Proofs Of Limit Theorems Research Articles

On Renewal Matrices Connected with Branching Processes with Tails of Distributions of Different Orders

We study irreducible renewal matrices generated by matrices whose rows are proportional to various distribution functions. Such matrices arise in studies of multi-dimensional critical Bellman–Harris branching processes. Proofs of limit theorems for such branching processes are based on asymptotic properties of a chosen family of renewal matrices. In the theory of branching processes, unsolved problems are known that correspond to the case in which the tails of some of the above mentioned distribution functions are integrable, while the other distributions lack this property.We assume that the heaviest tails are regularly varying at the infinity with parameter −β ∈ [−1, 0) and asymptotically proportional, while the other tails are infinitesimal with respect to them. Under a series of additional conditions, we describe asymptotic properties of the first and second order increments for the renewal matrices.

A Differential Analog of the Main Lemma of the Theory of Markov Branching Processes and Its Applications

We obtain a differential analog of the main lemma of the theory of Markov branching processes μ(t), t ≥ 0, with continuous time. We show that the results obtained can be used in the proof of limit theorems of the theory of branching processes by the known Stein-Tikhomirov method. Moreover, in contrast to the classical condition of nondegeneracy of the branching process {μ(t) > 0}, we consider the condition of its nondegeneracy in the distant future {μ(∞) > 0} and justify it in terms of generating functions. Under this condition, we study the asymptotic behavior of the trajectory of the process considered.

Elementary new proofs of classical limit theorems for Galton–Watson processes

Classical results describe the asymptotic behaviour of a Galton–Watson branching process conditioned on non-extinction. We give new proofs of limit theorems in critical and subcritical cases. The proofs are based on the representation of conditioned Galton–Watson generation sizes as a sum of independent increments which is derived from the decomposition of the conditioned Galton–Watson family tree along the line of descent of the left-most particle.

Elementary new proofs of classical limit theorems for Galton–Watson processes

Classical results describe the asymptotic behaviour of a Galton–Watson branching process conditioned on non-extinction. We give new proofs of limit theorems in critical and subcritical cases. The proofs are based on the representation of conditioned Galton–Watson generation sizes as a sum of independent increments which is derived from the decomposition of the conditioned Galton–Watson family tree along the line of descent of the left-most particle.

Elementary new proofs of classical limit theorems for Galton-Watson processes

Classical results describe the asymptotic behaviour of a Galton–Watson branching process conditioned on non-extinction. We give new proofs of limit theorems in critical and subcritical cases. The proofs are based on the representation of conditioned Galton–Watson generation sizes as a sum of independent increments which is derived from the decomposition of the conditioned Galton–Watson family tree along the line of descent of the left-most particle.

Oscillations of the random argument of a wiener process in the Skorokhod representation

A sequence of independent random variables with zero mean generates, by applying to it the A. V. Skorokhod representation, a sequence of random arguments of a Wiener process. At the proof of limit theorems for sums of independent random variables with the use of this representation, there arises the question of the stability of the sequence of the random arguments. In the paper the law of the iterated logarithms yields exact bounds for the oscillation of this sequence.

Proof of limit theorems for P�lya distributions using the generalized appell polynomials

Proof of limit theorems for P�lya distributions using the generalized appell polynomials

