Rough limit results for level-crossing probabilities

1994 ◽  
Vol 31 (2) ◽  
pp. 373-382 ◽  
Author(s):  
Harri Nyrhinen
Keyword(s):  
Stochastic Process ◽  
Large Deviations ◽  
Real Number ◽  
Rate Of Convergence ◽  
Generating Functions ◽  
Positive Real Number ◽  
Level Crossing ◽  
Positive Real ◽  
Large Deviations Theory ◽  
Crossing Probabilities

Let Y1, Y2, · ·· be a stochastic process and M a positive real number. Define TM = inf{n | Yn > M} (TM = + ∞ if for n = 1, 2, ···)· We are interested in the probabilities P(TM <∞) and in particular in the case when these tend to zero exponentially fast when M tends to infinity. The techniques of large deviations theory are used to obtain conditions for this and to find out the rate of convergence. The main hypotheses required are given in terms of the generating functions associated with the process (Yn).

Rough limit results for level-crossing probabilities

1994 ◽  
Vol 31 (02) ◽  
pp. 373-382 ◽  
Author(s):  
Harri Nyrhinen
Keyword(s):  
Stochastic Process ◽  
Large Deviations ◽  
Real Number ◽  
Rate Of Convergence ◽  
Generating Functions ◽  
Positive Real Number ◽  
Level Crossing ◽  
Positive Real ◽  
Large Deviations Theory ◽  
Crossing Probabilities

Let Y 1, Y 2, · ·· be a stochastic process and M a positive real number. Define TM = inf{n | Yn &gt; M} (TM = + ∞ if for n = 1, 2, ···)· We are interested in the probabilities P(TM &lt;∞) and in particular in the case when these tend to zero exponentially fast when M tends to infinity. The techniques of large deviations theory are used to obtain conditions for this and to find out the rate of convergence. The main hypotheses required are given in terms of the generating functions associated with the process (Yn ).

Rough descriptions of ruin for a general class of surplus processes

1998 ◽  
Vol 30 (04) ◽  
pp. 1008-1026 ◽  
Author(s):  
Harri Nyrhinen
Keyword(s):  
Stochastic Process ◽  
General Class ◽  
Positive Real Number ◽  
Ruin Probabilities ◽  
Positive Real ◽  
Exponential Estimates ◽  
Large Deviations Theory ◽  
Time Of Ruin ◽  
The Time Of Ruin ◽  
Time Point

Let {Y n | n = 1, 2,…} be a stochastic process and M a positive real number. Define the time of ruin by T = inf{n | Y n &gt; M} (T = +∞ if Y n ≤ M for n = 1, 2,…). Using the techniques of large deviations theory we obtain rough exponential estimates for ruin probabilities for a general class of processes. Special attention is given to the probability that ruin occurs up to a certain time point. We also generalize the concept of the safety loading and consider its importance to ruin probabilities.

Rough descriptions of ruin for a general class of surplus processes

1998 ◽  
Vol 30 (4) ◽  
pp. 1008-1026 ◽  
Author(s):  
Harri Nyrhinen
Keyword(s):  
Stochastic Process ◽  
General Class ◽  
Positive Real Number ◽  
Ruin Probabilities ◽  
Positive Real ◽  
Exponential Estimates ◽  
Large Deviations Theory ◽  
Time Of Ruin ◽  
The Time Of Ruin ◽  
Time Point

Let {Yn | n = 1, 2,…} be a stochastic process and M a positive real number. Define the time of ruin by T = inf{n | Yn > M} (T = +∞ if Yn ≤ M for n = 1, 2,…). Using the techniques of large deviations theory we obtain rough exponential estimates for ruin probabilities for a general class of processes. Special attention is given to the probability that ruin occurs up to a certain time point. We also generalize the concept of the safety loading and consider its importance to ruin probabilities.

Simulating level-crossing probabilities by importance sampling

1992 ◽  
Vol 24 (04) ◽  
pp. 858-874 ◽  
Author(s):  
T. Lehtonen ◽  
H. Nyrhinen
Keyword(s):  
Large Deviations ◽  
Importance Sampling ◽  
Asymptotic Optimality ◽  
Sampling Techniques ◽  
Mathematical Tool ◽  
Level Crossing ◽  
Large Deviations Theory ◽  
Importance Sampling Techniques ◽  
New Criterion ◽  
Crossing Probabilities

Let X 1, X 2, · ·· be independent and identically distributed random variables such that ΕΧ 1 &lt; 0 and P (X 1 ≥ 0) ≥ 0. Fix M ≥ 0 and let T = inf {n: X 1 + X 2 + · ·· + Xn ≥ M} (T = +∞, if for every n = 1,2, ···). In this paper we consider the estimation of the level-crossing probabilities P (T &lt;∞) and , by using Monte Carlo simulation and especially importance sampling techniques. When using importance sampling, precision and efficiency of the estimation depend crucially on the choice of the simulation distribution. For this choice we introduce a new criterion which is of the type of large deviations theory; consequently, the basic large deviations theory is the main mathematical tool of this paper. We allow a wide class of possible simulation distributions and, considering the case that M →∞, we prove asymptotic optimality results for the simulation of the probabilities P (T &lt;∞) and . The paper ends with an example.

