The Bivariate Generating Function and Two Problems in Discrete Stochastic Processes

SIAM Review ◽  
1962 ◽  
Vol 4 (2) ◽  
pp. 105-114 ◽  
Author(s):  
Irving Gerst
Keyword(s):  
Stochastic Processes ◽  
Generating Function ◽  
Bivariate Generating Function

Problèmes de premier passage résolubles par la méthode de séparation des variables

1995 ◽  
Vol 32 (02) ◽  
pp. 337-348
Author(s):  
Mario Lefebvre
Keyword(s):  
Brownian Motion ◽  
Stochastic Processes ◽  
Generating Function ◽  
Moment Generating Function ◽  
First Passage Times ◽  
Standard Brownian Motion ◽  
First Passage ◽  
The Moment ◽  
Passage Times

In this paper, bidimensional stochastic processes defined by ax(t) = y(t)dt and dy(t) = m(y)dt + [2v(y)]1/2 dW(t), where W(t) is a standard Brownian motion, are considered. In the first section, results are obtained that allow us to characterize the moment-generating function of first-passage times for processes of this type. In Sections 2 and 5, functions are computed, first by fixing the values of the infinitesimal parameters m(y) and v(y) then by the boundary of the stopping region.

scholarly journals On the Weiner-Masani Algorithm for Finding the Generating Function of Multivariate Stochastic Processes

1988 ◽  
Vol 16 (4) ◽  
pp. 1854-1859 ◽  
Author(s):  
A. G. Miamee
Keyword(s):  
Stochastic Processes ◽  
Generating Function

scholarly journals Special Cases of the Parking Functions Conjecture and Upper-Triangular Matrices

2011 ◽  
Vol DMTCS Proceedings vol. AO,... (Proceedings) ◽  
Author(s):  
Paul Levande
Keyword(s):  
Generating Function ◽  
Hilbert Series ◽  
Parking Functions ◽  
Combinatorial Interpretation ◽  
Forthcoming Article ◽  
Triangular Matrices ◽  
Upper Triangular Matrices ◽  
Diagonal Harmonics ◽  
Special Cases ◽  
Bivariate Generating Function

International audience We examine the $q=1$ and $t=0$ special cases of the parking functions conjecture. The parking functions conjecture states that the Hilbert series for the space of diagonal harmonics is equal to the bivariate generating function of $area$ and $dinv$ over the set of parking functions. Haglund recently proved that the Hilbert series for the space of diagonal harmonics is equal to a bivariate generating function over the set of Tesler matrices–upper-triangular matrices with every hook sum equal to one. We give a combinatorial interpretation of the Haglund generating function at $q=1$ and prove the corresponding case of the parking functions conjecture (first proven by Garsia and Haiman). We also discuss a possible proof of the $t = 0$ case consistent with this combinatorial interpretation. We conclude by briefly discussing possible refinements of the parking functions conjecture arising from this research and point of view. $\textbf{Note added in proof}$: We have since found such a proof of the $t = 0$ case and conjectured more detailed refinements. This research will most likely be presented in full in a forthcoming article. On examine les cas spéciaux $q=1$ et $t=0$ de la conjecture des fonctions de stationnement. Cette conjecture déclare que la série de Hilbert pour l'espace des harmoniques diagonaux est égale à la fonction génératrice bivariée (paramètres $area$ et $dinv$) sur l'ensemble des fonctions de stationnement. Haglund a prouvé récemment que la série de Hilbert pour l'espace des harmoniques diagonaux est égale à une fonction génératrice bivariée sur l'ensemble des matrices de Tesler triangulaires supérieures dont la somme de chaque équerre vaut un. On donne une interprétation combinatoire de la fonction génératrice de Haglund pour $q=1$ et on prouve le cas correspondant de la conjecture dans le cas des fonctions de stationnement (prouvé d'abord par Garsia et Haiman). On discute aussi d'une preuve possible du cas $t=0$, cohérente avec cette interprétation combinatoire. On conclut en discutant brièvement les raffinements possibles de la conjecture des fonctions de stationnement de ce point de vue. $\textbf{Note ajoutée sur épreuve}$: j'ai trouvé depuis cet article une preuve du cas $t=0$ et conjecturé des raffinements possibles. Ces résultats seront probablement présentés dans un article ultérieur.

