Pub Date : 2023-12-05DOI: 10.1007/s10986-023-09612-0
Jonas Kazys Sunklodas
In this paper, we estimate the difference |Eh(Zn) −Eh(Y)| between the expectations of real finite Lipschitz function h of the sum Zn = (X1 + ⋯ + Xn)/Bn, where ({B}_{n}^{2}) = E(X1 + ⋯ + Xn)2> 0, and a standard normal random variable Y, where real centered random variables X1,X2,… satisfy the φ-mixing condition, defined between the “past” and “ future”, or are m-dependent. In particular cases, under the condition ({sum }_{r=1}^{infty }rvarphi (r)<infty ) or ({sum }_{r=1}^{infty }{rvarphi }^{1/2}(r)<infty ), the obtained upper bounds for φ-mixing random variables are of order O(n−1/2). In addition, we refine the previously known upper bounds of order O((m + 1)1+δL2+δ,n), where L2+δ,n is the Lyapunov fraction of order 2 + δ, for m-dependent random variables, supplementing them with explicit constants. We also separately present the case of independent r.v.s.
{"title":"On normal approximation for φ-mixing and m-dependent random variables","authors":"Jonas Kazys Sunklodas","doi":"10.1007/s10986-023-09612-0","DOIUrl":"https://doi.org/10.1007/s10986-023-09612-0","url":null,"abstract":"<p>In this paper, we estimate the difference |<b>E</b><i>h</i>(<i>Z</i><sub><i>n</i></sub>) <i>−</i> <b>E</b><i>h</i>(<i>Y</i>)<i>|</i> between the expectations of real finite Lipschitz function <i>h</i> of the sum <i>Z</i><sub><i>n</i></sub> = (<i>X</i><sub>1</sub> + ⋯ + <i>X</i><sub><i>n</i></sub>)<i>/B</i><sub><i>n</i></sub>, where <span>({B}_{n}^{2})</span> = <b>E</b>(<i>X</i><sub>1</sub> + ⋯ + <i>X</i><sub><i>n</i></sub>)<sup>2</sup> <i>></i> 0, and a standard normal random variable <i>Y</i>, where real centered random variables <i>X</i><sub>1</sub><i>,X</i><sub>2</sub><i>,</i>… satisfy the <i>φ</i>-mixing condition, defined between the “past” and “ future”, or are <i>m</i>-dependent. In particular cases, under the condition <span>({sum }_{r=1}^{infty }rvarphi (r)<infty )</span> or <span>({sum }_{r=1}^{infty }{rvarphi }^{1/2}(r)<infty )</span>, the obtained upper bounds for <i>φ</i>-mixing random variables are of order <i>O</i>(<i>n</i><sup><i>−</i>1<i>/</i>2</sup>). In addition, we refine the previously known upper bounds of order <i>O</i>((<i>m</i> + 1)<sup>1+<i>δ</i></sup><i>L</i><sub>2+<i>δ,n</i></sub>), where <i>L</i><sub>2+<i>δ,n</i></sub> is the Lyapunov fraction of order 2 + <i>δ</i>, for <i>m</i>-dependent random variables, supplementing them with explicit constants. We also separately present the case of independent r.v.s.</p>","PeriodicalId":51108,"journal":{"name":"Lithuanian Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2023-12-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138492793","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 : 2023-11-28DOI: 10.1007/s10986-023-09609-9
Algirdas Ambrazevičius, Vladas Skakauskas
We investigate the existence, uniqueness, and long-time behavior of classical solutions to a coupled system of seven nonlinear parabolic equations. Four of them are determined in the interior of a region, and the other three are solved on a part of the boundary. In particular, such systems arise in modeling of surface reactions that involve the bulk diffusion of reactants toward and reaction products from the biocatalyst surface and surface diffusion of the intermediate reaction products.
