فهرست مطالب:

1.1.1 Convergence of Distributions and Moments of Some
Functionals of CPRs 1
1.1.2 CRPs with Stationary Increments 5
1.1.3 Strong Law of Large Numbers for a Simple Renewal
Process η(t) 7
1.1.4 Almost Sure Convergence of Some Functionals of
CRPs 7
1.2 First Moments of the Processes Z(t) and Y (t). Strong Laws
of Large Numbers 10
1.2.1 Asymptotics for First- and Second-Order Moments
of Z(t) and Y (t) 10
1.2.2 Strong Laws of Large Numbers 13
1.3 Central Limit Theorem and the Law of the Iterated Logarithm 13
1.3.1 Anscombe’s Theorem 13
1.3.2 Central Limit Theorem 14
1.3.3 Law of the Iterated Logarithm 16
1.4 Convergence to a Stable Law. Analog of the Law of the
Iterated Logarithm 17
1.4.1 Convergence to a Stable Law 17
1.4.2 Analog of the Law of the Iterated Logarithm 19
1.5 Invariance Principle 21
1.5.1 Introduction 21
1.5.2 Analog of Anscombe’s Theorem in the Case of
Convergence to a Continuous Process 22
1.5.3 Invariance Principle for Compound Renewal Processes 24
1.6 Convergence of Normalized Compound Renewal Processes
to Stable Processes in the Case when ξ has Infinite Variance 28
1.6.1 S-Convergence to Stable Processes 28
1.6.2 Absence of S-Convergence without Condition (1.6.5) 32
1.6.3 D-Convergence to Stable Processes 33
1.7 Limit Theorems for the First Passage Time of an Arbitrary
Boundary by a Compound Renewal Process 38
1.7.1 Introduction 38
1.7.2 Case of Finite Variance 39
1.7.3 Case of Infinite Variance 44
1.8 Main Limit Laws for Markov Additive Processes (for Sums
of Random Variables Defined on States of a Markov Chain) 49
1.8.1 Ergodic Theorems for Harris Markov Chains 49
1.8.2 Markov Additive Process 51
1.8.3 Main Limit Laws for Markov Additive Processes 53
2 Integro-Local Limit Theorems in the Normal Deviation Zone 57
2.1 Integro-Local Limit Theorems in the Case of Independent or
Linearly Dependent τ and ζ 57
2.1.1 Integro-Local Theorem for Random Walks 58
2.1.2 Integro-Local Theorems for Homogeneous CRPs in
the Case of Independent or Linearly Dependent τ and ζ 60
2.2 Refinement of Stone’s Integro-Local Theorem for Random
Walks 67
2.3 Integro-Local Theorems for Compound Renewal Processes
in the General Case 74
2.4 Extension of Results to the Inhomogeneous Case 84
2.5 Integro-Local Theorems for Markov Additive Processes 87
3 Large Deviation Principles for Compound Renewal Processes 90
3.1 Introduction 90
3.2 Relationship between Compound Renewal Processes and the
Renewal Measure. Deviation Function for the Renewal Measure 93
3.2.1 Renewal Measures and CRPs 93
3.2.2 Asymptotics for the Renewal Measure and the
Corresponding Deviation Function 95
3.2.3 Preliminary Version of the Local LDP for CRPs 98
3.3 Deviation Functions for the Renewal Measure and for
Compound Renewal Processes 99
3.3.1 Properties of the Function D(t, α) and of the
Deviation Functions for CRPs 99
3.4 Large Deviation Principles for Z(T) 105
3.4.1 The General Case 105
3.4.2 Homogeneous Processes and Processes with Stationary Increments 109
3.4.3 LDP for the Process Z(t), γ(t) and Its Consequences 111
3.5 Fundamental Functions and Their Properties. Further Properties of the Deviation Function D(α), D(α). On the Condition
λ+ < D(0) 112
3.5.1 Fundamental Functions and Their Properties. Further
Properties of the Function D(α) 112
3.5.2 Properties of the Functions μ(α) and μ(α) 124
3.5.3 Properties of the Deviation Function in Its General
Form and of the Corresponding Fundamental Function 130
3.5.4 Condition λ+ < D(α) and Strong Dependence
between τ and ζ in the Large Deviation Zone 135
3.5.5 Examples 137
3.6 On Large Deviation Principles for the Process Y (t) and for
Markov Additive Processes 141
3.6.1 LDP for the Process Y (t) on the Narrowing of the
Set Y (T) ∈ TΔ[α) 142
3.6.2 LDP for Y (t) when τ and ζ Are Independent 144
3.6.3 On Large Deviation Principles for Markov Additive
Processes 145
3.7 Rough Asymptotics for the Laplace Transform of the
Distribution of a Compound Renewal Process 147
4 Large Deviation Principles for Trajectories of Compound Renewal
Processes 154
4.1 Conditions for the Fulfillment of the LDP for the Increments
of a Process and for Finite-Dimensional Distributions 154
4.1.1 LDP for Increments of a CRP 154
4.1.2 Proof of Lemma 4.1.5 161
4.2 First Partial Local Large Deviation Principles for the
Trajectories of a Compound Renewal Process 163
4.2.1 Main Assertion and Its Proof 164
4.2.2 Proofs of Lemmas 4.2.2 and 4.2.4 170
4.3 Second Partial Local Large Deviation Principle 176
4.3.1 Main Results 176
4.3.2 On the Most Probable Trajectories 179
4.3.3 Auxiliary Assertions 181
4.3.4 Proof of Theorem 4.3.1 183
4.4 Complete Local Large Deviation Principle 186
4.5 Integral Large Deviation Principle for Trajectories of a Compound Renewal Process 189
4.5.1 Main Result and Its Proof 189
4.5.2 On the Relaxation of the Conditions of Theorem 4.5.1 196
4.6 Large Deviation Principles for the First Boundary Crossing