On linearly regressive processes

1972 ◽  
Vol 9 (01) ◽  
pp. 208-213 ◽  
Author(s):  
John F. Reynolds
Keyword(s):  
Stochastic Processes ◽  
Stationary Distribution ◽  
Generating Function ◽  
Probability Generating Function ◽  
Time Dependent ◽  
Correlation Structure ◽  
Important Class ◽  
Dependent Probability

We consider an important class of integer valued stochastic processes characterised by a certain form of time dependent probability generating function (P.G.F.). Certain results are obtained concerning the correlation structure of such processes, and it is shown that these are considerably simplified in the case of a stationary distribution having been attained. The results are illustrated in the latter case using some commonly occurring processes belonging to the class under consideration.

On a class of two-dimensional nearest-neighbour random walks

1994 ◽  
Vol 31 (A) ◽  
pp. 207-237 ◽  
Author(s):  
J. W. Cohen
Keyword(s):  
Random Walks ◽  
State Space ◽  
Generating Function ◽  
The State ◽  
Nearest Neighbour ◽  
North East ◽  
The North ◽  
One Step ◽  
Bivariate Generating Function ◽  
Positive Recurrent

For positive recurrent nearest-neighbour, semi-homogeneous random walks on the lattice {0, 1, 2, …} X {0, 1, 2, …} the bivariate generating function of the stationary distribution is analysed for the case where one-step transitions to the north, north-east and east at interior points of the state space all have zero probability. It is shown that this generating function can be represented by meromorphic functions. The construction of this representation is exposed for a variety of one-step transition vectors at the boundary points of the state space.

scholarly journals Euler–Catalan’s Number Triangle and Its Application

Symmetry ◽  
2020 ◽  
Vol 12 (4) ◽  
pp. 600 ◽  
Author(s):  
Yuriy Shablya ◽  
Dmitry Kruchinin
Keyword(s):  
Explicit Formula ◽  
Generating Function ◽  
Binary Trees ◽  
Combinatorial Objects ◽  
Left Branch ◽  
Bivariate Generating Function

In this paper, we study such combinatorial objects as labeled binary trees of size n with m ascents on the left branch and labeled Dyck n-paths with m ascents on return steps. For these combinatorial objects, we present the relation of the generated number triangle to Catalan’s and Euler’s triangles. On the basis of properties of Catalan’s and Euler’s triangles, we obtain an explicit formula that counts the total number of such combinatorial objects and a bivariate generating function. Combining the properties of these two number triangles allows us to obtain different combinatorial objects that may have a symmetry, for example, in their form or in their formulas.

Problèmes de premier passage résolubles par la méthode de séparation des variables

1995 ◽  
Vol 32 (2) ◽  
pp. 337-348 ◽  
Author(s):  
Mario Lefebvre
Keyword(s):  
Brownian Motion ◽  
Stochastic Processes ◽  
Generating Function ◽  
Moment Generating Function ◽  
First Passage Times ◽  
Standard Brownian Motion ◽  
First Passage ◽  
The Moment ◽  
Passage Times

In this paper, bidimensional stochastic processes defined by ax(t) = y(t)dt and dy(t) = m(y)dt + [2v(y)]1/2dW(t), where W(t) is a standard Brownian motion, are considered. In the first section, results are obtained that allow us to characterize the moment-generating function of first-passage times for processes of this type. In Sections 2 and 5, functions are computed, first by fixing the values of the infinitesimal parameters m(y) and v(y) then by the boundary of the stopping region.

scholarly journals Records in geometrically distributed words: Sum of positions

2008 ◽  
Vol 2 (2) ◽  
pp. 234-240 ◽  
Author(s):  
Helmut Prodinger
Keyword(s):  
Generating Function ◽  
Random Permutations ◽  
Bivariate Generating Function

Kortchemski introduced a new parameter for random permutations: the sum of the positions of the records. We investigate this parameter in the context of random words, where the letters are obtained by geometric probabilities. We find a relation for a bivariate generating function, from which we can obtain the expectation, exactly and asymptotically. In principle, one could get all moments from it, but the computations would be huge.

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