Logarithmic averages of stable random variables are asymptotically normal

A Survey on Simulating Stable Random Variables

In general case, Chambers et al. (1976) introduced the following algorithm for simulating any stable random variables $ X/sim(alpha, beta, gamma, delta) $ with four parameters. They use a nonlinear transformation of two independent uniform random variables for simulating an stable random variable... (to continue, click here)

Convergence of Weighted Averages of Random Variables Revisited

We show that for a large class of positive weights including the ones that are eventually monotone decreasing and those that are eventually monotone increasing but vary regularly, if the averages of random variables converge in some sense, then their corresponding weighted averages also converge in the same sense. We will also replace the sufficient conditions in the fundamental result of Jamis...

Prophet Inequalities for Averages of Independent Non-Negative Random Variables

The main purpose of this paper is to prove the following theorem, which sharpens results of Krengel and Sucheston [11, 12] in which the weaker constant 2(1 +-(3) was obtained. (Here EX is the expected value of the ran­ dom variable X, and ~ and T are the sets of stop rules ~ n, and of a.s. finite stop rules, respectively.) Theorem 1.1. If Xl' ... , X n are independent non-negative random variab...

Distributions of Functions of Normal Random Variables

The unit or standard normal random variable U is a normally distributed variable with mean zero and variance one, i. e. U ∼ N(0, 1). Note that if x ∼ N(µ, σ 2) that x − µ σ ∼ U ∼ N(0, 1) (1) Thus to simulate a normal random variable with mean µ and variance σ 2 , we can simply transform unit normals, as x ∼ µ + σU ∼ N(µ, σ 2) (2) Consider n independent random variables x i ∼ N(µ, σ 2), then x ∼...

Distributions of Functions of Normal Random Variables