P-SM with 0 and = -m, for some m N, after which it
P-SM with 0 and = -m, for some m N, and then it assumes that the GSK2646264 web parameter m is distributed based on an arbitrary distribution on N. This could be observed in Theorem 12 of Gnedin and Pitman [14] and Gnedin [15] by way of example. Nevertheless, differently from the Moveltipril Inhibitor definition of Gnedin and Pitman [14], in our context, the distribution of m depends on the sample size n. For (0, 1) and -, Pitman [5] first studied the big n asymptotic behaviour of Kn (, ). This could also be seen in Gnedin and Pitman [14] and also the references therein. Let a.s. – denote the practically sure convergence, and let S, be the scaled Mittag effler random variable defined above. Theorem 3.8 of Pitman [5] exploited a martingale convergence argument to show that: Kn (, ) a.s. – S, (20) n as n . The random variable S, is referred to as Pitman’s -diversity. For 0 and = -m for some m N, the big n asymptotic behaviour of Kn (, ) is trivial, that is: Kn (, ) – mw(21)as n . We refer to Dolera and Favaro [16,17] for Berry sseen sort refinements of (20) and to Favaro et al. [18,19] and Favaro and James [13] for generalisations of (20) with applications to Bayesian nonparametrics. This can also be observed in Pitman [5] (Chapter four) for a general therapy of (20). Based on Theorem 2, it really is all-natural to ask whether or not there exists an interplay in between Theorem 1 plus the significant n asymptotic behaviours (20) and (21). Hereafter, we show that: (i) (20), with all the just about confident convergence replaced by the convergence in distribution, arises by combining (6) with (i) of Theorem two; (ii) (8) arises by combining (21) with (ii) of Theorem two. This offers an alternative proof of Pitman’s -diversity. Theorem 3. Let Kn (, ) and K (, z, n) under the EP-SM and the NB-CPSM, respectively. As n : (i) For (0, 1) and -: Kn (, ) w – S, . n (ii) For 0 and z 0: K (, z, n) n 1–(22)-w(z) 1- . -(23)Proof. We show that (22) arises by combining (6) with statement (i) of Theorem 2. For any pair of N-valued random variables U and V, let dTV (U; V ) be the total variation distance involving the distribution of U as well as the distribution of V. In addition, let Computer denote a Poisson random variable with parameter c 0. For any (0, 1) and t 0, we show that as n : dTV (K (, tn , n); 1 Ptn ) 0. (24) This implies (22). The proof of (24) needs a cautious analysis from the probability creating function of K (, tn , n). In distinct, let us define (t; n, ) := tn 1 exactly where M (t) := =1 (m-1)! (m) sin(m) may be the Wright ainardi function m (Mainardi et al. [20]). Then, we apply Corollary 2 of Dolera and Favaro [16] to conclude that dTV (K (, tn , n); 1 P (t;n,) ) 0 as n . Lastly, we applied inequality (two.two) in Adell and Jodr[21] to get: tM (t) , M ( t )(-t)m-dTV (1 Ptn ; 1 P (t;n,) ) = dTV ( Ptn ; P (t;n,) )tM (t) min 1, M ( t )(2/e) (t; n, ) tnMathematics 2021, 9,10 ofSo that dTV (1 Ptn ; 1 P (t;n,) ) 0 as n , and (24) follows. Now, keeping and t fixed as above, we show that (24) entails (22). To this aim, we introduced the Kolmogorov distance dK which, for any pair of R -valued random variables U and V, is defined by dK (U; V ) := supx0 |Pr[U x ] – Pr[V x ]|. The claim to become verified is equivalent to: dK (Kn (, )/n ; S, ) 0 as n . We exploit statement (i) of Theorem 2. This leads to the distributional identity d Kn (, ) = K (, X,,n , n). Therefore, in view on the fundamental properties of the Kolmogorov distance: dK (Kn (, )/n ; S, ) dK (Kn (, ); K (, n S, , n)) (25) dK (K (, n S, , n); 1 Pn S, ) dK ([1 Pn S, ]/n ; S, ),where the P 0 is thought of.
