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Convergence in Total Variation of Random Sums

Author

Listed:
  • 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)

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.

Suggested Citation

  • Luca Pratelli & Pietro Rigo, 2021. "Convergence in Total Variation of Random Sums," Mathematics, MDPI, vol. 9(2), pages 1-11, January.
    • Handle: RePEc:gam:jmathe:v:9:y:2021:i:2:p:194-:d:482973
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    References listed on IDEAS

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    2. Korolev, V.Yu. & Chertok, A.V. & Korchagin, A.Yu. & Zeifman, A.I., 2015. "Modeling high-frequency order flow imbalance by functional limit theorems for two-sided risk processes," Applied Mathematics and Computation, Elsevier, vol. 253(C), pages 224-241.
    3. Christian Schluter & Mark Trede, 2016. "Weak convergence to the Student and Laplace distributions," Post-Print hal-01447853, HAL.
    4. Korolev, Victor & Zeifman, Alexander, 2021. "Bounds for convergence rate in laws of large numbers for mixed Poisson random sums," Statistics & Probability Letters, Elsevier, vol. 168(C).
    5. Pratelli, Luca & Rigo, Pietro, 2019. "Total variation bounds for Gaussian functionals," Stochastic Processes and their Applications, Elsevier, vol. 129(7), pages 2231-2248.
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