s, say $\\{y_l\\}$ also converge jointly and are asymptotically free. When $y=0$, the matrix $\\sqrt{np^{-1}}(C-\\rho I_p)$ converges to an elliptic variable with parameter $\\rho^2$. In particular, this elliptic variable is circular when $\\rho=0$ and is semi-circular when $\\rho=1$. If we take independent $C_l$, then the matrices $\\{\\sqrt{n_lp^{-1}}(C_l-\\rho_l I_p)\\}$ converge jointly and are also asymptotically free. As a consequence, the limiting spectral distribution of any symmetric matrix polynomial exists and has compact support.","authors":[{"id":"62e494fdd9f204418d6b638c","name":"Monika Bhattacharjee","org":"Indian Inst Technolgy Bombay, Dept Math, Mumbai, India","orgid":"5f71b28d1c455f439fe3ca7e"},{"email":"bosearu@gmail.com","id":"53f3a7b6dabfae4b34ae38ea","name":"Arup Bose","org":"Indian Stat Inst, Stat Math Unit, Kolkata, India"},{"id":"661ee048e6a7985c6e06ffe6","name":"Apratim Dey","org":"Stanford Univ, Dept Stat, Stanford, CA USA","orgid":"62331e330a6eb147dca8a6e8"}],"create_time":"2021-03-23T13:48:48.011Z","doi":"10.30757\u002Falea.v20-14","id":"6059d4be91e011ed950a5d2a","issn":"1980-0436","keywords":["Covariance Matrices"],"lang":"en","num_citation":0,"pages":{"end":"423","start":"395"},"pdf":"https:\u002F\u002Fcz5waila03cyo0tux1owpyofgoryroob.aminer.cn\u002FE5\u002FD1\u002FB5\u002FE5D1B5237E9478D8D868482A1D4AF99E.pdf","title":"Joint Convergence of Sample Cross-Covariance Matrices","update_times":{"u_a_t":"2024-12-26T03:00:58Z","u_v_t":"2024-12-26T03:00:58Z"},"urls":["https:\u002F\u002Fwww.semanticscholar.org\u002Fpaper\u002F6e1f2475723a5f17ad7238538e4d3ae60b8f4380"],"venue":{"info":{"name":"ALEA-LATIN AMERICAN JOURNAL OF PROBABILITY AND MATHEMATICAL STATISTICS"},"volume":"20"},"venue_hhb_id":"5eb81220edb6e7d53c0b5777","versions":[{"id":"61ca17ab5244ab9dcb0b090b","sid":"6e1f2475723a5f17ad7238538e4d3ae60b8f4380","src":"semanticscholar","vsid":"ALEA-LATIN AMERICAN JOURNAL OF PROBABILITY AND MATHEMATICAL STATISTICS","year":2023},{"id":"646c3f58d68f896efa6e3a4c","sid":"10.30757\u002Falea.v20-14","src":"crossref","year":2023},{"id":"6059d4be91e011ed950a5d2a","sid":"2103.11946","src":"arxiv","year":2021},{"id":"657820c0939a5f4082515943","sid":"W4287259023","src":"openalex","vsid":"S4306400194","year":2021},{"id":"657947b5939a5f408227278c","sid":"W3136645555","src":"openalex","vsid":"S2475989817","year":2023},{"id":"66c69f2f6c88b2fc286102ba","sid":"210b263de28a0b6fd51eefa4d68156be82993d4a","src":"semanticscholar","vsid":"381ce4db-9b8f-4267-aace-ce8042a89088","year":2021},{"id":"645d0498d68f896efa974d58","sid":"WOS:000960415300001","src":"wos","vsid":"ALEA-LATIN AMERICAN JOURNAL OF PROBABILITY AND MATHEMATICAL STATISTICS","year":2023}],"year":2023},{"abstract":"Spatial association measures for univariate static spatial data are widely used. Suppose the data is in the form of a collection of spatial vectors, say X_rt where r=1, … , R are the regions and t=1, … , T are the time points, in the same temporal domain of interest. Using Bergsma’s correlation coefficient ρ , we construct a measure of similarity between the regions’ series. Due to the special properties of ρ , unlike other spatial association measures which test for spatial randomness , our statistic can account for spatial pairwise independence . We have derived the asymptotic distribution of our statistic under null (independence of the regions) and alternate cases (the regions are dependent) when, across t the vector time series are assumed to be independent and identically distributed. The alternate scenario of spatial dependence is explored using simulations from the spatial autoregressive and moving average models. Finally, we provide application to modelling and testing for the presence of spatial association in COVID-19 incidence data, by using our statistic on the residuals