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th random variable $X_n$ is by a particular member $R_n$ of a given\nfamily of distributions, whose variance increases with $n$. The basic\nassumption is that the ratio of the characteristic function of $X_n$ and that\nof R_n$ converges to a limit in a prescribed fashion. Our results cover a\nnumber of classical examples in probability theory, combinatorics and number\ntheory.","abstract_zh":"","authors":[{"id":"53f42f74dabfaee43ebdf8b6","name":"A. D. Barbour"},{"id":"53f42e40dabfaee0d9b08334","name":"E. Kowalski"},{"id":"53f7c124dabfae8faa4b09fd","name":"A. Nikeghbali"}],"doi":"10.1007\u002Fs00440-013-0498-8","id":"53e99842b7602d970206d72d","lang":"en","num_citation":9,"order":2,"pages":{"end":"893","start":"859"},"pdf":"https:\u002F\u002Fstatic.aminer.cn\u002Fupload\u002Fpdf\u002Fprogram\u002F53e99842b7602d970206d72d_0.pdf","title":"Mod-discrete expansions","urls":["https:\u002F\u002Flink.springer.com\u002F10.1007\u002Fs00440-013-0498-8","https:\u002F\u002Farxiv.org\u002Fabs\u002F0912.1886"],"venue":{"info":{"name":"Probability Theory and Related Fields"},"issue":"3-4","volume":"158"},"versions":[{"id":"53e267c520f7fff677ab7016","sid":"14179670","src":"msra","year":2009},{"id":"56d901b2dabfae2eeedbb8d6","sid":"76995780","src":"mag","year":2013},{"id":"58d3f6c00cf221de035ccbaf","sid":"10.1007\u002Fs00440-013-0498-8","src":"springer","vsid":"440","year":2014},{"id":"5c6106f6da56297340ae5d18","sid":"0912.1886","src":"arxiv","year":2009}],"year":2014},{"authors":[{"id":"53f42e40dabfaee0d9b08334","name":"E. Kowalski"},{"id":"53f7c124dabfae8faa4b09fd","name":"A. Nikeghbali"}],"id":"53e99938b7602d970217763f","lang":"en","num_citation":0,"order":1,"pages":{"end":"319","start":"291"},"title":"Mod-Gaussian convergence and the value distribution of ζ(½ + it) and related quantities.","urls":["http:\u002F\u002Fdx.doi.org\u002F10.1112\u002Fjlms\u002Fjds003"],"venue":{"info":{"name":"J. London Math. Society"},"issue":"1","volume":"86"},"versions":[{"id":"599c79af601a182cd265982a","sid":"journals\u002Fjlms\u002FKowalskiN12","src":"dblp","vsid":"journals\u002Fjlms","year":2012}],"year":2012},{"abstract":"In the context of mod-Gaussian convergence, as defined previously in our work with Jacod, we obtain asymptotic formulas and lower bounds for local probabilities for a sequence of random vectors which are approximately Gaussian in this sense, with increasing covariance matrix. This is motivated by the conjecture concerning the density of the set of values of the Riemann zeta function on the critical line. We obtain evidence for this fact, and derive unconditional results for random matrices in compact classical groups, as well as for certain families of L-functions over finite fields.","abstract_zh":"","authors":[{"id":"53f42e40dabfaee0d9b08334","name":"E. Kowalski"},{"id":"53f7c124dabfae8faa4b09fd","name":"A. Nikeghbali"}],"doi":"10.1112\u002Fjlms\u002Fjds003","id":"53e9a593b7602d9702eb8c2b","lang":"en","num_citation":0,"order":1,"pages":{"end":"319","start":"291"},"pdf":"https:\u002F\u002Fstatic.aminer.cn\u002Fstorage\u002Fpdf\u002Farxiv\u002F09\u002F0912\u002F0912.3237.pdf","title":"Mod-Gaussian convergence and the value distribution of $\\zeta(1\u002F2+it)$ and related quantities","urls":["http:\u002F\u002Fdx.doi.org\u002F10.1112\u002Fjlms\u002Fjds003","https:\u002F\u002Farxiv.org\u002Fabs\u002F0912.3237","http:\u002F\u002Fwww.webofknowledge.com\u002F"],"venue":{"info":{"name":"JOURNAL