123 research outputs found
Exponential bounds for the tail probability of the supremum of an inhomogeneous random walk
Let 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
can be bounded above by
with some positive constants and
. 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/
Randomly stopped maximum and maximum of sums with consistently varying distributions
Let be a sequence of independent random variables,
and be a counting random variable independent of this sequence. In
addition, let and for .
We consider conditions for random variables and
under which the distribution functions of the random maximum
and of the random
maximum of sums belong to the
class of consistently varying distributions. In our consideration the random
variables 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/
Random convolution of inhomogeneous distributions with -exponential tail
Let be a sequence of independent random variables
(not necessarily identically distributed), and be a counting random
variable independent of this sequence. We obtain sufficient conditions on
and under which the distribution function of
the random sum belongs to the class of
-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/
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
A Lundberg-type inequality for an inhomogeneous renewal risk model
We obtain a Lundberg-type inequality in the case of an inhomogeneous renewal
risk model. We consider the model with independent, but not necessarily
identically distributed, claim sizes and the interoccurrence times. In order to
prove the main theorem, we first formulate and prove an auxiliary lemma on
large values of a sum of random variables asymptotically drifted in the
negative direction.Comment: Published at http://dx.doi.org/10.15559/15-VMSTA30 in the Modern
Stochastics: Theory and Applications (https://www.i-journals.org/vtxpp/VMSTA)
by VTeX (http://www.vtex.lt/
On the Discounted Penalty Function for Claims Having Mixed Exponential
It is considered the classical risk model with mixed exponential claim sizes. Using known results it is obtained the explicit expression of the GerberShiu discounted penalty function ψ(x,δ) = E e −δT 1(T < ∞) , by some infinite series. Here δ > 0 is the force of interest, x – the initial reserve and T – ruin time. The dependance of the discounted penalty function on the main parameters x, θ, λ, δ, α, σ, ν is presented in diagrams, where λ > 0 is the parameter of Poisson process, θ > 0 is the safety loading coefficient, 0 ≤ α ≤ 1 and σ, ν > 0 are the parameters of the mixed exponential distributio
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
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
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 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
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
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