obtained after model fitting.","authors":[{"email":"kapparadivya@gmail.com","id":"640169e3e0aa0262c78e71b8","name":"Divya Kappara","org":"University of Hyderabad","orgid":"5f71b33f1c455f439fe4188a"},{"id":"53f3a7b6dabfae4b34ae38ea","name":"Arup Bose","org":"Indian Statistical Institute","orgid":"5f71b30f1c455f439fe402d2"},{"id":"53f46a13dabfaeb22f54ce90","name":"Madhuchhanda Bhattacharjee","org":"University of Hyderabad","orgid":"5f71b33f1c455f439fe4188a"}],"create_time":"2024-03-13T05:57:34.276Z","doi":"10.1007\u002Fs00184-023-00939-9","id":"658931d5939a5f40821d1e6b","issn":"0026-1335","keywords":["Bergsma’s correlation","Spatial association measure","U-statistic","Spatial autoregressive model","Spatial moving average model"],"lang":"en","num_citation":0,"title":"An Association Measure for Spatio-Temporal Time Series","update_times":{"u_a_t":"2024-12-25T20:24:16Z","u_v_t":"2024-12-25T20:24:16Z"},"urls":["https:\u002F\u002Flink.springer.com\u002Farticle\u002F10.1007\u002Fs00184-023-00939-9"],"venue":{"info":{"name":"METRIKA"}},"versions":[{"id":"658931d5939a5f40821d1e6b","sid":"10.1007\u002Fs00184-023-00939-9","src":"springernature","vsid":"METRIKA","year":2023},{"id":"658d65b0939a5f40825fa703","sid":"10.1007\u002Fs00184-023-00939-9","src":"springer","vsid":"184","year":2023},{"id":"65d7a151939a5f40826a4a76","sid":"W4390145628","src":"openalex","vsid":"S97293958","year":2023},{"id":"666415dd01d2a3fbfc7d2f78","sid":"10.1007\u002Fs00184-023-00939-9","src":"crossref","year":2023},{"id":"65cf8200939a5f4082ce3edf","sid":"WOS:001130433400001","src":"wos","vsid":"METRIKA","year":2023}],"year":2023},{"abstract":" We consider the ``visible'' Wigner matrix, a Wigner matrix whose $(i, j)$-th entry is coerced to zero if $i, j$ are co-prime. Using a recent result from elementary number theory on co-primality patterns in integers, we show that the limiting spectral distribution of this matrix exists, and give explicit descriptions of its moments in terms of infinite products over primes $p$ of certain polynomials evaluated at $1\u002Fp$. We also consider the complementary ``invisible'' Wigner matrix. ","authors":[{"id":"53f3a7b6dabfae4b34ae38ea","name":"Arup Bose"},{"id":"6183314a8672f1a6df29487c","name":"Soumendu Sundar Mukherjee"}],"create_time":"2023-12-20T02:33:03.511Z","hashs":{"h1":"vwm"},"id":"6582516d939a5f4082afa5a8","num_citation":0,"pdf":"https:\u002F\u002Fcz5waila03cyo0tux1owpyofgoryroob.aminer.cn\u002F35\u002F93\u002FCC\u002F3593CC2FB5CFB21F407F786A4CC4D9DE.pdf","title":"The \"visible\" Wigner matrix","urls":["https:\u002F\u002Farxiv.org\u002Fabs\u002F2312.12428"],"venue":{},"versions":[{"id":"6582516d939a5f4082afa5a8","sid":"2312.12428","src":"arxiv","year":2023}],"year":2023},{"abstract":" Spatial association measures for univariate static spatial data are widely used. When the data is in the form of a collection of spatial vectors with the same temporal domain of interest, we construct a measure of similarity between the regions' series, using Bergsma's correlation coefficient $\\rho$. Due to the special properties of $\\rho$, unlike other spatial association measures which test for spatial randomness, our statistic can account for spatial pairwise independence. We have derived the asymptotic behavior of our statistic under null (independence of the regions) and alternate cases (the regions are dependent). We explore the alternate scenario of spatial dependence further, using simulations for the SAR and SMA dependence models. Finally, we provide application to modelling and testing for the presence of spatial association in COVID-19 incidence data, by using our statistic on the residuals obtained after model fitting. ","authors":[{"id":"640169e3e0aa0262c78e71b8","name":"Divya Kappara"},{"id":"53f3a7b6dabfae4b34ae38ea","name":"Arup