Asymptotic Distribution Of Sum Research Articles

Proved Random Numbers Obtained from Hardware Devices

In a previous paper, we have showed how to obtain sequences of number proved random. With this aim, we used sequences of noises yn such that the conditional probabilities have Lipschitz coefficients not too large. We transformed them using Fibonacci congruences. Then, we obtained sequences xn which admit the IID model for correct model. This method consisted to value the work of Marsaglia in order to build his CD-ROM. But we did not use Rap Music (as Marsaglia), but texts files. This method also uses an extractor and at the same time the notion of correct models. In this paper, we apply this method to numbers provided by machines or chips. Unfortunately, it is less sure than they have Lipschtiz coefficient not too large. But we can solve this problem: it suffices to use the Central Limit Theorem. We do it modulo 1. In this case, we use a new limit theorem, the XOR Limit theorem : asymptotic distribution of sum of random vectors modulo 1 are asymptotically independent. Then Lipschtiz coefficient of associated sequences are not too large and we can obtain IID sequences by using Fibonacci congruences.

Asymptotic distribution of sum and maximum for Gaussian processes

Let {Xn, n ≥ 0} be a stationary Gaussian sequence of standard normal random variables with covariance function r(n) = EX0Xn. Let Under some mild regularity conditions on r(n) and the condition that r(n)lnn = o(1) or (r(n)lnn)−1 = O(1), the asymptotic distribution of is obtained. Continuous-time results are also presented as well as a tube formula tail area approximation to the joint distribution of the sum and maximum.

Asymptotic distribution of sum and maximum for Gaussian processes

Asymptotic distribution of sum and maximum for Gaussian processes

Asymptotic distribution of sum and maximum for Gaussian processes

Let {X n , n ≥ 0} be a stationary Gaussian sequence of standard normal random variables with covariance function r(n) = E X 0 X n . Let Under some mild regularity conditions on r(n) and the condition that r(n)lnn = o(1) or (r(n)lnn)−1 = O(1), the asymptotic distribution of is obtained. Continuous-time results are also presented as well as a tube formula tail area approximation to the joint distribution of the sum and maximum.

A remark on the asymptotic theory of sums with random size

The aim of the present note is to point to a strange fact, through an example, in connexion with the asymptotic distribution of sums of a random number of independent and identically distributed (i.i.d.) random variables. We shall formulate our conclusion after the introduction of some notations.