Problem 197
4.6.1 Level Lines 198
4.6.2 Inequalities for the Distribution of the Maximum
Value of a CRP 201
4.6.3 Large Deviation Principles for the First Boundary
Crossing Problem 204
4.7 Large Deviation Principles for the Second Boundary Crossing Problem 208
4.7.1 Most Probable (Shortest) Trajectories 208
4.7.2 The Second Boundary Crossing Problem 211
4.8 Moderately Large Deviation Principles for Trajectories of
Compound Renewal Processes 215
4.8.1 Main Results 215
4.8.2 Proofs 218
4.8.3 Rough (Logarithmic) Invariance Principle for CRPs
in the Moderately Large Deviation Zone 224
5 Integro-Local Limit Theorems under the Cramér Moment
Condition 225
5.1 Introduction 225
5.2 Main Results 226
5.2.1 Integro-Local Theorem for the Process Z(t) 226
5.2.2 Integro-Local Theorem for the Process Y (t) 231
5.2.3 Integro-Local Theorem for Finite-Dimensional
Distributions of the Process Z(t) 233
5.2.4 Normal and Moderately Large Deviations 235
5.3 Integro-Local Theorems for the Renewal Measure 237
5.4 Proof of Theorem 5.2.1 and Its Generalization 249
5.4.1 Proof of Theorem 5.2.1 249
5.4.2 Extension of Results to the Case when the Distribution of (τ1, ζ1) Depends on a Parameter 254
5.5 Proofs of Theorems 5.2.10–5.2.14 256
5.5.1 Proof of Theorem 5.2.10 256
5.5.2 Proof of Theorem 5.2.13 on Finite-Dimensional
Distributions 262
5.5.3 Proof of Theorem 5.2.14 263
5.6 Exact Asymptotics of the Laplace Transform of the Distribution of a Compound Renewal Process and Related
Problems 264
5.6.1 Main Result 264
5.6.2 Refinement of the Inequalities of Theorem 4.6.3 for
the Distribution of Z(T) 267
5.6.3 Exact Asymptotics of the Moments of a CRP 269
5.7 Integro-Local Theorems for Markov Additive Processes under
the Cramér Conditions 272
6 Exact Asymptotics in Boundary Crossing Problems for Compound
Renewal Processes 275
6.1 Asymptotics of Distributions of the Maximal Value of
a Compound Renewal Process with Linear Drift. First
Passage Time of a High Level 275
6.1.1 Preliminaries 275
6.1.2 Distribution of the Maximal Value of a CRP with Drift 278
6.1.3 Distribution of the First Passage Time of a High Level 281
6.2 Limit Theorems under the Cramér Condition for the Conditional Distribution of Jumps when the Trajectory Has a Fixed
End 287
6.2.1 Limit Conditional Distribution of Jumps 287
6.2.2 On the Distribution of the Vector αξ 290
6.3 Integro-Local Theorems for the First Passage Time of a High
Level by the Trajectory of a Compound Renewal Process 290
6.4 Integral Theorems for the Distribution of Z(T) = maxt ≤T Z(t) 295
6.4.1 The Case a < 0, α > 0 295
6.4.2 The Case α > a ≥ 0 302
6.4.3 The Case a > 0, α ∼ a as T → ∞ 304
6.4.4 Asymptotics of the Probability that the Trajectory of
a CRP Does Not Cross a High Level x for α = Tx < a 306
6.5 Integro-Local Theorems in Boundary Crossing Problems for
Compound Renewal Processes 308
6.5.1 Integro-Local Theorems for the First Boundary
Crossing Problem 308
6.5.2 Integro-Local Theorems for the Ruin Probability
Problem 314
6.6 Integral Theorems in Boundary Crossing Problems 315
6.6.1 Integral Theorems in the First Boundary Crossing
Problem 315
6.6.2 On the Second Boundary Crossing Problem 317
6.7 Applications to the Ruin Probability Problem for Insurance
Companies 319
7 Extension of the Invariance Principle to the Zones of Moderately
Large and Small Deviations 325
7.1 Strong Approximation of a CRP by a Wiener Process 325
7.2 Extension of the Invariance Principle to the Zone of Moderately Large Deviations 329
7.3 First Boundary Crossing Problem for Moderately Large
Deviations 331
7.4 Extension of the Invariance Principle for Lipschitz Functionals
to the Zone of Moderately Large Deviations 335
7.5 Extension of the Invariance Principle to the Zone of Moderately Small Deviations 336
7.5.1 Extension of the Invariance Principle for the FirstType Sets 336
7.5.2 Extension of the Invariance Principle to the Zone of
Small Deviations for Second-Type Sets 337
7.5.3 Second Boundary Crossing Problem in the Small
Deviation Zone 339
Appendix A On Boundary Crossing Problems for Compound Renewal
Processes when the Cramér Condition Is Not Fulfilled 341
A.1 Distribution of the Maximal Value on the Whole Half-Axis
of a Compound Renewal Process with Drift 341
A.1.1 Distribution of the Maximal Value of a Compound
Renewal Process when the Cramér Condition Is Not
Met 341
A.1.2 Second-Order Approximation for the Distribution of
the Maximal Value of the CRP Z (q)(t) 343
A.1.3 Transient Phenomena for CRPs. First- and SecondOrder Asymptotics 344
A.2 Asymptotics of the Distributions of Z0(T) = Z(T) − aT
and Z 0(T) = maxt ≤T Z0(t) when the Distributions of Jumps
Are Regularly Varying 346
A.3 First Boundary Crossing Problem in the Case of Regular
Variation of Jump Distributions 349
Basic Notation 352
References 357
Index 363

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