Bose"},{"id":"53f46a13dabfaeb22f54ce90","name":"Madhuchhanda Bhattacharjee"}],"create_time":"2023-03-30T19:16:06.367Z","doi":"10.48550\u002Farxiv.2303.16824","id":"6424fe3490e50fcafd78b907","keywords":["spatial association","spatial independence","short time course data"],"num_citation":0,"pdf":"https:\u002F\u002Fcz5waila03cyo0tux1owpyofgoryroob.aminer.cn\u002F2C\u002FC8\u002FB2\u002F2CC8B270A6690ABCD4A13A2CF66EC585.pdf","title":"Measuring spatial association and testing spatial independence based on\n short time course data","urls":["https:\u002F\u002Fopenalex.org\u002FW4361807116","https:\u002F\u002Farxiv.org\u002Fabs\u002F2303.16824"],"venue":{"info":{"name":"arXiv (Cornell University)"}},"versions":[{"id":"6424fe3490e50fcafd78b907","sid":"2303.16824","src":"arxiv","year":2023},{"id":"6578fb24939a5f4082ab6ce2","sid":"W4361807116","src":"openalex","year":2023}],"year":2023},{"abstract":"Bergsma (A new correlation coefficient, its orthogonal decomposition and associated tests of independence, arXiv preprint arXiv:math\u002F0604627 , 2006) proposed a covariance κ (X,Y) between random variables X and Y , and gave two estimates for it, based on n i.i.d. samples. He derived the asymptotic distributions of these estimates under the assumption of independence between X and Y . Our main focus is on the dependent case. This measure turns out to be same as the distance covariance (dCov) measure for multivariate X and Y , when we specialize to real-valued X and Y . We first derive several alternate expressions for κ , which are useful to understand the properties of κ and its estimates better. One of the alternate expressions for κ leads to a very intuitive third estimator of κ that is a nice function of four U -statistics. We establish the exact finite sample algebraic relation between the three estimates. This yields the relation between the bias of these estimators. In the dependent case, using the U statistics central limit theorem, it is easy to show that our estimate is asymptotic normal. The relation between the three estimates is then used to show that Bergsma’s two estimates have the same limit distribution in the dependent case. When X and Y are independent, the above limit is degenerate. With a higher scaling, the non-degenerate limit distribution of all three estimators is obtained using the theory of degenerate U -statistics and the above algebraic relations. In particular, the known asymptotic distribution results for the two estimates of Bergsma for the independent case follow. For specific parametric bivariate distributions, the value of κ can be derived in terms of the natural dependence parameters of these distributions. In particular, we derive the formula for κ when ( X , Y ) are distributed as Gumbel’s bivariate exponential. We bring out various aspects of these estimators through extensive simulations from several prominent bivariate distributions. In particular, we investigate the empirical relationship between κ and the dependence parameters, the distributional properties of the estimators, and the accuracy of these estimators. We also investigate the finite sample powers of these measures for testing independence, compare these among themselves, and with other well known such measures. Based on these exercises, the proposed estimator seems as good or better than its competitors both in terms of power and computing efficiency.","authors":[{"email":"bosearu@gmail.com","id":"53f3a7b6dabfae4b34ae38ea","name":"Arup Bose","org":"Stat.