Sentence examples for any extra conditions from inspiring English sources

So Theorem 2.3 extends corresponding results for independent random variables without necessarily adding any extra conditions.

As a result, we extend some complete convergence theorems for independent random variables to the case of negatively dependent random variables without necessarily imposing any extra conditions.

We extend the classical Jamison convergence theorem and the Marcinkiewicz strong law of large numbers for independent sequences of random variables to -mixing sequences of random variables without necessarily adding any extra conditions.

As a result, we extend some complete convergence and complete moment convergence theorems for independent random variables to the case of negatively associated random variables without necessarily imposing any extra conditions.

In this paper, based on Gut and Stadtmüller (2011 [1]) and Qiu and Chen (2014 [2]), we extend the complete convergence and complete moment theorems for independent random variables to the negatively associated sequences of random variables without necessarily imposing any extra conditions, which extend the corresponding results of Gut and Stadtmüller (2011 [1]) and Qiu and Chen (2014 [2]).

When ({R_{0}^{T}}<1), the disease modeled by (4) will almost surely go extinct without any extra condition.

Finally, we want to address one conjecture on extinction of the disease, i.e., When ({R_{0}^{T}}<1), the disease modeled by (4) will almost surely go extinct without any extra condition.

Since (3.20) are all linear boundary value problems, it is not difficult to prove the existence of solution and the exponential decaying of solution without imposing any extra condition.

This kind of ordering he called a 'well-ordering', which we now define as a total-ordering with an extra condition, namely that any subset has a least element in the ordering.

Mr. Akinmusire imposed an extra condition for himself.

Abbas has an extra condition; he insists that Hamas should restore things in Gaza the way they were before June 14 , 2007