the universal Monte Carlo method for lookback option pricing is supported by numerical experiments.","PeriodicalId":51193,"journal":{"name":"Theory of Probability and its Applications","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2024-08-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142198631","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-08-14DOI: 10.1137/s0040585x97t99188x
I. A. Ibragimov, N. V. Smorodina, M. M. Faddeev
Theory of Probability &Its Applications, Volume 69, Issue 2, Page 227-242, August 2024. In the classical Itô formula, we propose replacing the second derivative (understood in the usual sense) by the second derivative in the sense of differentiation of distributions. In particular, we show that this can be done if the first derivative lies in the class $L_{2,mathrm{loc}}(mathbf{R})$. Earlier, Föllmer, Protter, and Shiryayev [Bernoulli, 1 (1995), pp. 149--169] obtained a different form of the last term in the Itô formula under the same conditions.
概率论及其应用》（Theory of Probability &Its Applications），第 69 卷，第 2 期，第 227-242 页，2024 年 8 月。 在经典的伊托公式中，我们建议用分布微分意义上的二阶导数代替二阶导数（通常意义上的理解）。我们特别指出，如果一阶导数位于类$L_{2,mathrm{loc}}(mathbf{R})$中，就可以做到这一点。早些时候，Föllmer、Protter 和 Shiryayev [Bernoulli, 1 (1995), pp.
{"title":"A Remark on the Itô Formula","authors":"I. A. Ibragimov, N. V. Smorodina, M. M. Faddeev","doi":"10.1137/s0040585x97t99188x","DOIUrl":"https://doi.org/10.1137/s0040585x97t99188x","url":null,"abstract":"Theory of Probability &Its Applications, Volume 69, Issue 2, Page 227-242, August 2024. <br/> In the classical Itô formula, we propose replacing the second derivative (understood in the usual sense) by the second derivative in the sense of differentiation of distributions. In particular, we show that this can be done if the first derivative lies in the class $L_{2,mathrm{loc}}(mathbf{R})$. Earlier, Föllmer, Protter, and Shiryayev [Bernoulli, 1 (1995), pp. 149--169] obtained a different form of the last term in the Itô formula under the same conditions.","PeriodicalId":51193,"journal":{"name":"Theory of Probability and its Applications","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2024-08-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142198632","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-08-14DOI: 10.1137/s0040585x97t991830
A. N. Shiryaev
Theory of Probability &Its Applications, Volume 69, Issue 2, Page 169-172, August 2024. This year, 2024, the Russian Academy of Sciences celebrates the significant date of the 300th anniversary of its creation. The paper provides brief information about the formation of the Academy and describes its current state.
概率论及其应用》（Theory of Probability &Its Applications），第 69 卷，第 2 期，第 169-172 页，2024 年 8 月。 今年（2024 年）是俄罗斯科学院成立 300 周年这一重要日子。本文简要介绍了该科学院的成立过程并描述了其现状。
{"title":"300 Years of the Russian Academy of Sciences","authors":"A. N. Shiryaev","doi":"10.1137/s0040585x97t991830","DOIUrl":"https://doi.org/10.1137/s0040585x97t991830","url":null,"abstract":"Theory of Probability &Its Applications, Volume 69, Issue 2, Page 169-172, August 2024. <br/> This year, 2024, the Russian Academy of Sciences celebrates the significant date of the 300th anniversary of its creation. The paper provides brief information about the formation of the Academy and describes its current state.","PeriodicalId":51193,"journal":{"name":"Theory of Probability and its Applications","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2024-08-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142198637","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-08-14DOI: 10.1137/s0040585x97t991957
A. I. Bufetov, I. A. Ibragimov, M. A. Lifshits, A. V. Malyutin, F. V. Petrov, N. V. Smorodina, A. N. Shiryaev, Yu. V. Yakubovich
Theory of Probability &Its Applications, Volume 69, Issue 2, Page 331-335, August 2024. A remembrance of the life and accomplishments of outstanding mathematician Anatolii Moiseevich Vershik, who passed away on February 14, 2024.
