Precise large deviations for the difference of two sums of WUOD and non identically distributed random variables with dominatedly varying tails

ABSTRACT

In this article, we study large deviations for non random difference ∑^n₁(t)_{j = 1}X_1j − ∑^n₂(t)_{j = 1}X_2j and random difference ∑^N₁(t)_{j = 1}X_1j − ∑^N₂(t)_{j = 1}X_2j, where {X_1j, j ⩾ 1} is a sequence of widely upper orthant dependent (WUOD) random variables with non identical distributions {F_1j(x), j ⩾ 1}, {X_2j, j ⩾ 1} is a sequence of independent identically distributed random variables, n₁(t) and n₂(t) are two positive integer-valued functions, and {N_i(t), t ⩾ 0}²_{i = 1} with EN_i(t) = λ_i(t) are two counting processes independent of {X_ij, j ⩾ 1}²_{i = 1}. Under several assumptions, some results of precise large deviations for non random difference and random difference are derived, and some corresponding results are extended.

KEYWORDS:

• Dominatedly varying tail
• Non-identical distributions
• Precise large deviations
• Widely upper orthant dependence

MATHEMATICS SUBJECT CLASSIFICATION:

• 60F10
• 60F05
• 60G50

Funding

The first author is supported by the National Natural Science Foundation of China (grant Nos. 11371077 and 61175041). The third author is supported by the National Natural Science Foundation of China (grant Nos. 11101061 and 11371077).