# Asymptotic negligibility

2010 Mathematics Subject Classification: Primary: 60F99 [MSN][ZBL]

A property of random variables indicating that their individual contribution as components of a sum is small. This concept is important, for example, in the so-called triangular array. Let the random variables $X _ {nk }$( $n=1, 2 ,\dots$; $k=1 \dots k _ {n}$) be mutually independent for each $n$, and let

$$S _ {n} = X _ {n1} + \dots +X _ {n k _ {n} } .$$

If for all $\epsilon > 0$ and $\delta > 0$, at sufficiently large values of $n$, the inequality

$$\tag{1 } \max _ {1 \leq k \leq k _ {n} } \ {\mathsf P} ( | X _ {nk} | > \epsilon ) < \delta$$

is satisfied, the individual terms $X _ {nk}$ are called asymptotically negligible (the variables $X _ {nk }$ then form a so-called zero triangular array). If condition (1) is met, one obtains the following important result: The class of limit distributions for $S _ {n} - A _ {n}$( $A _ {n}$ are certain "centering" constants) coincides with the class of infinitely-divisible distributions (cf. Infinitely-divisible distribution). If the distributions of $S _ {n}$ converge to a limit distribution, $k _ {n} \rightarrow \infty$, and the terms are identically distributed, condition (1) is automatically met. If the requirement for asymptotic negligibility is strengthened by assuming that for all $\epsilon > 0$ and $\delta > 0$ for all sufficiently large $n$ one has

$$\tag{2 } {\mathsf P} \left ( \max _ {1 \leq k \leq k _ {n} } \ | X _ {nk} | > \epsilon \right ) < \delta ,$$

then the following statement is valid: If (2) is met, the limit distribution for $S _ {n} - A _ {n}$ can only be a normal distribution (in particular with variance equal to zero, i.e. a degenerate distribution).