Zone of normal attraction

A domain of the form $ 0 \leq x \leq \psi ( n) $ for which

$$ \frac{ {\mathsf P} \{ Z _ {n} \geq x \} }{ {\mathsf P} \{ Y \geq x \} } \rightarrow 1 \ \textrm{ or } \ \ \frac{ {\mathsf P} \{ Z _ {n} \leq - x \} }{ {\mathsf P} \{ Y \leq - x \} } \rightarrow 1 $$

as $ n \rightarrow \infty $, where $ \{ \psi ( n) \} \uparrow \infty $, $ \{ Z _ {n} \} $ is a sequence of random variables and $ Y $ is a random variable with a normal distribution. Zones of normal attraction have been studied for the case

$$ Z _ {n} = \frac{1}{\sigma \sqrt n } \sum _ { j= } 1 ^ { n } ( X _ {j} - a ) , $$

where $ \{ X _ {n} \} $ is a sequence of independent, identically-distributed random variables with mathematical expectation $ a $ and finite positive variance $ \sigma ^ {2} $.

References

[1]	I.A. Ibragimov, Yu.V. Linnik, "Independent and stationary sequences of random variables" , Wolters-Noordhoff (1971) (Translated from Russian)

Comments

A rather general formulation of the problems of large deviations is as follows, [a3]. Suppose that for a family of stochastic processes $ \xi ^ {a} ( t) $ a result of the law-of-large-numbers type holds (cf. Law of large numbers), $ \xi ^ {a} \rightarrow x $ as $ a \rightarrow \infty $. Problems on large deviations of the process $ \xi ^ {a} ( t) $ from its most probable path $ x( t) $ for large values of $ a $ are concerned with the limiting behaviour as $ a \rightarrow \infty $ of the infinitesimal probabilities $ p ^ {a} ( \xi ^ {a} \in A ) $ for measurable sets $ A $ that are at a positive distance from the non-random limiting function $ x $( in a suitable function space (space of paths)). Problems concerning the asymptotics as $ a \rightarrow \infty $ of expectations of the form $ {\mathsf E} ^ {a} [ f ^ { a } ( \xi ^ {a} )] $ also form part of large deviation theory if the main part of these expectations for large values of $ a $ comes from the low probability values of $ \xi ^ {a} $.

References

[a1]	V.V. Petrov, "Sums of independent random variables" , Springer (1975) (Translated from Russian)
[a2]	R.J. Serfling, "Approximation theorems of mathematical statistics" , Wiley (1980) pp. 6, 96
[a3]	A.D. [A.D. Ventsel'] Wentzell, "Limit theorems on large deviations for Markov stochastic processes" , Kluwer (1990) (Translated from Russian)
[a4]	L. Saulis, V.A. Statulevicius, "Limit theorems for large deviations" , Kluwer (1991) (Translated from Russian)