Advances in Combinatorial Methods and Applications to Probability and Statistics

I-Lattice Paths and Combinatorial Methods.- 1 Lattice Paths and Faber Polynomials.- 1.1 Introduction.- 1.2 Faber Polynomials.- 1.3 Counting Paths.- 1.4 A Positivity Result.- 1.5 Examples.- References.- 2 Lattice Path Enumeration and Umbral Calculus.- 2.1 Introduction.- 2.1.1 Notation.- 2.2 Initial Value Problems.- 2.2.1 The role of ex.- 2.2.2 Piecewise affine boundaries.- 2.2.3 Applications: Bounded paths.- 2.3 Systems of Operator Equations.- 2.3.1 Applications: Lattice paths with several step directions.- 2.4 Symmetric Sheffer Sequences.- 2.4.1 Applications: Weighted left turns.- 2.4.2 Paths inside a band.- 2.5 Geometric Sheffer Sequences.- 2.5.1 Applications: Crossings.- References.- 3 The Enumeration of Lattice Paths With Respect to Their Number of Turns.- 3.1 Introduction.- 3.2 Notation.- 3.3 Motivating Examples.- 3.4 Turn Enumeration of (Single) Lattice Paths.- 3.5 Applications.- 3.6 Nonintersecting Lattice Paths and Turns.- References.- 4 Lattice Path Counting Simple Random Walk Statistics, and Randomizations: An Analytic Approach.- 4.1 Introduction.- 4.2 Lattice Paths.- 4.3 Simple Random Walks.- 4.4 Randomized Random Walks.- References.- 5 Combinatorial Identities: A Generalization of Dougall's Identity.- 5.1 Introduction.- 5.2 The Generalized Pfaff-Saalschutz Formula.- 5.3 A Modified Pfaff-Saalschiitz Sum of Type II(4,4,1)N.- 5.4 A Well-Balanced II(5,5,1)N Identity.- 5.5 A Generalization of Dougall's Weil-Balanced II(7 7,1)N Identity.- References.- 6 A Comparison of Two Methods for Random Labelling of Balls by Vectors of Integers.- 6.1 First Way.- 6.2 Second Way.- 6.3 Variance and Standard Deviation.- 6.4 Analysis of the Second Way.- References.- II-Applications to Probability Problems.- 7 On the Ballot Theorems.- 7.1 Introduction.- 7.2 The Classical Ballot Theorem.- 7.3 The Original Proofs of Theorem 7.2.1.- 7.4 Historical Background.- 7.5 The General Ballot Theorem.- 7.6 Some Combinatorial Identities.- 7.7 Another Extension of The Classical Ballot Theorem.- References.- 8 Some Results for Two-Dimensional Random Walk.- 8.1 Introduction.- 8.2 Identities and Distributions.- 8.3 Pairs of LRW Paths.- References.- 9 Random Walks on SL(2, F2) and Jacobi Symbols of Quadratic Residues.- 9.1 Introduction.- 9.2 Preliminaries.- 9.3 A Calculation of the Character ?(?M,m)and Its Relation.- References.- 10 Rank Order Statistics Related to a Generalized Random Walk.- 10.1 Introduction.- 10.2 Some Auxiliary Results.- 10.3 The Technique.- 10.4 Definitions of Rank Order Statistics.- 10.5 Distributions of N?,n+* (a) and R?, n+*(a).- 10.6 Distributions of ??,n+ (a) and Rf?,n+(a).- 10.7 Distributions of N?,n* (a) and R?,n* (a).- References.- 11 On a Subset Sum Algorithm and Its Probabilistic and Other Applications.- 11.1 Introduction.- 11.2 A Derivation of the Algorithm.- 11.3 A Class of Discrete Probability Distributions.- 11.4 A Remark on a Summation Procedure When Constructing Partitions.- References.- 12 I and J Polynomials in a Potpourri of Probability Problems.- 12.1 Introduction.- 12.2 Guide to the Problems of this Paper.- 12.3 Triangular Network with Common Failure Probability q for Each Unit.- 12.4 Duality Levels in a Square with Diagonals That Do Not Intersect: Problem 12.5.- References.- 13 Stirling Numbers and Records.- 13.1 Stirling Numbers.- 13.2 Generalized Stirling Numbers.