which is the Central Limit Theorem. In principle, all the random variables X1, X2, · · · , XN can be of order 1, hence SN ∼ 1 as well, but the probability of having such a rare event is incredibly small. We can even estimate the bound on the probability for the rare event from the large deviation principle. A similar phenomenon happens when we form a large matrix from i.i.d. random variables an...

The stochastic processes of almost exclusive interest in modeling channel noise are the Gaussian processes. Gaussian processes are stochastic processes for which the random variables N(t1), N(t2), . . . , N(tk) are jointly Gaussian for all t1, . . . , tk and all k > 0. Today we start by giving a more complete discussion of jointly Gaussian random variables. We restrict our attention to zero mea...

We derive multi-level concentration inequalities for polynomials in independent random variables with an α-sub-exponential tail decay. A particularly interesting case is given by quadratic forms f(X1,…,Xn)=⟨X,AX⟩, which we prove Hanson–Wright-type explicit dependence on various norms of the matrix A. consequence these a two-level inequality variables, such as Poisson chaos. provide applications...

for various types of risk process X = (X(t), t ≥ 0) and functions f(t). The similar problem of finding the buffer overflow probability appears in the queuing theory for different communication network models. The tasks of such type were solved for many types of processes, including Gaussian ones and aforementioned FBM (see, for example, Norros [1], Michna [2], Baldi and Pacchiarotti [3], etc.)....

A new synthesis scheme is proposed to generate a random vector with prescribed joint density that induces a (latent) Gaussian tree structure. The quality of synthesis is shown by vanishing total variation distance between the synthesized and desired statistics. The proposed layered and successive synthesis scheme relies on the learned structure of tree to use sufficient number of common random ...