Sherzod M. Mirakhmedov

ESAIM: PS 25 (2021) 53–54 ESAIM: Probability and Statistics

https://doi.org/10.1051/ps/2021002 www.esaim-ps.org

CORRECTION

THE PROBABILITIES OF LARGE DEVIATIONS FOR A CERTAIN CLASS OF STATISTICS ASSOCIATED WITH MULTINOMIAL

DISTRIBUTIONS

Sherzod M. Mirakhmedov

Received December 28, 2020. Accepted January 8, 2021.

There are misprints in the statements of Theorem 2.1 and 2.2 of the article.

ESAIM: PS (2020)24581–606

In the relations (2.7) and (2.8) it must be written RN(η) instead ofRN, andO_1+x

√ N

N

instead ofo_1+x

√ N

N

. The initial sentence of condition (ii) of Theorem 2.2 was omitted. The corrected variant is

Theorem 2.2. Letn→ ∞,N → ∞,p_max→0 and the functionsh_m(·) be non-negative. If (i) For each integers∈[3, k_n] there exist non-negativea₁,a₂ andb₁,b₂ such that

E

(ξ_m−np_m)²h^s_m(ξ_m)

=O

s^a(np_m)^bEh^s_m(ξ_m)

, (2.9)

where a=a₁,b=b₁ for allm∈ N ⊆(1,2, . . . ,N), anda=a₂,b=b₂ for allm∈(1,2, . . . ,N)\N, and kn=o min(p⁻¹_max, Kn(a1, b1),Kn(a2, b2))

, (2.10)

(ii) For each integer s∈[3, k_n] there existsω²_m>0 and a sequence of positive numbers V_n such that for some ν ≥1

|Cs(hm(ξm))| ≤(s!)^1+νV_n^s−2ω_m² , m= 1, . . . ,N, (2.11)

Institute of Mathematics, Academy of Sciences of Uzbekistan, Mirzo Ulugbek str., 81, 100125 Tashkent, Uzbekistan.

* Corresponding author:[email protected]

Article published by EDP Sciences c EDP Sciences, SMAI 2021

54 S.M. MIRAKHMEDOV

then for allxN such that

0≤xN =o min

W_N^1/(1+2ν), k_n^1/2

, (2.12)

where WN =V_n⁻¹σNmin(1,σ²_N/eω²_N),ωe²_N =ω₁²+...+ω²_N, it holds P{RN(η)> xNσN+AN}= (1−Φ(xN))

1 +O

(xN + 1)W_N^−1/(1+2ν)

, (2.13)

and

P{RN(η)<−xNσN +AN}= Φ(−xN) 1 +O

(xN+ 1)W_N^−1/(1+2ν)

. (2.14)

At last, Corollary 3.6 should be read as Theorem 3.6, proof of which follows from Theorem 2.2 alike to proofs of Theorems 3.5 and 3.7.