mathematics Article Convergence in Total Variation of Random Sums Luca Pratelli 1 and Pietro Rigo 2,* 1 Accademia Navale, viale Italia 72, 57100 Livorno, Italy; [email protected] 2 Dipartimento di Scienze Statistiche “P. Fortunati”, Università di Bologna, via delle Belle Arti 41, 40126 Bologna, Italy * Correspondence: [email protected] Abstract: Let (Xn) be a sequence of real random variables, (Tn) a sequence of random indices, and Tn (tn) a sequence of constants such that tn ! ¥. The asymptotic behavior of Ln = (1/tn) ∑i=1 Xi, as n ! ¥, is investigated when (X ) is exchangeable and independent of (T ). We give conditions for p n n Mn = tn (Ln − L) −! M in distribution, where L and M are suitable random variables. Moreover, ( ) j ( 2 ) − ( 2 )j ≤ when Xn is i.i.d., we find constants an and bn such that supA2B(R) P Ln A P L A an j ( 2 ) − ( 2 )j ≤ ! ! and supA2B(R) P Mn A P M A bn for every n. In particular, Ln L or Mn M in total variation distance provided an ! 0 or bn ! 0, as it happens in some situations. Keywords: exchangeability; random sum; rate of convergence; stable convergence; total variation distance MSC: 60F05; 60G50; 60B10; 60G09 1. Introduction All random elements appearing in this paper are defined on the same probability space, say (W, A, P). Tn A random sum is a quantity such as ∑i=1 Xi, where (Xn : n ≥ 1) is a sequence of real Citation: Pratelli, L.; Rigo, P. random variables and (Tn : n ≥ 1) a sequence of N-valued random indices. In the sequel, Convergence in Total Variation of in addition to (Xn) and (Tn), we fix a sequence (tn : n ≥ 1) of positive constants such that Random Sums. Mathematics 2021, 9, tn ! ¥ and we let 194. https://doi.org/10.3390/ math9020194 Tn ∑i=1 Xi Ln = . tn Received: 21 December 2020 Accepted: 15 January 2021 Random sums find applications in a number of frameworks, including statistical Published: 19 January 2021 inference, risk theory and insurance, reliability theory, economics, finance, and forecasting of market changes. Accordingly, the asymptotic behavior of Ln, as n ! ¥, is a classical Publisher’s Note: MDPI stays neu- topic in probability theory. The related literature is huge and we do not try to summarize it tral with regard to jurisdictional clai- here. We just mention a general text book [1] and some useful recent references: [2–10]. ms in published maps and institutio- In this paper, the asymptotic behavior of Ln is investigated in the (important) special nal affiliations. case where (Xn) is exchangeable and independent of (Tn). More precisely, we assume that: (i) (Xn) is exchangeable; (ii) (Xn) is independent of (Tn); Copyright: © 2021 by the authors. Li- P censee MDPI, Basel, Switzerland. (iii) Tn −! V for some random variable V > 0. This article is an open access article tn distributed under the terms and con- Under such conditions, we prove a weak law of large numbers (WLLN), a central ditions of the Creative Commons At- limit theorem (CLT), and we investigate the rate of convergence with respect to the total tribution (CC BY) license (https:// variation distance. creativecommons.org/licenses/by/ 4.0/). Mathematics 2021, 9, 194. https://doi.org/10.3390/math9020194 https://www.mdpi.com/journal/mathematics Mathematics 2021, 9, 194 2 of 11 Suppose in fact EjX1j < ¥ and conditions (i)-(ii)-(iii) hold. Define p L = VE(X1 j T ) and Mn = tn (Ln − L), where V is the random variable involved in condition (iii) and T the tail s-field of (Xn). P Then, it is not hard to show that Ln −! L. To obtain a CLT, instead, is not straightforward. In Section3, we prove that Mn ! M in distribution, where M is a suitable random variable, p n o provided E(X2) < ¥ and t Tn − V converges stably. Finally, in Section4, assuming 1 n tn (Xn) i.i.d. and some additional conditions, we find constants an and bn such that sup jP(Ln 2 A) − P(L 2 A)j ≤ an and A2B(R) sup jP(Mn 2 A) − P(M 2 A)j ≤ bn for every n ≥ 1. A2B(R) In particular, Ln ! L or Mn ! M in total variation distance provided an ! 0 or bn ! 