On limit theorems for fields of martingale differences

We prove a central limit theorem for stationary multiple (random) fields of martingale differences $f\circ T_{\underline{i}}$, $\underline{i}\in \Bbb Z^d$, where $T_{\underline{i}}$ is a $\Bbb Z^d$ action. In most cases the multiple (random) fields of martingale differences is given by a completely commuting filtration. A central limit theorem proving convergence to a normal law has been known for Bernoulli random fields and in [V15] this result was extended to random fields where one of generating transformations is ergodic. In the present paper it is proved that a convergence takes place always and the limit law is a mixture of normal laws. If the $\Bbb Z^d$ action is ergodic and $d\geq 2$, the limit law need not be normal. For proving the result mentioned above, a generalisation of McLeish's CLT for arrays $(X_{n,i})$ of martingale differences is used. More precisely, sufficient conditions for a CLT are found in the case when the sums $\sum_i X_{n,i}^2$ converge only in distribution. The CLT is followed by a weak invariance principle. It is shown that central limit theorems and invariance principles using martingale approximation remain valid in the non-ergodic case

Martingale-coboundary decomposition for stationary random fields

We prove a martingale-coboundary representation for random fields with a completely commuting filtration. For random variables in L2 we present a necessary and sufficient condition which is a generalization of Heyde's condition for one dimensional processes from 1975. For Lp spaces with 2 \leq p < \infty we give a necessary and sufficient condition which extends Volny's result from 1993 to random fields and improves condition of El Machkouri and Giraudo from 2016 (arXiv:1410.3062). In application, new weak invariance principle and estimates of large deviations are found.Comment: Stochastics and Dynamics 201

Policy Reform Under Electoral Uncertainty

How does uncertainty affect the process of policy reform? Our investigation identifies two types of uncertainties, one at the electoral level and another at the implementation level. When voters abstain from the electoral process, electoral uncertainty emerges. Implementation uncertainty arises whenever the politician is unable to guarantee a positive outcome from a policy implementation. Using a political agency model where two groups of voters delegate to a politician the decision to implement reform or maintain the status quo of the economy, we show that both implementation uncertainty and electoral uncertainty affect policy implementation in different ways. Implementation uncertainty might introduce disagreement between voters about the (ex-ante) convenience of implementing the project. On the other hand, with electoral uncertainty in the political system, political power may be detached from the group’s relative size, thus linking it to the citizens’ probability of being the decisive vote. In short, a highly disciplined minority group could gather enough political power to impose their preferred policies over a less disciplined majority group.

Local limit theorem in deterministic systems

We show that for every ergodic and aperiodic probability preserving system, there exists a $\mathbb{Z}$ valued, square integrable function $f$ such that the partial sums process of the time series $\left\{f\circ T^i\right\}_{i=0}^\infty$ satisfies the lattice local limit theorem.Comment: 17 page