THE UNIVARIATE GAMMA (CHI-SQUARED) SUPERSTATISTICS HAS BEEN USED IN SEVERAL APPLICATIONS BY ASSUMING INDEPENDENCE BETWEEN SYSTEMS. HOWEVER, IN SOME CASES IT SEEMS MORE REASONABLE TO CONSIDER A DEPENDENCE STRUCTURE. THIS FACT MOTIVATES THE INTRODUCTION OF A FAMILY OF BIVARIATE SUPERSTATISTICS BASED ON AN EXTENSION OF THE GAMMA DISTRIBUTION, DEFINED BY GENERALIZED HYPERGEOMETRIC FUNCTIONS. THE PARTICULAR CASES INCLUDE BOLTZMANN AND OTHER STATISTICAL WEIGHTING FACTORS IN THE LITERATURE. NUMERICAL ILLUSTRATIONS SHOW THE BEHAVIOUR OF THE PROPOSED SUPERSTATISTICS.

FROM A BIVARIATE SUPERSTATISTICS, WE SHOW THAT IT IS POSSIBLE TO OBTAIN A GENERALIZED KAPPA DISTRIBUTION, CURRENTLY KNOWN FOR THEIR GREAT PERFORMANCE DESCRIBING MANY ANISOTROPIC HIGH-ENERGY TAIL PLASMAS. SOME PARTICULAR CASES OBTAINED THROUGH THIS PROCEDURE ARE SHOWN IN THIS PAPER, AND, ON THE OTHER HAND, THE BIMODALITY EFFECT OF MARGINAL ON THE STATIONARY SUPERSTATISTICAL DISTRIBUTION IS EXPLORED.

WE STUDY A FERROMAGNETIC ISING MODEL WITH A STAGGERED CELL-BOARD MAGNETIC FIELD PREVIOUSLY PROPOSED FOR IMAGE PROCESSING [MARUANI ET AL., MARKOV PROCESSES RELAT. FIELDS 1, 419?442 (1995)]. WE COMPLEMENT PREVIOUS RESULTS ON THE EXISTENCE OF PHASE TRANSITIONS AT LOW TEMPERATURE [GONZÁLEZ-NAVARRETE ET AL., J. STAT. PHYS. 162, 139?161 (2016)] BY DETERMINING BOUNDS TO THE REGION OF UNIQUENESS OF GIBBS MEASURES. WE ESTABLISH SUFFICIENT RIGOROUS UNIQUENESS CONDITIONS DERIVED FROM THREE DIFFERENT CRITERIA: (1) DOBRUSHIN CRITERION [R. DOBRUSHIN, THEORY PROBAB. APPL. 13, 197?224 (1968)], (2) DISAGREEMENT PERCOLATION [J. VAN DEN BERG AND C. MAES, ANN. PROBAB. 22, 749?763 (1994)], AND (3) DOBRUSHIN?SHLOSMAN CRITERIA [R. DOBRUSHIN AND S. SHLOSMAN, IN STATISTICAL PHYSICS AND DYNAMICAL SYSTEMS: RIGOROUS RESULTS, EDITED BY J. FRITZ, A. JAFFE, AND D. SZASZ (BIRKHAUSER, BASEL, 1985)]. THESE CONDITIONS ARE SUBSEQUENTLY SOLVED NUMERICALLY AND THE RESULTING UNIQUENESS REGIONS ARE COMPARED.

WE INTRODUCE A MULTIDIMENSIONAL WALK WITH MEMORY AND RANDOM TENDENCY. THE ASYMPTOTIC BEHAVIOUR IS CHARACTERIZED, PROVING A LAW OF LARGE NUMBERS AND SHOWING A PHASE TRANSITION FROM DIFFUSIVE TO SUPERDIFFUSIVE REGIMES. IN FIRST CASE, WE OBTAIN A FUNCTIONAL LIMIT THEOREM TO GAUSSIAN VECTORS. IN SUPERDIFFUSIVE REGIME, WE OBTAIN STRONG CONVERGENCE TO A NON-GAUSSIAN RANDOM VECTOR AND CHARACTERIZE ITS MOMENTS.

