Exponential bounds for the tail probability of the supremum of an inhomogeneous random walk

Let $\{\xi_1,\xi_2,\ldots\}$ be a sequence of independent but not necessarily identically distributed random variables. In this paper, the sufficient conditions are found under which the tail probability $\mathbb{P}(\sup_{n\geqslant0}\sum_{i=1}^n\xi_i>x)$ can be bounded above by $\varrho_1\exp\{-\varrho_2x\}$ with some positive constants $\varrho_1$ and $\varrho_2$. A way to calculate these two constants is presented. The application of the derived bound is discussed and a Lundberg-type inequality is obtained for the ultimate ruin probability in the inhomogeneous renewal risk model satisfying the net profit condition on average.Comment: Published at https://doi.org/10.15559/18-VMSTA99 in the Modern Stochastics: Theory and Applications (https://www.i-journals.org/vtxpp/VMSTA) by VTeX (http://www.vtex.lt/

Asymptotic formulas for the left truncated moments of sums with consistently varying distributed increments

In this paper, we consider the sum Snξ = ξ1 + ... + ξn of possibly dependent and nonidentically distributed real-valued random variables ξ1, ... , ξn with consistently varying distributions. By assuming that collection {ξ1, ... , ξn} follows the dependence structure, similar to the asymptotic independence, we obtain the asymptotic relations for E((Snξ)α1(Snξ > x)) and E((Snξ – x)+)α, where α is an arbitrary nonnegative real number. The obtained results have applications in various fields of applied probability, including risk theory and random walks

Random convolution of inhomogeneous distributions with O\mathcal{O}-exponential tail

Let $\{\xi_1,\xi_2,\ldots\}$ be a sequence of independent random variables (not necessarily identically distributed), and $\eta$ be a counting random variable independent of this sequence. We obtain sufficient conditions on $\{\xi_1,\xi_2,\ldots\}$ and $\eta$ under which the distribution function of the random sum $S_{\eta}=\xi_1+\xi_2+\cdots+\xi_{\eta}$ belongs to the class of $\mathcal{O}$-exponential distributions.Comment: Published at http://dx.doi.org/10.15559/16-VMSTA52 in the Modern Stochastics: Theory and Applications (https://www.i-journals.org/vtxpp/VMSTA) by VTeX (http://www.vtex.lt/

Regularly distributed randomly stopped sum, minimum, and maximum

Let {ξ1,ξ2,...} be a sequence of independent real-valued, possibly nonidentically distributed, random variables, and let η be a nonnegative, nondegenerate at 0, and integer-valued random variable, which is independent of {ξ1,ξ2,...}. We consider conditions for {ξ1,ξ2,...} and η under which the distributions of the randomly stopped minimum, maximum, and sum are regularly varying

Randomly stopped maximum and maximum of sums with consistently varying distributions

Let $\{\xi_1,\xi_2,\ldots\}$ be a sequence of independent random variables, and $\eta$ be a counting random variable independent of this sequence. In addition, let $S_0:=0$ and $S_n:=\xi_1+\xi_2+\cdots+\xi_n$ for $n\geqslant1$. We consider conditions for random variables $\{\xi_1,\xi_2,\ldots\}$ and $\eta$ under which the distribution functions of the random maximum $\xi_{(\eta)}:=\max\{0,\xi_1,\xi_2,\ldots,\xi_{\eta}\}$ and of the random maximum of sums $S_{(\eta)}:=\max\{S_0,S_1,S_2,\ldots,S_{\eta}\}$ belong to the class of consistently varying distributions. In our consideration the random variables $\{\xi_1,\xi_2,\ldots\}$ are not necessarily identically distributed.Comment: Published at http://dx.doi.org/10.15559/17-VMSTA74 in the Modern Stochastics: Theory and Applications (https://www.i-journals.org/vtxpp/VMSTA) by VTeX (http://www.vtex.lt/

Asymptotic behavior of the Gerber–Shiu discounted penalty function in the Erlang(2) risk process with subexponential claims

We investigate the asymptotic behavior of the Gerber–Shiu discounted penalty function ɸ(u) = E(e−δT 1{T <∞} | U(0) = u), where T denotes the time to ruin in the Erlang(2) risk process. We obtain an asymptotic expression for the discounted penalty function when claim sizes are subexponentially distributed

Ruin probability for renewal risk models with neutral net profit condition

In ruin theory, the net profit condition intuitively means that the sizes of the incurred random claims are on average less than the premiums gained between the successive interoccurrence times. The breach of the net profit condition causes guaranteed ruin in few but simple cases when both the claims’ interoccurrence time and random claims are degenerate. In this work, we give a simplified argumentation for the unavoidable ruin when the incurred claims are on average equal to the premiums gained between the successive interoccurrence times. We study the discrete-time risk model with N ∈ N periodically occurring independent distributions, the classical risk model, also known as the Cramér–Lundberg risk process, and the more general Sparre Andersen model

Ruin probability for renewal risk models with neutral net profit condition

In ruin theory, the net profit condition intuitively means that the incurred random claims on average do not occur more often than premiums are gained. The breach of the net profit condition causes guaranteed ruin in few but simple cases when both the claims' inter-occurrence time and random claims are degenerate. In this work, we give a simplified argumentation for the unavoidable ruin when the incurred claims on average occur equally as the premiums are gained. We study the discrete-time risk model with $N\in\mathbb{N}$ periodically occurring independent distributions, the classical risk model, also known as the Cram\'er-Lundberg risk process, and the more general E. Sparre Andersen model

A note on product-convolution for generalized subexponential distributions

In this paper, we consider the stability property of the class of generalized subexponential distributions with respect to product-convolution. Assuming that the primary distribution is in the class of generalized subexponential distributions, we find conditions for the second distribution in order that their product-convolution belongs to the class of generalized subexponential distributions as well. The similar problem for the class of generalized subexponential positively decreasing-tailed distributions is considered