OF THE LONDON MATHEMATICAL SOCIETY-SECOND SERIES"},"issue":"","volume":"86"},"versions":[{"id":"53e27fdd20f7fff67830cb2c","sid":"27661763","src":"msra","year":2009},{"id":"56d91e4bdabfae2eee8d84c0","sid":"2156978754","src":"mag","year":2009},{"id":"5c6106f8da56297340ae62c3","sid":"0912.3237","src":"arxiv","year":2010},{"id":"5fc7100de8bf8c1045388662","sid":"WOS:000306970100015","src":"wos","vsid":"JOURNAL OF THE LONDON MATHEMATICAL SOCIETY-SECOND SERIES","year":2012}],"year":2012},{"abstract":"Analogously to the space of virtual permutations, we define projective limits\nof isometries: these sequences of unitary operators are natural in the sense\nthat they minimize the rank norm between successive matrices of increasing\nsizes. The space of virtual isometries we construct this way may be viewed as a\nnatural extension of the space of virtual permutations of Kerov, Olshanski and\nVershik as well as the space of virtual isometries of Neretin. We then derive\nwith purely probabilistic methods an almost sure convergence for these random\nmatrices under the Haar measure: for a coherent Haar measure on virtual\nisometries, the smallest normalized eigenangles converge almost surely to a\npoint process whose correlation function is given by the sine kernel. This\nalmost sure convergence actually holds for a larger class of measures as is\nproved by Borodin and Olshanski. We give a different proof, probabilist in the\nsense that it makes use of martingale arguments and shows how the eigenangles\ninterlace when going from dimension n to n+1. Our method also proves that for\nsome universal constant epsilon\u003E0, the rate of convergence is almost surely\ndominated by n^{-epsilon} when the dimension n goes to infinity.","authors":[{"id":"53f4584fdabfaeb22f5087e7","name":"P. Bourgade"},{"id":"53f45f4bdabfaee2a1d94194","name":"J. Najnudel"},{"id":"53f7c124dabfae8faa4b09fd","name":"A. Nikeghbali"}],"doi":"","id":"53e9bc10b7602d9704877d6a","lang":"en","num_citation":5,"order":2,"pdf":"https:\u002F\u002Fstatic.aminer.cn\u002Fstorage\u002Fpdf\u002Farxiv\u002F11\u002F1102\u002F1102.2633.pdf","title":"A unitary extension of virtual permutations","versions":[{"id":"53e27fa720f7fff6782fdffb","sid":"27601554","src":"msra","year":2011},{"id":"5c610750da56297340afc342","sid":"1102.2633","src":"arxiv","year":2011}],"year":2011},{"abstract":"Building on earlier work introducing the notion of “mod-Gaussian” convergence of sequences of random variables, which arises naturally in Random Matrix Theory and number theory, we discuss the analogue notion of “mod-Poisson” convergence. We show in particular how it occurs naturally in analytic number theory in the classical Erdős– Kac Theorem. In fact, this case reveals deep connections and anal...","authors":[{"id":"53f42e40dabfaee0d9b08334","name":"E. Kowalski"},{"id":"53f7c124dabfae8faa4b09fd","name":"A. Nikeghbali"}],"doi":"10.1093\u002Fimrn\u002Frnq019","id":"53e9b0eeb7602d9703b6c0ed","lang":"en","num_citation":16,"order":1,"pages":{"end":"3587","start":"3549"},"pdf":"https:\u002F\u002Fstatic.aminer.cn\u002Fstorage\u002Fpdf\u002Farxiv\u002F09\u002F0905\u002F0905.0318.pdf","title":"Mod-Poisson convergence in probability and number theory","venue":{"info":{"name":"International Mathematics Research