概率论及其应用》（Theory of Probability &Its Applications），第 69 卷，第 2 期，第 331-335 页，2024 年 8 月。 缅怀2024年2月14日逝世的杰出数学家阿纳托利-莫伊谢耶维奇-弗尔希克的生平和成就。
{"title":"In Memory of A. M. Vershik (12.28.1933--02.14.2024)","authors":"A. I. Bufetov, I. A. Ibragimov, M. A. Lifshits, A. V. Malyutin, F. V. Petrov, N. V. Smorodina, A. N. Shiryaev, Yu. V. Yakubovich","doi":"10.1137/s0040585x97t991957","DOIUrl":"https://doi.org/10.1137/s0040585x97t991957","url":null,"abstract":"Theory of Probability &Its Applications, Volume 69, Issue 2, Page 331-335, August 2024. <br/> A remembrance of the life and accomplishments of outstanding mathematician Anatolii Moiseevich Vershik, who passed away on February 14, 2024.","PeriodicalId":51193,"journal":{"name":"Theory of Probability and its Applications","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2024-08-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142198603","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-08-14DOI: 10.1137/s0040585x97t991878
A. Yu. Zaitsev
Theory of Probability &Its Applications, Volume 69, Issue 2, Page 217-226, August 2024. Let $X, X_1,dots, X_n,dots$ be independent identically distributed $d$-dimensional random vectors with common distribution $F$. Let $F_{(n)}$ be the distribution of the normalized random vector $X/sqrt{n}$. Then $(X_1+dots+X_n)/sqrt{n}$ has distribution $F_{(n)}^n$ (the power is understood in the convolution sense). Let $pi(,{cdot},,{cdot},)$ be the Prokhorov distance. We show that, for any $d$-dimensional distribution $F$, there exist $c_1(F)>0$ and $c_2(F)>0$ depending only on $F$ such that $pi(F_{(n)}^n, F_{(n)}^{n+1})leqslant c_1(F)/sqrt n$ and $(F^n){A} le (F^{n+1}){A^{c_2(F)}}+c_2(F)/sqrt{n}$, $(F^{n+1}){A} leq (F^n){A^{c_2(F)}}+c_2(F)/sqrt{n}$ for each Borel set $A$ and for all natural numbers $n$ (here, $A^{varepsilon}$ denotes the $varepsilon$-neighborhood of a set $A$).
{"title":"On Proximity of Distributions of Successive Sums with Respect to the Prokhorov Distance","authors":"A. Yu. Zaitsev","doi":"10.1137/s0040585x97t991878","DOIUrl":"https://doi.org/10.1137/s0040585x97t991878","url":null,"abstract":"Theory of Probability &Its Applications, Volume 69, Issue 2, Page 217-226, August 2024. <br/> Let $X, X_1,dots, X_n,dots$ be independent identically distributed $d$-dimensional random vectors with common distribution $F$. Let $F_{(n)}$ be the distribution of the normalized random vector $X/sqrt{n}$. Then $(X_1+dots+X_n)/sqrt{n}$ has distribution $F_{(n)}^n$ (the power is understood in the convolution sense). Let $pi(,{cdot},,{cdot},)$ be the Prokhorov distance. We show that, for any $d$-dimensional distribution $F$, there exist $c_1(F)>0$ and $c_2(F)>0$ depending only on $F$ such that $pi(F_{(n)}^n, F_{(n)}^{n+1})leqslant c_1(F)/sqrt n$ and $(F^n){A} le (F^{n+1}){A^{c_2(F)}}+c_2(F)/sqrt{n}$, $(F^{n+1}){A} leq (F^n){A^{c_2(F)}}+c_2(F)/sqrt{n}$ for each Borel set $A$ and for all natural numbers $n$ (here, $A^{varepsilon}$ denotes the $varepsilon$-neighborhood of a set $A$).","PeriodicalId":51193,"journal":{"name":"Theory of Probability and its Applications","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2024-08-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142198633","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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