-Math. Unit, Indian Statistical Institute","orgid":"5f71b30f1c455f439fe402d2"},{"id":"640169e3e0aa0262c78e71b8","name":"Divya Kappara","org":"University of Hyderabad","orgid":"5f71b33f1c455f439fe4188a"},{"id":"53f46a13dabfaeb22f54ce90","name":"Madhuchhanda Bhattacharjee","org":"University of Hyderabad","orgid":"5f71b33f1c455f439fe4188a"}],"create_time":"2023-12-14T09:49:11.54Z","doi":"10.1007\u002Fs42952-023-00236-1","id":"65372f9c939a5f40823eb45a","issn":"1226-3192","keywords":["Bergsma’s covariance","Eigenvalues","U and V-statistic","Degenerate U-statistics","Measures of dependence","Distance covariance","Powers of tests of independence"],"lang":"en","num_citation":0,"pages":{"end":"1054","start":"1025"},"title":"Estimation of Bergsma’s Covariance","update_times":{"u_a_t":"2024-12-25T20:21:16Z","u_v_t":"2024-12-25T20:21:16Z"},"urls":["https:\u002F\u002Flink.springer.com\u002Farticle\u002F10.1007\u002Fs42952-023-00236-1"],"venue":{"info":{"name":"JOURNAL OF THE KOREAN STATISTICAL SOCIETY"},"issue":"4","volume":"52"},"versions":[{"id":"65372f9c939a5f40823eb45a","sid":"10.1007\u002Fs42952-023-00236-1","src":"springernature","vsid":"JOURNAL OF THE KOREAN STATISTICAL SOCIETY","year":2023},{"id":"656d7a82939a5f4082922c96","sid":"10.1007\u002Fs42952-023-00236-1","src":"crossref","year":2023},{"id":"65790cfd939a5f4082c8474c","sid":"W4387730770","src":"openalex","vsid":"S86794028","year":2023},{"id":"65856a82939a5f408262e036","sid":"10.1007\u002Fs42952-023-00236-1","src":"springer","vsid":"42952","year":2023},{"id":"667b07f7a925a558d5361173","sid":"cfa0194a6de48e9368a09355e32fd59aa120dbf4","src":"semanticscholar","vsid":"337b33c4-610d-47a8-85bc-ea316e914fe6","year":2023},{"id":"65734a57939a5f408235c27b","sid":"WOS:001089306700001","src":"wos","vsid":"JOURNAL OF THE KOREAN STATISTICAL SOCIETY","year":2023}],"year":2023},{"abstract":"Consider the empirical autocovariance matrix at a given non-zero time lag based on observations from a multivariate complex Gaussian stationary time series. The spectral analysis of these autocovariance matrices can be useful in certain statistical problems, such as those related to testing for white noise. We study the behavior of their spectral measures in the asymptotic regime where the time series dimension and the observation window length both grow to infinity, and at the same rate. Following a general framework in the field of the spectral analysis of large random non-Hermitian matrices, at first the probabilistic behavior of the small singular values of the shifted versions of the autocovariance matrix are obtained. This is then used to infer about the large sample behaviour of the empirical spectral measure of the autocovariance matrices at any lag. Matrix orthogonal polynomials on the unit circle play a crucial role in our study.","authors":[{"email":"bosearu@gmail.com","id":"53f3a7b6dabfae4b34ae38ea","name":"Arup Bose","org":"Indian Stat Inst, Stat & Math Unit, Kolkata, India"},{"email":"walid.hachem@univ-eiffel.fr","id":"53f429c9dabfaeb2acfb8aba","name":"Walid Hachem","org":"Univ Gustave Eiffel, ESIEE Paris, CNRS, LIGM, F-77454 Marne La Vallee, France","orgid":"61e69da6689627346574227c"}],"create_time":"2024-08-09T15:11:45.091Z","doi":"10.1142\u002Fs2010326322500538","id":"616e37435244ab9dcbd1a7b0","issn":"2010-3263","keywords":["High-dimensional times series analysis","large non-Hermitian matrix theory","limit spectral distribution","matrix orthogonal polynomials","multivariate stationary processes","small singular values"],"lang":"en","num_citation":1,"pdf":"https:\u002F\u002Fcz5waila03cyo0tux1owpyofgoryroob.aminer.cn\u002F6F\u002F5A\u002FCF\u002F6F5ACF076DB8A999B56FDB6F93E3F10F.pdf","title":"Spectral Measure of Empirical Autocovariance Matrices of High Dimensional Gaussian Stationary Processes","update_times":{"u_a_t":"2024-12-26T03:53:54Z","u_v_t":"2024-12-26T03:53:54Z"},"urls":["http:\u002F\u002Farxiv.org\u002Fabs\u002F2110.08523"],"venue":{"info":{"name":"RANDOM MATRICES-THEORY AND APPLICATIONS"},"issue":"02","volume":"12"},"versions":[{"id":"616e37435244ab9dcbd1a7b0","sid":"2110.08523","src":"arxiv","vsid":"RANDOM MATRICES-THEORY AND