On complete convergence of moments of i.i.d.r.v. with finite variances

L. V. Rozovsky

Zapiski Nauchnykh Seminarov POMI, Probability and statistics. Part 34, Tome 525 (2023), pp. 109-121

Résumé

Let $\{X_n\}, n\ge 1,$ be a sequence of independent random variables with common distribution functions, zero means and unit variances, $\bar{S}_n =( X_1 +\cdots + X_n)/\sqrt n$. The main goal of this note is a study of the behavior of sums $$ \Sigma_r(\varepsilon) = \sum\limits_{n\ge 1} n^s \mathbf{E} \bar S^r_n I[\bar S_n\ge \varepsilon n^\delta], $$ as $\varepsilon\to +0$ under optimal (that is, necessary) moment assumptions, where $\delta, s, r$ are some constants, such that $\delta> 0$ and $s+1$ and $r$ are non-negative. In particular, it is shown that if $s>-1/2$ and $(2-r) \delta = s+1$, then $$ \varepsilon^{2-r} \Sigma_r(\varepsilon) = \dfrac{1}{2\delta (2-r)} + O \big(\lambda(\rho)\big),\ \rho=\varepsilon^{-1/2\delta}, \lambda(\rho)=\mathbf{E} X_1^2 \Big(1 \land \dfrac{| X_1|}{\rho}\Big). $$ A similar estimate with a more complicated formulation holds also in the case $-1<s\le -1/2$. Thus, for $\delta=1/2$ we generalize the pioneering result of Heyde (Appl. Probab., 1975) and most its refinements (e.g. due to He and Xie (Acta Math. Appl. Sin., 2013)), as well as the corresponding statements of Liu and Lin (Statist. Probab. Lett. 2006) and Kong and Dai (Stoch. Dynamics, 2017).

Bibliographie

[1] C. C. Heyde, “A supplement to the strong law of large numbers”, J. Appl. Probab., 12 (1975), 173–175

[2] R. Chen, “A remark on the tail probability of a distribution”, J. Multivariate Anal., 8 (1978), 328–333

[3] V. V. Buldygin, O. I. Klesov, J. G. Steinebach, “Precise asymptotics over a small parameter for a series of large deviation probabilities”, Theory Stoch. Process., 13:1 (2007), 44–56

[4] A. Gut, J. Steinebach, “Precise asymptotics a general approach”, Acta. Math. Hungar., 138 (2013), 365–385

[5] J. He, T. Xie, “Asymptotic property for some series of probability”, Acta Math. Appl. Sin., 29 (2013), 179–186

[6] L. T. Kong, H. S. Dai, “Convergence rates in precise asymptotics for a kind of complete moment convergence”, Stoch. Dynam., 17:2 (2017), 1750015, 18 pp.

[7] L. V. Rozovskii, “O tochnoi asimptotike v slabom zakone bolshikh chisel dlya summ nezavisimykh sluchainykh velichin s obschei funktsiei raspredeleniya iz oblasti prityazheniya ustoichivogo zakona. II”, Teoriya veroyatn. i ee primen., 49:4 (2004), 803–813

[8] Y. Zhang, “A note on the convergence rates in precise asymptotics”, J. Ineq. Appl., 2019 (2019), 15

[9] L. V. Rozovskii, “Nekotorye predelnye teoremy dlya bolshikh uklonenii summ nezavisimykh sluchainykh velichin s obschei funktsiei raspredeleniya iz oblasti prityazheniya ustoichivogo zakona”, Zap. nauchn. semin. POMI, 495, 2020, 250–266

[10] L. V. Rozovsky, “One more on the convergence rates in precise asymptotics”, Statist. Probab. Lett., 171 (2021)

[11] L. V. Rozovskii, “O skorosti skhodimosti v “tochnykh asimptotikakh” dlya sluchainykh velichin s ustoichivym raspredeleniem”, Zap. nauchn. semin. POMI, 501, 2021, 259–275

[12] L. V. Rozovskii, “O polnoi skhodimosti momentov v tochnykh asimptotikakh”, Zap. nauchn. semin. POMI, 515, 2022, 180–188

[13] L. V. Rozovskii, “K voprosu o skorosti skhodimosti v tochnykh asimptotikakh”, Teoriya veroyatn. i ee primen., 68:1 (2023), 57–74

[14] L. V. Rozovskii, “Otsenka snizu veroyatnostei bolshikh uklonenii summ nezavisimykh sluchainykh velichin s konechnymi dispersiyami”, Zap. nauchn. semin. POMI, 260, 1999, 218–239

[15] L. V. Rozovskii, “Summy nezavisimykh sluchainykh velichin s konechnymi dispersiyami - umerennye ukloneniya i neravnomernye otsenki v TsPT”, Zap. nauchn. semin. POMI, 311, 2004, 242–259

[16] L. V. Rozovskii, “Otsenki skorosti skhodimosti v “intervalnoi” TsPT dlya summ nezavisimykh sluchainykh vektorov”, Vestnik SpbGU, 4(62):3 (2017), 466–476

[17] P. Hall, “Bounds on the rate of convergence of moments in the central limit theorem”, Ann. Probab., 10:4 (1982), 1004–1018