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We consider a sequence $X^n=(X^n_t)_{t\ge 0},n\ge 1$ of semimartingales. Each $X^n$ is a weak solution to an It\^o equation with respect to a Wiener process and a Poissonian martingale measure and is in general non-Markovian process. For this sequence, we prove the large deviation principle in the Skorokhod space $D=D_{[0,\infty)}$. We use a new approach based on of exponential tightness. This allows us to establish the large deviation principle under weaker assumptions than before.

In this paper we obtain central limit theorems for some dependent random variables which are used to describe the distribution of magnetic spins present in a ferromagnet crystal. A ferromagnet crystal consists of a large number of sites. At site i there is some amount of magnetic spin present. The magnetic spin present at any site interacts with the magnetic spins at its neighboring sites and hence gives rise to some dependency among the magnetic spin present.

In this article we obtain central limit theorems for some dependent random variables which are used to describe the distribution of magnetic spins present in a ferromagnet crystal. A ferromagnet crystal consists of a large number of sites. At site 1 there is some amount of magnetic spins present. The magnetic spin present at any site interacts with the magnetic spins at its neighboring sites and hence gives rise to some dependency among the magnetic spin present.