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The limiting behaviour of sums and maximums of iid random variables from the viewpoint of different observers

Krajka T. K., Rychlik Z.

Probability and Mathematical Statistics

2014

Vol. 34, Fasc. 2

237--252

The limiting behaviour of observed and all random variables in the max limit schema was considered by Mladenović and Piterbarg (2006) and Krajka (2011). Here those results are generalised in two directions: we allow more than one observer and one superobserver; we consider the max limit schema as well as the sum limit schema.

Limit theorems for products of sums of independent random variables

Krajka T. K., Rychlik Z.

Probability and Mathematical Statistics

2010

Vol. 30, Fasc. 1

73--85

Let {Xn; n ≥ 1} be a sequence of independent random variables with finite second moments and {Nn; n ≥ 1} be a sequence of positive integer-valued random variables. Write Sn. =Σnk-1(Xk−EXk); n ≥1; and let N be a standard normal random variable. In the paper the convergences...[formula], are considered for some sequences {an} and {γn} of positive integer numbers such that Sn + an≥ 0 a.e. The case when γn are random variables is also considered. The main results generalize the main theorems presented by Pang et al. [3].