ABSTRACT WE PROPOSE AN APPROACH TO CONSTRUCT BERNOULLI TRIALS {XI, I ? 1} COMBINING DEPENDENCE AND INDEPENDENCE PERIODS, AND WE CALL IT THE BERNOULLI SEQUENCE WITH RANDOM DEPENDENCE (BSRD). THE STRUCTURE OF DEPENDENCE, IN THE PAST SI = X1 + ? + XI, DEFINES A CLASS OF NON-MARKOVIAN RANDOM WALKS OF RECENT INTEREST IN THE LITERATURE. IN THIS PAPER, THE DEPENDENCE IS ACTIVATED BY AN AUXILIARY COLLECTION OF BERNOULLI TRIALS {YI, I ? 1}, CALLED MEMORY SWITCH SEQUENCE. WE INTRODUCE THE CONCEPT OF MEMORY LAPSE PROPERTY, WHICH IS CHARACTERIZED BY INTERVALS OF CONSECUTIVE INDEPENDENT STEPS IN BSRD. THE MAIN RESULTS INCLUDE CLASSICAL LIMIT THEOREMS FOR A CLASS OF LINEAR BSRD. IN PARTICULAR, WE OBTAIN A CENTRAL LIMIT THEOREM FOR A CLASS OF BSRD WHICH GENERALIZES SOME PREVIOUS RESULTS IN THE LITERATURE. ALONG THE PAPER, SEVERAL EXAMPLES OF POTENTIAL APPLICATIONS ARE PROVIDED.

EN ESTE TRABAJO PRESENTAMOS UNA PROPUESTA DIDÁCTICA DE ANÁLISIS EXPLORATORIO DE DATOS ASOCIADOS A LA EVOLUCIÓN DE LA PANDEMIA EN LAS COMUNAS DE CHILE. LAS ACTIVIDADES SON FORMULADAS UTILIZANDO EL SOFTWARE GEOGEBRA, EN SU VERSIÓN CLÁSICA 6.0. SE SELECCIONARON 15 COMUNAS DE 3 REGIONES PARA INCLUIR UN ANÁLISIS UNIVARIADO DE LOS CASOS ACTIVOS COMUNALES, UN ANÁLISIS DE REGRESIÓN PARA IDENTIFICAR COMUNAS CUYAS EVOLUCIONES SEAN SIMILARES Y ANÁLISIS MULTIVARIADO PARA COMPARAR COMUNAS EN GRUPOS. FINALMENTE SE EJEMPLIFICA UN MÉTODO DE AGRUPAMIENTO PARA CLASIFICAR COMUNAS EN FUNCIÓN DE SUS CASOS ACTIVOS POR CADA 100 MIL HABITANTES.

WE INTRODUCE A ONE-DIMENSIONAL RANDOM WALK, WHICH AT EACH STEP PERFORMS A REINFORCED DYNAMICS WITH PROBABILITY ? AND WITH PROBABILITY 1??, THE RANDOM WALK PERFORMS A STEP INDEPENDENT OF THE PAST. WE ANALYSE ITS ASYMPTOTIC BEHAVIOUR, SHOWING A LAW OF LARGE NUMBERS AND CHARACTERIZING THE DIFFUSIVE AND SUPERDIFFUSIVE REGIONS. WE PROVE CENTRAL LIMIT THEOREMS AND LAW OF ITERATED LOGARITHM BASED ON THE MARTINGALE APPROACH.

WE PROPOSE A MODEL FOR DIFFUSION OF TWO OPPOSITE OPINIONS. HERE, THE DECISION TO BE TAKEN BY EACH INDIVIDUAL IS A RANDOM VARIABLE, WHICH DEPENDS ON THE TENDENCY OF THE POPULATION AS WELL AS ON ITS OWN TREND CHARACTERISTIC. THE INFLUENCE OF THE POPULATION TREND CAN BE POSITIVE, NEGATIVE, OR NONEXISTENT IN A RANDOM FORM. WE PROVE A PHASE TRANSITION IN THE BEHAVIOR OF THE PROPORTION OF EACH OPINION. SPECIFICALLY, THE MEAN SQUARE PROPORTIONS ARE LINEAR FUNCTIONS OF TIME IN THE DIFFUSIVE CASE BUT ARE GIVEN BY A POWER LAW IN THE SUPERDIFFUSIVE REGIME.