Notices"},"issue":"18","volume":"2010"},"versions":[{"id":"53e26b5920f7fff677c27fb5","sid":"16421116","src":"msra","year":2010},{"id":"53e2662020f7fff677a3c5fa","sid":"13535318","src":"msra","year":2009},{"id":"5d9edb6247c8f76646017626","sid":"2962780129","src":"mag","vsid":"111381568","year":2010},{"id":"5c6106c5da56297340adb0bd","sid":"0905.0318","src":"arxiv","year":2009},{"id":"5f2ca2529fced0a24b5060ce","sid":"8189893","src":"ieee","vsid":"8016802","year":2010}],"year":2010},{"abstract":"We prove a multidimensional extension of Selberg’s central limit theorem for the logarithm of the Riemann zeta function on\n the critical line. The limit is a totally disordered process, whose coordinates are all independent and Gaussian.","abstract_zh":"","authors":[{"id":"53f4447fdabfaee4dc7c98eb","name":"C. P. Hughes"},{"id":"53f7c124dabfae8faa4b09fd","name":"A. Nikeghbali"},{"id":"53f4313cdabfaee0d9b2de5b","name":"M. Yor"}],"doi":"10.1007\u002Fs00440-007-0079-9","id":"53e9a627b7602d9702f52df6","lang":"en","num_citation":15,"order":1,"pages":{"end":"59","start":"47"},"pdf":"https:\u002F\u002Fstatic.aminer.cn\u002Fstorage\u002Fpdf\u002Farxiv\u002F06\u002F0612\u002Fmath0612195.pdf","title":"An arithmetic model for the total disorder process","urls":["http:\u002F\u002Fdx.doi.org\u002F10.1007\u002Fs00440-007-0079-9","https:\u002F\u002Flink.springer.com\u002F10.1007\u002Fs00440-007-0079-9","https:\u002F\u002Farxiv.org\u002Fabs\u002Fmath\u002F0612195","http:\u002F\u002Fwww.webofknowledge.com\u002F"],"venue":{"info":{"name":"Probability Theory and Related Fields"},"issue":"1","volume":"141"},"versions":[{"id":"53e2560e20f7fff677687a57","sid":"5031043","src":"msra","year":2008},{"id":"56d901b3dabfae2eeedbc2ac","sid":"2052851329","src":"mag","year":2007},{"id":"58d1f6540cf221de03525ee4","sid":"10.1007\u002Fs00440-007-0079-9","src":"springer","vsid":"440","year":2008},{"id":"5c610688da56297340ac65f3","sid":"math\u002F0612195","src":"arxiv","year":2006},{"id":"5ff1dd2919519e9c397756db","sid":"WOS:000252807600003","src":"wos","vsid":"PROBABILITY THEORY AND RELATED FIELDS","year":2008}],"year":2008},{"abstract":"In this paper we deduce a universal result about the asymptotic distribution\nof roots of random polynomials, which can be seen as a complement to an old and\nfamous result of Erdos and Turan. More precisely, given a sequence of random\npolynomials, we show that, under some very general conditions, the roots tend\nto cluster near the unit circle, and their angles are uniformly distributed.\nThe method we use is deterministic: in particular, we do not assume\nindependence or equidistribution of the coefficients of the polynomial.","abstract_zh":"","authors":[{"id":"53f4447fdabfaee4dc7c98eb","name":"C. P. Hughes"},{"id":"53f7c124dabfae8faa4b09fd","name":"A. Nikeghbali"}],"doi":"10.1112\u002FS0010437X07003302","id":"53e9afb3b7602d9703a02131","lang":"en","num_citation":83,"order":1,"pages":{"end":"","start":""},"pdf":"https:\u002F\u002Fstatic.aminer.cn\u002Fstorage\u002Fpdf\u002Farxiv\u002F04\u002F0406\u002Fmath0406376.pdf","title":"The zeros of random polynomials cluster uniformly near the unit circle","urls":["http:\u002F\u002Fdx.doi.org\u002F10.1112\u002FS0010437X07003302","https:\u002F\u002Farxiv.org\u002Fabs\u002Fmath\u002F0406376"],"venue":{"info":{"name":"Compositio