On packing dimension preservation by distribution functions of random variables with independent $\tilde{Q}$-digits

Random variables, joint distribution functions, and copulas

If G is an n-dimensional joint distribution function with 1-dimensional margins F1,...,Fn, then there exists a function C (called an "w-copula") from the unit »-cube to the unit interval such that G(xx, ..., xn) = C(F1(x1), ..., Fn(xn)) for all real n-tuples (xlt ..., xn). The paper is devoted to an investigation of the structure and properties of n-copulas and their connection with random vari...

A Dimension Gap for Continued Fractions with Independent Digits

Moment inequalities for functions of independent random variables

A general method for obtaining moment inequalities for functions of independent random variables is presented. It is a generalization of the entropy method which has been used to derive concentration inequalities for such functions [7], and is based on a generalized tensorization inequality due to Lata la and Oleszkiewicz [25]. The new inequalities prove to be a versatile tool in a wide range o...

Local Tail Bounds for Functions of Independent Random Variables

It is shown that functions defined on {0, 1, . . . , r − 1}n satisfying certain conditions of bounded differences that guarantee subgaussian tail behavior also satisfy a much stronger “local” subgaussian property. For self-bounding and configuration functions we derive analogous locally subexponential behavior. The key tool is Talagrand’s (1994) variance inequality for functions defined on the ...

Pairwise Independent Random Variables

In this lecture we discuss how to derandomize algorithms. We will see a brute force algorithm (enumeration) for derandomization. We will also see that some random algorithms do not need true randomness. Specifically, we will see an example where only pairwise random bits are needed. Next, we will see how we can generate pairwise random values and how this conservation on the amount of randomnes...