Let {Xi, i = 1, 2, . . .} be i.i.d. standard gaussian variables. Let Sn = X1 + . . . + Xn be the sequence of partial sums and Ln = max 0≤i<j≤n Sj − Si √ j − i . We show that the distribution of Ln, appropriately normalized, converges as n → ∞ to the Gumbel distribution. In some sense, the the random variable Ln, being the maximum of n(n+1)/2 dependent standard gaussian variables, behaves like t...

We study the convergence of densities of a sequence of random variables to a normal density. The random variables considered are nonlinear functionals of a Gaussian process. The tool we are using is the Malliavin calculus, in particular, the integration by parts formula and the Stein’s method. Applications to the convergence of densities of the least square estimator for the drift parameter in ...

We here provide additional proofs, definitions, lemmas and derivations omitted in the paper. Note that material contained in the latter are referred to by the captions used there (e.g. Theorem 1), whereas auxiliary statements contained exclusively in this supplement are preceded by a capital Roman letter (e.g. Theorem A.1). A Sub-Gaussian random variables and concentration inequalities A random...

In this paper, the two parameter ADK entropy, as a generalized of Re'nyi entropy, is considered and some properties of it, are investigated. We will see that the ADK entropy for continuous random variables is invariant under a location and is not invariant under a scale transformation of the random variable. Furthermore, the joint ADK entropy, conditional ADK entropy, and chain rule of this ent...

We consider quadratic forms Q n = X 1j<kn a jk X j X k ; where X j are i.i.d. random variables with nite third moment. We obtain optimal bounds for the Kolmogorov distance between the distribution of Q n and the distribution of the same quadratic forms with X j replaced by corresponding Gaussian random variables.

We consider the generalized differential entropy of normalized sums of independent and identically distributed (IID) continuous random variables. We prove that the Rényi entropy and Tsallis entropy of order α (α > 0) of the normalized sum of IID continuous random variables with bounded moments are convergent to the corresponding Rényi entropy and Tsallis entropy of the Gaussian limit, and obtai...

We construct an independent increments Gaussian process associated to a class of multicolor urn models. The construction uses random variables from the urn model which are different from the random variables for which central limit theorems are available in the two color case.

The formula for the expectation of the product of four scalar real Gaussian random variables is generalized to matrix-valued (real or complex) Gaussian random variables. As an application of the extended formula, we present a simple derivation of the covariance matrix of instrumental variable (IV) estimates of parameters in multivariate linear regression models. (*) Faculty of Electrical Engine...

Bayesian networks provide a method of rep­ resenting conditional independence between random variables and computing the prob­ ability distributions associated with these random variables. In this paper, we ex­ tend Bayesian network structures to compute probability density functions for continuous random variables. We make this extension by approximating prior and conditional den­ sities using...

This paper presents a useful theorem for non-linear transformations of the sum of independent, zeromean, Gaussian random variables. It is proved that the linear regression coefficient of the non-linear transformation output with respect to the overall input is identical to the linear regression coefficient with respect to any Gaussian random variable that is part of the input. As a side-result,...