Mathematica"},"issue":"03","volume":"144"},"versions":[{"id":"53e253a320f7fff6775deddc","sid":"3982831","src":"msra","year":2008},{"id":"56d826f4dabfae2eeee33f7c","sid":"2103995763","src":"mag","year":2004},{"id":"5c610678da56297340ac0c46","sid":"math\u002F0406376","src":"arxiv","year":2007}],"year":2008},{"abstract":"Take a generic subgroup G, endowed with its Haar mea- sure, from U(n,K), the unitary group of dimension n over the field K of real, complex or quaternion numbers. We give some equalities in law for Z := det(Id G), G 2 G : under some general conditions, Z can be decomposed as a product of independent random variables, whose laws are explicitly known (Section 2). Consequently G, endowed with a generalization of its Haar measure (the Hua-Pickrell measure), can be generated as a product of independent reflections. This constitutes a generalization of the well known Ewens sampling formula, correspond- ing to G = Sn, the n-dimensional symmetric group (Section 3). Finally, explicit determinantal point processes can be associated to the spec- trum induced by the Hua-Pickrell measures, implying asymptotics on correlation functions (Section 4).","abstract_zh":"","authors":[{"id":"53f4584fdabfaeb22f5087e7","name":"P. BOURGADE"},{"id":"53f7c124dabfae8faa4b09fd","name":"A. NIKEGHBALI"},{"id":"540888a5dabfae8faa64b47c","name":"A. ROUAULT"}],"doi":"","id":"53e9b12ab7602d9703baa807","lang":"en","num_citation":4,"order":1,"pages":{"end":"","start":""},"title":"HUA-PICKRELL MEASURES ON GENERAL COMPACT GROUPS","venue":{"issue":"","volume":""},"versions":[{"id":"53e25c2820f7fff6777b0ffe","sid":"6930869","src":"msra","year":2007},{"id":"56d8560bdabfae2eee2921a4","sid":"1479900166","src":"mag","year":2007}],"year":2007},{"abstract":"","abstract_zh":"","authors":[{"id":"53f47611dabfaec09f27d34e","name":"V Durrleman"},{"id":"53f7c124dabfae8faa4b09fd","name":"A Nikeghbali"},{"id":"53f4c89cdabfaee57677d874","name":"T Roncalli"}],"doi":"10.2139\u002Fssrn.1032545","id":"53e998fdb7602d970213b39d","lang":"en","num_citation":69,"order":1,"pages":{"end":"","start":""},"pdf":"https:\u002F\u002Fstatic.aminer.cn\u002Fupload\u002Fpdf\u002Fprogram\u002F53e998fdb7602d970213b39d_0.pdf","title":"Which copula is the right one","urls":["http:\u002F\u002Fdx.doi.org\u002F10.2139\u002Fssrn.1032545"],"venue":{"issue":"","volume":""},"versions":[{"id":"53e253c420f7fff6775e842e","sid":"4092844","src":"msra","year":2000},{"id":"56d89f83dabfae2eee59f393","sid":"1607575316","src":"mag","year":2000}],"year":2000},{"abstract":"","abstract_zh":"","authors":[{"id":"53f43a3adabfaee4dc7a64f3","name":"E. Bouye"},{"id":"53f47611dabfaec09f27d34e","name":"V. Durrleman"},{"id":"53f7c124dabfae8faa4b09fd","name":"A. Nikeghbali"},{"id":"53f430f8dabfaedf43544130","name":"G. Riboulet"},{"id":"53f4c89cdabfaee57677d874","name":"T. Roncalli"}],"doi":"10.2139\u002Fssrn.1032533","id":"53e9b1f8b7602d9703c8d6d5","lang":"en","num_citation":185,"order":2,"pages":{"end":"","start":""},"title":"Copulas for Finance: A Reading Guide and Some Applications","urls":["http:\u002F\u002Fdx.doi.org\u002F10.2139\u002Fssrn.1032533"],"venue":{"issue":"","volume":""},"versions":[{"id":"53e2521920f7fff67757b7e7","sid":"2901457","src":"msra","year":2000},{"id":"56d8b96fdabfae2eee22fc34","sid":"2163402412","src":"mag","year":2000}],"year":2000}],"profilePubsTotal":10,"profilePatentsPage":1,"profilePatents":[],"profilePatentsTotal":0,"profilePatentsEnd":true,"profileProjectsPage":0,"profileProjects":null,"profileProjectsTotal":null,"newInfo":null,"checkDelPubs":[]}};