Geodesic
Normal limit law for protected node profile of random recursive trees
Teoriâ veroâtnostej i ee primeneniâ, Tome 67 (2022) no. 3, pp. 563-578

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Protected nodes, i.e., nodes with distance at least 2 to each leaf, have been studied in various classes of random rooted trees. In this short note, we investigate the protected node profile, i.e., the number of protected nodes with the same distance from the root in random recursive trees. Here, when the limit ratio of the level and logarithm of tree size is zero, we present the asymptotic expectations, variances, and covariance of the protected node profile and the nonprotected node profile in random recursive trees. We also show that protected node and nonprotected node profiles have a bivariate normal limiting distribution via the joint characteristic function and singularity analysis.
Keywords: random recursive trees, profile, protected node, characteristic function, singularity analysis, Berry–Esseen inequality.
Mots-clés : bivariate normal distribution 
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     author = {J. Toofanpour and M. Javanian and R. Imany-Nabiyyi},
     title = {Normal limit law for protected node profile of random recursive trees},
     journal = {Teori\^a vero\^atnostej i ee primeneni\^a},
     pages = {563--578},
     publisher = {mathdoc},
     volume = {67},
     number = {3},
     year = {2022},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TVP_2022_67_3_a7/}
}
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J. Toofanpour; M. Javanian; R. Imany-Nabiyyi. Normal limit law for protected node profile of random recursive trees. Teoriâ veroâtnostej i ee primeneniâ, Tome 67 (2022) no. 3, pp. 563-578. http://geodesic.mathdoc.fr/item/TVP_2022_67_3_a7/