APPLICATIONS","year":2023},{"id":"61c82d355244ab9dcb6dd8af","sid":"5d1752e2c6ce54d8dc114b5ffc0015db666a1862","src":"semanticscholar","year":2021},{"id":"656faa50939a5f408221573a","sid":"10.1142\u002Fs2010326322500538","src":"crossref","year":2022},{"id":"65777f1c939a5f4082920291","sid":"W4308897346","src":"openalex","vsid":"S4306400194","year":2021},{"id":"65778aba939a5f4082d02ec3","sid":"W3205767612","src":"openalex","vsid":"S4306402512","year":2021},{"id":"65778c81939a5f4082d93405","sid":"W4226164998","src":"openalex","vsid":"S4306402512","year":2021},{"id":"65787696939a5f4082da5670","sid":"W4285593235","src":"openalex","vsid":"S4210169430","year":2022},{"id":"6678b3c1da8b09851ec8605c","sid":"33b10b42b5a6f09c46e9d152da686adea539a44e","src":"semanticscholar","vsid":"9f7a71e7-9b23-4200-8843-8e1f6a926fd5","year":2021},{"id":"645bd87ef11ef10fc6348568","sid":"WOS:000848873400001","src":"wos","vsid":"RANDOM MATRICES-THEORY AND APPLICATIONS","year":2023}],"year":2023},{"abstract":" Bergsma (2006) proposed a covariance $\\kappa$(X,Y) between random variables X and Y. He derived their asymptotic distributions under the null hypothesis of independence between X and Y . The non-null (dependent) case does not seem to have been studied in the literature. We derive several alternate expressions for $\\kappa$. One of them leads us to a very intuitive estimator of $\\kappa$(X,Y) that is a nice function of four naturally arising U-statistics. We derive the exact finite sample relation between all three estimates. The asymptotic distribution of our estimator, and hence also of the other two estimators, in the non-null (dependence) case, is then obtained by using the U-statistics central limit theorem. The exact value of $\\kappa$(X,Y) is hard to calculate for most bivariate distributions. Here we provide detailed derivation of $\\kappa$ for two well known parametric families, namely, the bivariate exponential and the bivariate normal distributions. Using these we carry out extensive simulation to study the properties of these estimates with a focus on the non-null case. In the null case, the limit is known to be degenerate. However with a higher scaling, the non-degenerate limit distribution of our estimator is again obtained using the theory of degenerate U-statistics. This quickly leads us also to the known asymptotic distribution results for the two estimates of Bergsma in the null case. We used simulation techniques for the null case to investigate the accuracies of the discrete approximation method. ","authors":[{"id":"640169e3e0aa0262c78e71b8","name":"Divya Kappara"},{"id":"53f3a7b6dabfae4b34ae38ea","name":"Arup Bose"},{"id":"53f46a13dabfaeb22f54ce90","name":"Madhuchhanda Bhattacharjee"}],"create_time":"2022-12-20T08:58:26.151Z","doi":"10.48550\u002Farxiv.2212.08921","hashs":{"h1":"abirb","h3":"c"},"id":"63a1751690e50fcafd1f4511","keywords":["bivariate independence","covariance","revisiting bergsma"],"lang":"en","num_citation":0,"pdf":"https:\u002F\u002Fcz5waila03cyo0tux1owpyofgoryroob.aminer.cn\u002F22\u002F9F\u002FA4\u002F229FA46D91C650CFA847ED25B4149D14.pdf","title":"Assessing bivariate independence: Revisiting Bergsma's covariance","update_times":{"u_a_t":"2023-01-20T07:05:56.513Z"},"urls":["https:\u002F\u002Farxiv.org\u002Fabs\u002F2212.08921"],"venue":{"info":{"name":"arXiv (Cornell University)","publisher":"Cornell University"}},"versions":[{"id":"63a1751690e50fcafd1f4511","sid":"2212.08921","src":"arxiv","year":2022},{"id":"657862d4939a5f4082bc5f8b","sid":"W4312046954","src":"openalex","vsid":"S4306400194","year":2022}],"year":2022}],"profilePubsTotal":174,"profilePatentsPage":0,"profilePatents":null,"profilePatentsTotal":null,"profilePatentsEnd":false,"profileProjectsPage":0,"profileProjects":null,"profileProjectsTotal":null,"newInfo":null,"checkDelPubs":[]}};