{"title":"Solvability of a nonlinear parabolic problem arising in modeling surface reactions","authors":"Algirdas Ambrazevičius, Vladas Skakauskas","doi":"10.1007/s10986-023-09609-9","DOIUrl":"https://doi.org/10.1007/s10986-023-09609-9","url":null,"abstract":"<p>We investigate the existence, uniqueness, and long-time behavior of classical solutions to a coupled system of seven nonlinear parabolic equations. Four of them are determined in the interior of a region, and the other three are solved on a part of the boundary. In particular, such systems arise in modeling of surface reactions that involve the bulk diffusion of reactants toward and reaction products from the biocatalyst surface and surface diffusion of the intermediate reaction products.</p>","PeriodicalId":51108,"journal":{"name":"Lithuanian Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2023-11-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138516943","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 : 2023-11-08DOI: 10.1007/s10986-023-09610-2
Lei Shi, Muhammad Arif
{"title":"Sharp bounds on the third Hankel determinant for the Ozaki close-to-convex and convex functions","authors":"Lei Shi, Muhammad Arif","doi":"10.1007/s10986-023-09610-2","DOIUrl":"https://doi.org/10.1007/s10986-023-09610-2","url":null,"abstract":"","PeriodicalId":51108,"journal":{"name":"Lithuanian Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-11-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135341060","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 : 2023-11-08DOI: 10.1007/s10986-023-09611-1
Wei Zhang
{"title":"On general sums involving the floor function with applications to k-free numbers","authors":"Wei Zhang","doi":"10.1007/s10986-023-09611-1","DOIUrl":"https://doi.org/10.1007/s10986-023-09611-1","url":null,"abstract":"","PeriodicalId":51108,"journal":{"name":"Lithuanian Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-11-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135391014","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 : 2023-07-01DOI: 10.1007/s10986-023-09601-3
Danijel Krizmanić
{"title":"A functional limit theorem for self-normalized linear processes with random coefficients and i.i.d. heavy-tailed innovations","authors":"Danijel Krizmanić","doi":"10.1007/s10986-023-09601-3","DOIUrl":"https://doi.org/10.1007/s10986-023-09601-3","url":null,"abstract":"","PeriodicalId":51108,"journal":{"name":"Lithuanian Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2023-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42842797","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 : 2023-07-01DOI: 10.1007/s10986-023-09602-2
B. Jasiulis-Gołdyn, Alicja Lechańska, Jolanta KrystynaMisiewicz
{"title":"Cramér–Lundberg model for some classes of extremal Markov sequences","authors":"B. Jasiulis-Gołdyn, Alicja Lechańska, Jolanta KrystynaMisiewicz","doi":"10.1007/s10986-023-09602-2","DOIUrl":"https://doi.org/10.1007/s10986-023-09602-2","url":null,"abstract":"","PeriodicalId":51108,"journal":{"name":"Lithuanian Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2023-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48818112","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 : 2023-07-01DOI: 10.1007/s10986-023-09607-x
Zbigniew J. Jurek
Abstract In the probability theory, selfdecomposable or class L 0 distributions play an important role as they are limit distributions of normalized partial sums of sequences of independent, not necessarily identically distributed, random variables. The class L 0 is quite large and includes many known classical distributions. For this note, the most important feature of the selfdecomposable variables are their random integral representation with respect to a Lévy process. From those random integral representations we get the equality of logarithms of some characteristic functions. These allow us to get formulas for some definite integrals; some of them were previously unknown, and some are rarely quoted in popular tables of integrals and series.
{"title":"Some definite integrals arising from selfdecomposable characteristic functions","authors":"Zbigniew J. Jurek","doi":"10.1007/s10986-023-09607-x","DOIUrl":"https://doi.org/10.1007/s10986-023-09607-x","url":null,"abstract":"Abstract In the probability theory, selfdecomposable or class L 0 distributions play an important role as they are limit distributions of normalized partial sums of sequences of independent, not necessarily identically distributed, random variables. The class L 0 is quite large and includes many known classical distributions. For this note, the most important feature of the selfdecomposable variables are their random integral representation with respect to a Lévy process. From those random integral representations we get the equality of logarithms of some characteristic functions. These allow us to get formulas for some definite integrals; some of them were previously unknown, and some are rarely quoted in popular tables of integrals and series.","PeriodicalId":51108,"journal":{"name":"Lithuanian Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135805387","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 : 2023-07-01DOI: 10.1007/s10986-023-09604-0
A. Alahmadi, O. Klurman, F. Luca, H. Shoaib
{"title":"On the arguments of the roots of the generalized Fibonacci polynomial","authors":"A. Alahmadi, O. Klurman, F. Luca, H. Shoaib","doi":"10.1007/s10986-023-09604-0","DOIUrl":"https://doi.org/10.1007/s10986-023-09604-0","url":null,"abstract":"","PeriodicalId":51108,"journal":{"name":"Lithuanian Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2023-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42730345","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 : 2023-07-01DOI: 10.1007/s10986-023-09606-y
Mantas Dirma, Neda Nakliuda, J. Šiaulys
{"title":"Generalized moments of sums with heavy-tailed random summands","authors":"Mantas Dirma, Neda Nakliuda, J. Šiaulys","doi":"10.1007/s10986-023-09606-y","DOIUrl":"https://doi.org/10.1007/s10986-023-09606-y","url":null,"abstract":"","PeriodicalId":51108,"journal":{"name":"Lithuanian Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2023-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42647852","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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