Product of exponentials and spectral radius of random k-circulants
Annales de l'I.H.P. Probabilités et statistiques, Volume 48 (2012) no. 2, p. 424-443

We consider n × n random k-circulant matrices with n → ∞ and k = k(n) whose input sequence {al}l≥0 is independent and identically distributed (i.i.d.) random variables with finite (2 + δ) moment. We study the asymptotic distribution of the spectral radius, when n = kg + 1. For this, we first derive the tail behaviour of the g fold product of i.i.d. exponential random variables. Then using this tail behaviour result and appropriate normal approximation techniques, we show that with appropriate scaling and centering, the asymptotic distribution of the spectral radius is Gumbel. We also identify the centering and scaling constants explicitly.

Nous considérons des matrices aléatoires k-circulantes de taille n × n avec n → ∞ et k = k(n), dont les entrées {al}l≥0 sont des variables aléatoires, de moment (2 + δ) fini, indépendantes et identiquement distribuées. Nous étudions la distribution asymptotique du rayon spectral, lorsque n = kg + 1. Pour établir cette distribution asymptotique, nous calculons d'abord le comportement de la queue du produit de g variables aléatoires exponentielles i.i.d. Ensuite, en utilisant un résultat sur le comportement des queues et les techniques appropriées d'approximation normale, nous montrons que, après renormalisation et recentrage, la distribution limite est une distribution de Gumbel. Nous identifions explicitement les constantes de recentrage et de remise à l'échelle.

DOI : https://doi.org/10.1214/10-AIHP404
Classification:  Primary 60B20,  secondary,  60B10,  60F05,  62E20,  62G32,  15A52,  60F99,  60F05
Keywords: eigenvalues, Gumbel distribution, k-circulant matrix, Laplace asymptotics, large dimensional random matrix, linear process, normal approximation, spectral radius, spectral density, tail of product
@article{AIHPB_2012__48_2_424_0,
     author = {Bose, Arup and Hazra, Rajat Subhra and Saha, Koushik},
     title = {Product of exponentials and spectral radius of random $k$-circulants},
     journal = {Annales de l'I.H.P. Probabilit\'es et statistiques},
     publisher = {Gauthier-Villars},
     volume = {48},
     number = {2},
     year = {2012},
     pages = {424-443},
     doi = {10.1214/10-AIHP404},
     zbl = {1244.60010},
     language = {en},
     url = {http://www.numdam.org/item/AIHPB_2012__48_2_424_0}
}
Product of exponentials and spectral radius of random $k$-circulants. Annales de l'I.H.P. Probabilités et statistiques, Volume 48 (2012) no. 2, pp. 424-443. doi : 10.1214/10-AIHP404. http://www.numdam.org/item/AIHPB_2012__48_2_424_0/

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