- 13.3 Stirling Numbers and Records.- 13.4 Generalized Stirling Numbers and Records in the F?-scheme.- 13.5 Record Values from Discrete Distributions and Generalized Stirling Numbers.- References.- III-Applications to Urn Models.- 14 Advances in Urn Models During The Past Two Decades.- 14.1 Introduction.- 14.2 Polya-Eggenberger Urns and Their Generalizations and Modifications.- 14.3 Generalizations of the Classical Occupancy Model.- 14.4 Ehrenfest Urn Model.- 14.5 Polya Urn Model with a Continuum of Colors.- 14.6 Stopping Problems in Urns.- 14.7 Limit Theorems for Urns with Random Drawings.- 14.8 Limit Theorems for Sequential Occupancy.- 14.9 Limit Theorems for Infinite Urn Models.- 14.10 Urn Models with Indistinguishable Balls (Bose-Einstein Statistics).- 14.11 Ewens Sampling Formula and Coalescent Urn Models.- 14.12 Reinforcement-Depletion (Compartmental) Urn Models.- 14.13 Urn Models for Interpretation of Mathematical and Probabilistic Concepts and Engineering and Statistical Applications.- References.- 15 A Unified Derivation of Occupancy and Sequential Occupancy Distributions.- 15.1 Introduction.- 15.2 Occupancy Distributions.- 15.3 Sequential Occupancy Distributions.- References.- 16 Moments Binomial Moments and Combinatorics.- 16.1 Basic Relations.- 16.2 Linear Inequalities in Sk, pr and qr.- 16.3 A Statistical Paradox and an Urn Model with Applications.- 16.4 Quadratic Inequalities.- References.- IV-Applications to Queueing Theory.- 17 Nonintersecting Paths and Applications to Queueing Theory.- 17.1 Introduction.- 17.2 Dissimilar Bernoulli Processes.- 17.3 The r-Node Series Jackson Network.- 17.4 The Dummy Path Lemma for Poisson Processes.- 17.5 A Special Variant of D/M/l Queues.- References.- 18 Transient Busy Period Analysis of Initially Non-Empty M/G/l Queues-Lattice Path Approach.- 18.1 Introduction.- 18.2 Lattice Path Approach.- 18.3 Discretized M/C2/l Model.- 18.3.1 Transition probabilities.- 18.3.2 Counting of lattice paths.- 18.3.3 Busy period probability.- 18.4 Continuous M/C2/l Model.- 18.5 Particular Cases.- References.- 19 Single Server Queueing System with Poisson Input: A Review of Some Recent Developments.- 19.1 Introduction.- 19.2 Exceptional Service for the First Unit in Each Busy Period.- 19.3 M/G/l With Random Setup Time 5.- 19.4 M/G/l System Under N-Policy.- 19.5 M/G/l Under N-Policy and With Setup Time.- 19.6 Queues With Vacation: M/G/l Queueing System With Vacation.- 19.7 M/G/l - Vm System.- 19.8 M/G/l - Vm With Exceptional First Vacation.- 19.9 M/G/l - Vs System.- 19.10 M/G/l System With Vacation and Under N-Policy (With Threshold N).- 19.11 Mx/G/1 System With Batch Arrival.- 19.12 Mx/G/1 Under N-Policy.- 19.13 Mx/G/1-Vm and Mx/G/1 - Vs.- 19.14 Mx/G/1 Vacation Queues Under N-Policy.- 19.15 Concluding Remarks.- References.- 20 Recent Advances in the Analysis of Polling Systems.- 20.1 Introduction.- 20.2 Notations and Preliminaries.- 20.3 Main Results.- 20.4 Some Related Models.- 20.4.1 Customer routing.- 20.4.2 Stopping only at a preferred station.- 20.4.3 Gated or mixed service policy.- 20.4.4 State-dependent setups.- 20.4.5 Periodic monitoring during idle period.- 20.5 Insights.- 20.6 Future Directions.- References.- V-Applications to Waiting Time Problems.- 21 Waiting Times and Number of Appearances of Events in a Sequence of Discrete Random Variables.- 21.1 Introduction.- 21.2 Definitions and Notations.- 21.3 General Results.