Some Theorems for Inverse Uncertainty Distribution of Uncertain Processes

In real life, indeterminacy and determinacy are symmetric, while indeterminacy is absolute. We are devoted to studying indeterminacy through uncertainty theory. Within the framework of uncertainty theory, uncertain processes are used to model the evolution of uncertain phenomena. The uncertainty distribution and inverse uncertainty distribution of uncertain processes are important tools to describe uncertain processes. An independent increment process is a special uncertain process with independent increments. An important conjecture about inverse uncertainty distribution of an independent increment process has not been solved yet. In this paper, the conjecture is proven, and therefore, a theorem is obtained. Based on this theorem, some other theorems for inverse uncertainty distribution of the monotone function of independent increment processes are investigated. Meanwhile, some examples are given to illustrate the results.

Discounted payments theorems for large deviations

Let Z(t) = Σ j=1N(t) Xj, t ≥ 0, be a stochastic process, where Xj are independent identically distributed random variables, and N(t) is non-negative integer-valued process with independent increments. Throughout, we assume that N(t) and Xj are independent. The paper considers normal approximation to the distribution of properly centered and normed random variable Zδ =∫0∞e- δt dZ(t), δ > 0, taking into consideration large deviations both in the Cramér zone and the power Linnik zones. Also, we obtain a nonuniform estimate in the Berry–Essen inequality.

Large deviations for random evolutions with independent increments in the scheme of Lévy approximation with split and double merging

Asymptotic analysis of the problem of large deviations for random evolutions with independent increments in the circuit of Lévy approximation with split and double merging is carried out. Large deviations for random evolutions in the circuit of Lévy approximation with split and double merging are determined by the exponential generator for the jumping process with independent increments.

Remarks on the absolute maximum of a Lévy process

Asymptotic behaviour of the distribution of the absolute maximum of a process with independent increments is studied depending on the properties of the Lévy measure of the process. Some applications to the risk process are also considered.

Remarks on the absolute maximum of a Lévy process

Asymptotic behaviour of the distribution of the absolute maximum of a process with independent increments is studied depending on the properties of the Lévy measure of the process. Some applications to the risk process are also considered.