Convergence in Total Variation of Random Sums

Luca Pratelli and Pietro Rigo
Additional contact information
Luca Pratelli: Accademia Navale, viale Italia 72, 57100 Livorno, Italy
Pietro Rigo: Dipartimento di Scienze Statistiche “P. Fortunati”, Università di Bologna, via delle Belle Arti 41, 40126 Bologna, Italy

Mathematics, 2021, vol. 9, issue 2, 1-11

Abstract: Let ( X n ) be a sequence of real random variables, ( T n ) a sequence of random indices, and ( τ n ) a sequence of constants such that τ n → ∞ . The asymptotic behavior of L n = ( 1 / τ n ) ∑ i = 1 T n X i , as n → ∞ , is investigated when ( X n ) is exchangeable and independent of ( T n ) . We give conditions for M n = τ n ( L n − L ) ? M in distribution, where L and M are suitable random variables. Moreover, when ( X n ) is i.i.d., we find constants a n and b n such that sup A ∈ B ( R ) | P ( L n ∈ A ) − P ( L ∈ A ) | ≤ a n and sup A ∈ B ( R ) | P ( M n ∈ A ) − P ( M ∈ A ) | ≤ b n for every n . In particular, L n → L or M n → M in total variation distance provided a n → 0 or b n → 0 , as it happens in some situations.

Keywords: exchangeability; random sum; rate of convergence; stable convergence; total variation distance (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2021
References: View references in EconPapers View complete reference list from CitEc
Citations:

Downloads: (external link)
https://www.mdpi.com/2227-7390/9/2/194/pdf (application/pdf)
https://www.mdpi.com/2227-7390/9/2/194/ (text/html)

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:gam:jmathe:v:9:y:2021:i:2:p:194-:d:482973

Access Statistics for this article

Mathematics is currently edited by Ms. Emma He

More articles in Mathematics from MDPI
Bibliographic data for series maintained by MDPI Indexing Manager ().