Calculus and Analytic Geometry

(NOTE: Every chapter ends with Questions to Guide Your Review, Practice Exercises, and Additional Exercises.) P. Preliminaries. Real Numbers and the Real Line. Coordinates, Lines, and Increments. Functions. Shifting Graphs. Trigonometric Functions. 1. Limits and Continuity. Rates of Change and Limits. Rules for Finding Limits. Target Values and Formal Definitions of Limits. Extensions of the Limit Concept. Continuity. Tangent Lines. 2. Derivatives. The Derivative of a Function. Differentiation Rules. Rates of Change. Derivatives of Trigonometric Functions. The Chain Rule. Implicit Differentiation and Rational Exponents. Related Rates of Change. 3. Applications of Derivatives. Extreme Values of Functions. The Mean Value Theorem. The First Derivative Test for Local Extreme Values. Graphing with y e and y . Limits as x a a, Asymptotes, and Dominant Terms. Optimization. Linearization and Differentials. Newton's Method. 4. Integration. Indefinite Integrals. Differential Equations, Initial Value Problems, and Mathematical Modeling. Integration by Substitution--Running the Chain Rule Backward. Estimating with Finite Sums. Riemann Sums and Definite Integrals. Properties, Area, and the Mean Value Theorem. Substitution in Definite Integrals. Numerical Integration. 5. Applications of Integrals. Areas Between Curves. Finding Volumes by Slicing. Volumes of Solids of Revolution--Disks and Washers. Cylindrical Shells. Lengths of Plan Curves. Areas of Surfaces of Revolution. Moments and Centers of Mass. Work. Fluid Pressures and Forces. The Basic Pattern and Other Modeling Applications. 6. Transcendental Functions. Inverse Functions and Their Derivatives. Natural Logarithms. The Exponential Function. ax and logax. Growth and Decay. L'Hopital's Rule. Relative Rates of Growth. Inverse Trigonomic Functions. Derivatives of Inverse Trigonometric Functions Integrals. Hyperbolic Functions. First Order Differential Equations. Euler's Numerical Method Slope Fields. 7. Techniques of Integration. Basic Integration Formulas. Integration by Parts. Partial Fractions. Trigonometric Substitutions. Integral Tables and CAS. Improper Integrals. 8. Infinite Series. Limits of Sequences of Numbers. Theorems for Calculating Limits of Sequences. Infinite Series. The Integral Test for Series of Nonnegative Terms. Comparison Tests for Series of Nonnegative Terms. The Ratio and Root Tests for Series of Nonnegative Terms. Alternating Series, Absolute and Conditional Convergence. Power Series. Taylor and Maclaurin Series. Convergence of Taylor Series Error Estimates. Applications of Power Series. 9. Conic Sections, Parametrized Curves, and Polar Coordinates. Conic Sections and Quadratic Equations. Classifying Conic Sections by Eccentricity. Quadratic Equations and Rotations. Parametrizations of Plan Curves. Calculus with Parametrized Curves. Polar Coordinates. Graphing in Polar Coordinates. Polar Equations for Conic Sections. Integration in Polar Coordinates. 10. Vectors and Analytic Geometry in Space. Vectors in the Plane. Cartesian (Rectangular) Coordinates and Vectors in Space. Dot Products. Cross Products. Lines and Planes in Space. Cylinders and Quadric Surfaces. Cylindrical and Spherical Coordinates. 11. Vector-Valued Functions and Motion in Space. Vector-Valued Functions and Space Curves. Modeling Projectile Motion. Arc Length and the Unit Tangent Vector T. Curvature, Torison, and the TNB Frame. Planetary Motion and Satellites. 12. Multivariable Functions and Partial Derivatives. Functions of Several Variables. Limits and Continuity. Partial Derivatives. Differentiability, Linearization, and Differentials. The Chain Rule. Partial Derivatives with Constrained Variables. Directional Derivatives, Gradient Vectors, and Tangent Planes. Extreme Values and Saddle Points. Lagrange Multipliers. Taylor's Formula. 13. Multiple Integrals. Double Integrals. Areas, Moments, and Centers of Mass. Double Integrals in Polar Form. Triple Integrals in Rectangular Coordinates. Masses and Moments in Three Dimensions. Triple Integrals in Cylindrical and Spherical Coordinates. Substitutions in Multiple Integrals. 14. Integration in Vector Fields. Line Integrals. Vector Fields, Work, Circulation, and Flux. Path Independence, Potential Functions, and Conservative Fields. Green's Theorem in the Plane. Surface Area and Surface Integrals. Parametrized Surfaces. Stokes's Theorem. The Divergence Theorem and a Unified Theory. Appendices. Mathematical Induction. Proofs of Limit Theorems in Section 1.2. Complex Numbers. Simpson's One-Third Rule. Cauchy's Mean Value Theorem and the Stronger Form of L'Hopital's Rule. Limits that Arise Frequently. The Distributive Law for Vector Cross Products. Determinants and Cramer's Rule. Euler's Theorem and the Increment Theorem.