Simulating level-crossing probabilities by importance sampling

1992 ◽  
Vol 24 (4) ◽  
pp. 858-874 ◽  
Author(s):  
T. Lehtonen ◽  
H. Nyrhinen
Keyword(s):  
Large Deviations ◽  
Importance Sampling ◽  
Asymptotic Optimality ◽  
Sampling Techniques ◽  
Mathematical Tool ◽  
Level Crossing ◽  
Large Deviations Theory ◽  
Importance Sampling Techniques ◽  
New Criterion ◽  
Crossing Probabilities

Let X1, X2, · ·· be independent and identically distributed random variables such that ΕΧ1 < 0 and P(X1 ≥ 0) ≥ 0. Fix M ≥ 0 and let T = inf {n: X1 + X2 + · ·· + Xn ≥ M} (T = +∞, if for every n = 1,2, ···). In this paper we consider the estimation of the level-crossing probabilities P(T <∞) and , by using Monte Carlo simulation and especially importance sampling techniques. When using importance sampling, precision and efficiency of the estimation depend crucially on the choice of the simulation distribution. For this choice we introduce a new criterion which is of the type of large deviations theory; consequently, the basic large deviations theory is the main mathematical tool of this paper. We allow a wide class of possible simulation distributions and, considering the case that M →∞, we prove asymptotic optimality results for the simulation of the probabilities P(T <∞) and . The paper ends with an example.

scholarly journals Finding of principle square root of a real number by using interpolation method

2018 ◽  
Vol 7 (1) ◽  
pp. 77-83
Author(s):  
Rajendra Prasad Regmi
Keyword(s):  
Real Number ◽  
Positive Real Number ◽  
Interpolation Method ◽  
Square Root ◽  
Real Numbers ◽  
Positive Real ◽  
Unknown Quantity ◽  
Square Roots

There are various methods of finding the square roots of positive real number. This paper deals with finding the principle square root of positive real numbers by using Lagrange’s and Newton’s interpolation method. The interpolation method is the process of finding the values of unknown quantity (y) between two known quantities.

Some remarks on the solvability of non-local elliptic problems with the Hardy potential

2014 ◽  
Vol 16 (04) ◽  
pp. 1350046 ◽  
Author(s):  
B. Barrios ◽  
M. Medina ◽  
I. Peral
Keyword(s):  
Real Number ◽  
Sobolev Inequality ◽  
Positive Real Number ◽  
Fractional Power ◽  
Existence Of Solution ◽  
Sharp Constant ◽  
Hardy Potential ◽  
Positive Real ◽  
Existence And Multiplicity ◽  
Non Local

The aim of this paper is to study the solvability of the following problem, [Formula: see text] where (-Δ)s, with s ∈ (0, 1), is a fractional power of the positive operator -Δ, Ω ⊂ ℝN, N > 2s, is a Lipschitz bounded domain such that 0 ∈ Ω, μ is a positive real number, λ < ΛN,s, the sharp constant of the Hardy–Sobolev inequality, 0 < q < 1 and [Formula: see text], with αλ a parameter depending on λ and satisfying [Formula: see text]. We will discuss the existence and multiplicity of solutions depending on the value of p, proving in particular that p(λ, s) is the threshold for the existence of solution to problem (Pμ).

Ultradiversities and their spherical completeness

2020 ◽  
Vol 26 (2) ◽  
pp. 231-240
Author(s):  
Gholamreza H. Mehrabani ◽  
Kourosh Nourouzi
Keyword(s):  
Real Number ◽  
Finite Subset ◽  
Positive Real Number ◽  
Metric Spaces ◽  
Nonexpansive Mappings ◽  
Ultrametric Spaces ◽  
Positive Real

AbstractDiversities are a generalization of metric spaces which associate a positive real number to every finite subset of the space. In this paper, we introduce ultradiversities which are themselves simultaneously diversities and a sort of generalization of ultrametric spaces. We also give the notion of spherical completeness for ultradiversities based on the balls defined in such spaces. In particular, with the help of nonexpansive mappings defined between ultradiversities, we show that an ultradiversity is spherically complete if and only if it is injective.

Ballots, queues and random graphs

1989 ◽  
Vol 26 (1) ◽  
pp. 103-112 ◽  
Author(s):  
Lajos Takács
Keyword(s):  
Real Number ◽  
Random Graph ◽  
Random Graphs ◽  
Limit Distribution ◽  
Positive Real Number ◽  
Positive Real ◽  
Ballot Theorem

This paper demonstrates how a simple ballot theorem leads, through the interjection of a queuing process, to the solution of a problem in the theory of random graphs connected with a study of polymers in chemistry. Let Γn(p) denote a random graph with n vertices in which any two vertices, independently of the others, are connected by an edge with probability p where 0 < p < 1. Denote by ρ n(s) the number of vertices in the union of all those components of Γn(p) which contain at least one vertex of a given set of s vertices. This paper is concerned with the determination of the distribution of ρ n(s) and the limit distribution of ρ n(s) as n → ∞and ρ → 0 in such a way that np → a where a is a positive real number.

AN EXACT FORMULA FOR THE HARMONIC CONTINUED FRACTION

2020 ◽  
pp. 1-11
Author(s):  
MARTIN BUNDER ◽  
PETER NICKOLAS ◽  
JOSEPH TONIEN
Keyword(s):  
Real Number ◽  
Continued Fraction ◽  
Positive Real Number ◽  
Exact Formula ◽  
Positive Real ◽  
Image Position

For a positive real number $t$ , define the harmonic continued fraction $$\begin{eqnarray}\text{HCF}(t)=\biggl[\frac{t}{1},\frac{t}{2},\frac{t}{3},\ldots \biggr].\end{eqnarray}$$ We prove that $$\begin{eqnarray}\text{HCF}(t)=\frac{1}{1-2t(\frac{1}{t+2}-\frac{1}{t+4}+\frac{1}{t+6}-\cdots \,)}.\end{eqnarray}$$

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