Sums Of Real-valued Random Variables Research Articles

Large Deviations for Sums of Random Vectors Attracted to Operator Semi-Stable Laws

Let \(\{X_i, i\ge 1\}\) be i.i.d. \(\mathbb {R}^d\)-valued random vectors attracted to operator semi-stable laws and write \(S_n=\sum _{i=1}^{n}X_i\). This paper investigates precise large deviations for both the partial sums \(S_n\) and the random sums \(S_{N(t)}\), where N(t) is a counting process independent of the sequence \(\{X_i, i\ge 1\}\). In particular, we show for all unit vectors \(\theta \) the asymptotics $$\begin{aligned} {\mathbb P}(|\langle S_n,\theta \rangle |>x)\sim n{\mathbb P}(|\langle X,\theta \rangle |>x) \end{aligned}$$ which holds uniformly for x-region \([\gamma _n, \infty )\), where \(\langle \cdot , \cdot \rangle \) is the standard inner product on \(\mathbb {R}^d\) and \(\{\gamma _n\}\) is some monotone sequence of positive numbers. As applications, the precise large deviations for random sums of real-valued random variables with regularly varying tails and \(\mathbb {R}^d\)-valued random vectors with weakly negatively associated occurrences are proposed. The obtained results improve some related classical ones.

The Generalized Trotter Operator and Weak Convergence Of Dependent Random Variables in Different Probability Metrics

The Trotter operator-theoretic method for establishing weak convergence of sums of real-valued random variables X i i∈N on a probability space (Ω,A,P) is extended to the situation that the Xi are dependent. For this goal, the generalized Trotter-operator is defined for functions f∈Cb and a sub- σ algebra G ⊂ A by VX¦G f (y) ≔ ∫Rf(x+y) dPX¦G(x,w). General results concerning the weak convergence of dependent random variables (r.vs.) with rates are presented. Further, the distance V between the distributions P,Q of two random variables X,Y with respect to a function f and a sub-σ algebra G, namely the Trotter-distance V(P,Q;f), defined via the generalized Trotter-operator by V(P,Q;f) = supy¦∫Rf(x+y) d(P−q)(x)¦, is compared with other well-known probability-metrics, which metrize weak convergence, such as the Zolotarev-metric ξ, the Gudynas-metric η, the Levy-metric L or the Prohorov-metric ϱ. Lastly, the Trotter-distance with rates is shown to be equivalent to weak convergence with rates.