- 21.4 Waiting Times and Number of Occurrences of Delayed Recurrent Events.- 21.5 Distribution of the Number of Success Runs in a Two-State Markov Chain.- 21.5.1 Non-overlapping success runs.- 21.5.2 Success runs of length at least k.- 21.5.3 Overlapping success runs.- 21.5.4 Number of non-overlapping windows of length at most k containing exactly 2 successes.- 21.6 Conclusions.- References.- 22 On Sooner and Later Problems Between Success and Failure Runs.- 22.1 Introduction.- 22.2 Number of Ocurrences of the Sooner Event Until the Later Waiting Time.- 22.3 Joint Distribution of Numbers of Runs.- References.- 23 Distributions of Numbers of Success-Runs Until the First Consecutive k Successes in Higher Order Markov Dependent Trials.- 23.1 Introduction.- 23.2 Numbers of Success-Runs in Higher Order Markov Chain.- 23.3 Case l < m.- References.- 24 On Multivariate Distributions of Various Orders Obtained by Waiting for the r-th Success Run of Length k in Trials With Multiple Outcomes.- 24.1 Introduction.- 24.2 Independent Trials.- 24.3 Generalized Sequence of Order k.- References.- 25 A Multivariate Negative Binomial Distribution of Order k Arising When Success Runs are Allowed to Overlap.- 25.1 Introduction.- 25.2 Multivariate Negative Binomial Distribution of Order k Type III.- 25.3 Characteristics and Distributional Properties of $$overline {MNB} _{k,III} (r;q_1 ldots ,q_m ),$$.- References.- VI-Applications to Distribution Theory.- 26 The Joint Energy Distributions of the Bose-Einstein and of the Fermi-Dirac Particles.- 26.1 Introduction.- 26.2 Derivation of the Joint Distribution and of the Joint Entropy.- 26.2.1 On the method.- 26.2.2 Joint distribution of the number of particles in energy intervals.- 26.3 Determination of the Limit Distributions.- 26.4 Discussion.- References.- 27 On Modified g-Bessel Functions and Some Statistical Applications.- 27.1 Introduction.- 27.2 Notation.- 27.3 The Distribution of the Difference of Two Euler Random Variables.- 27.4 The Distribution of the Difference of Two Heine Random Variables.- 27.5 Comments on the Distribution of the Difference of Two Generalized Euler Random Variables.- References.- 28 A g-Logarithmic Distribution.- 28.1 Introduction.- 28.2 A q-Logarithmic Distribution.- 28.3 A Group Size Model for the Distribution.- References.- 29 Bernoulli Learning Models: Uppuluri Numbers.- 29.1 Introduction.- 29.2 The General Model.- 29.2.1 Special cases of the general probabilistic model.- 29.3 Waiting Time Learning Models.- 29.3.1 Special cases of waiting time learning models.- References.- VII-Applications to Nonparametric Statistics.- 30 Linear Nonparametric Tests Against Restricted Alternatives: The Simple-Tree Order and The Simple Order.- 30.1 Introduction.- 30.2 Background.- 30.3 Objectives.- 30.4 Exploration and Reformulation.- 30.5 Test for the Simple-Tree Problem.- 30.5.1 Some particular cases.- 30.5.2 Derivation of the MSSMP test.- 30.6 Test for the Simple Order Problem.- 30.6.1 Derivation of the (A)MSSMP test.- 30.6.2 Power comparisons.- 30.7 Extending the Class of SMP Tests.- References.- 31 Nonparametric Estimation of the Ratio of Variance Components.- 31.1 Introduction.- 31.2 Proposed Estimation Procedure.- 31.3 Monte Carlo Comparison.- 31.4 Adjustment for Bias.- References.- 32 Limit Theorems for M-Processes Via Rank Statistics Processes.- 32.1 Introduction.- 32.2 Case ?1 =...= ?n.- 32.3 Change Point Alternatives.- References.- Author Index.