Asymptotic negligibility

2020 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).

References

[F]	W. Feller, "An introduction to probability theory and its applications", 2 , Wiley (1966) pp. 210