The development of a general inferential theory for nonlinear models with cross-sectionally or spatially dependent data has been hampered by a lack of appropriate limit theorems. To facilitate a general asymptotic inference theory relevant to economic applications, this paper first extends the...

Sample measures of top centile contributions to the total (concentration) are downward biased, unstable estimators, extremely sensitive to sample size and concave in accounting for large deviations. It makes them particularly unfit in domains with power law tails, especially for low values of...

Consider an i.i.d. random field {Xk:k∈Z+d}, together with a sequence of unboundedly increasing nested sets Sj=⋃k=1jHk,j≥1, where the sets Hj are disjoint. The canonical example consists of the hyperbolas Hj={k∈Z+d:|k|=j}. We are interested in the number of “hyperbolas” Hj that...

Let X1,…,Xn be i.i.d. copies of random variable X where 0E|X|∞ and let X̄=1n∑i=1nXi. One can show that X1−X̄,…,Xn−X̄ are exchangeable, and as a result identically distributed, but not independent. We use this result to prove that for n≥3, X is symmetric about a point if and only...

We consider a new family of convex weakly compact valued integrable random sets which is called an adapted array of convex weakly compact valued integrable random variables of type p (1⩽p⩽2). By this concept, more general laws of large numbers will be established. Some illustrative examples...

In this paper we prove exponential inequalities (also called Bernstein’s inequality) for fractional martingales. As an immediate corollary, we will discuss a weak law of large numbers for fractional martingales under a divergence assumption on the β-variation of the fractional martingale. A...

Nicolas Bernoulli suggested the St Petersburg game, nearly 300 years ago, which is widely believed to produce a paradox in decision theory. This belief stems from a long standing mathematical error in the original calculation of the expected value of the game. This article argues that, in...

We analyze whether sliding window time averages applied to stationary increment processes converge to a limit in probability. The question centers on averages, correlations, and densities constructed via time averages of the increment x(t,T)=x(t+T)−x(t), e.g. x(t,T)=ln(p(t+T)/p(t)) in finance...

A weak law of large numbers is established for a sequence of systems of N classical point particles with logarithmic pair potential in Rn, or Sn,n∈N, which are distributed according to the configurational microcanonical measure δ(E−H), or rather some regularization thereof, where H is the...

We prove a law of large numbers for a class of Zd-valued random walks in dynamic random environments, including non-elliptic examples. We assume for the random environment a mixing property called conditional cone-mixing and that the random walk tends to stay inside wide enough space–time...