0, as it happens in some situations. A last note is that, to our knowledge, random sums have been rarely investigated when (Xn) is exchangeable. Similarly, convergence of Ln or Mn in total variation distance is usually not taken into account. This paper contributes to fill this gap. 2. Preliminaries In the sequel, the probability distribution of any random element U is denoted by L(U). If S is a topological space, B(S) is the Borel s-field on S and Cb(S) the space of real bounded continuous functions on S. The total variation distance between two probability measures on B(S), say m and n, is dTV (m, n) = sup jm(A) − n(A)j. A2B(S) With a slight abuse of notation, if X and Y are S-valued random variables, we write dTV (X, Y) instead of dTV L(X), L(Y) , namely dTV (X, Y) = sup jP(X 2 A) − P(Y 2 A)j. A2B(S) If X is a real random variable, we say that L(X) is absolutely continuous to mean that L(X) is absolutely continuous with respect to Lebesgue measure. The following technical fact is useful in Section4. Lemma 1. Let X be a strictly positive random variable. Then, p lim dTV X + qn X, X = 0 n provided the qn are constants such that qn ! 0 and L(X) is absolutely continuous. R ¥ Proof. Let f be a density of X. Since limn −¥j fn(x) − f (x)j dx = 0, for some sequence fn of continuous densities, it can be assumed that f is continuous. Furthermore, since X > 0, for each e > 0 there is b > 0 such that P(X < b) < e. For such a b, one obtains p p dTV X + qn X, X ≤ e + sup P X + qn X 2 A j X ≥ b − P X 2 A j X ≥ b . A2B(R) Hence, it can be also assumed X ≥ b a.s. for some b > 0. Mathematics 2021, 9, 194 3 of 11 p Let gn be a density of X + qn X. Since p ¥ ¥ Z + Z + dTV X + qn X, X = f (x) − gn(x) dx = f (x) − gn(x) dx, −¥ b it suffices to showp that f (x) = limn gn(x) for each x > b. To prove the latter fact, define 2 0 fn(x) = x + qn x. For large n, one obtains 4 qn < b. In this case, fn > 0 on (b, ¥) and gn can be written as q 2 f−1(x) −1 n gn(x) = f fn (x) q . −1 qn + 2 fn (x) Therefore, f (x) = limn gn(x) follows from the continuity of f and q2 q q f−1(x) = x + n − n q2 + 4x −! x. n 2 2 n 2.1. Stable Convergence Stable convergence, introduced by Renyi in [11], is a strong form of convergence in distribution. It actually occurs in a number of frameworks, including the classical CLT, and thus it quickly became popular; see, e.g., [12] and references therein. Here, we just recall the basic definition. Let S be a metric space, (Yn) a sequence of S-valued random variables, and K a kernel (or a random probability measure) on S. The latter is a map K on W such that K(w) is a probability measure on B(S), for each w 2 W, and w 7! K(w)(B) is A-measurable for each B 2 B(S). Say that Yn converges stably to K if lim E f (Yn) j H = E K(·)( f ) j H , (1) n R for all f 2 Cb(S) and H 2 A with P(H) > 0, where K(·)( f ) = f (x) K(·)(dx). More generally, take a sub-s-field G ⊂ A and suppose K is G-measurable (i.e., w 7! K(w)(B) is G-measurable for fixed B 2 B(S)). Then, Yn converges G-stably to K if condition (1) holds whenever H 2 G and P(H) > 0. An important special case is when K is a trivial kernel, in the sense that K(w) = n for all w 2 W where n is a fixed probability measure on B(S). In this case, Yn converges G-stably to n if and only if Z lim E G f (Yn) = E(G) f dn n whenever f 2 Cb(S) and G : W ! R is bounded and G-measurable. 3. WLLN and CLT for Random Sums In this section, we still let Tn ∑i=1 Xi p Ln = , L = VE(X1 j T ) and Mn = tn (Ln − L), tn where V is the random variable involved in condition (iii) and \ T = s(Xn, Xn+1,...) n Mathematics 2021, 9, 194 4 of 11 is the tail s-field of (Xn). Recall that V > 0. Recall also that, by de Finetti’s theorem, (Xn) is exchangeable if and only if is i.i.d. conditionally on T , namely n P X1 2 A1,..., Xn 2 An j T = ∏ P X1 2 Ai j T a.s. i=1 for all n ≥ 1 and all A1,..., An 2 B(R). The following WLLN is straightforward. P Theorem 1. If EjX1j < ¥ and conditions (i) and (iii) hold, then Ln −! L. P Proof.