A q -Random Walk Approximated by a q -Brownian Motion Malvina - PowerPoint PPT Presentation

Introduction Main Results Selected References A q -Random Walk Approximated by a q -Brownian Motion Malvina Vamvakari Department of Informatics and Telematics Harokopio University, Athens GASCom 2016, La Marana, Corsica Malvina Vamvakari A q -Random Walk Approximated by a q -Brownian Motion

Introduction Main Results Selected References Table of contents 1 Introduction q -Series Preliminaries, 0 < q < 1 The q -Binomial and the q -Poisson Distributions 2 Main Results A q -Poisson Process Distributions of the Interarrival and Waiting Times Fitting Interarrival Times of Pageviews on Harokopio University’s Web to a q -Exponential Distribution A q -Random Walk 3 Selected References Malvina Vamvakari A q -Random Walk Approximated by a q -Brownian Motion

Introduction Main Results Selected References Basic Definitions The q -shifted factorials are n � ( 1 − aq k − 1 ) , n = 1 , 2 , . . . , or ∞ ( a ; q ) 0 := 1 , ( a ; q ) n := k = 1 Malvina Vamvakari A q -Random Walk Approximated by a q -Brownian Motion

Introduction Main Results Selected References Basic Definitions The q -shifted factorials are n � ( 1 − aq k − 1 ) , n = 1 , 2 , . . . , or ∞ ( a ; q ) 0 := 1 , ( a ; q ) n := k = 1 The multiple q -shifted factorials are defined by k � ( a 1 , a 2 , . . . , a k ; q ) n := ( a j ; q ) n j = 1 Malvina Vamvakari A q -Random Walk Approximated by a q -Brownian Motion

Introduction Main Results Selected References Basic Definitions The q -shifted factorials are n � ( 1 − aq k − 1 ) , n = 1 , 2 , . . . , or ∞ ( a ; q ) 0 := 1 , ( a ; q ) n := k = 1 The multiple q -shifted factorials are defined by k � ( a 1 , a 2 , . . . , a k ; q ) n := ( a j ; q ) n j = 1 The q -binomial coefficient is � n � ( q ; q ) n := k ( q ; q ) k ( q ; q ) n − k q Malvina Vamvakari A q -Random Walk Approximated by a q -Brownian Motion

Introduction Main Results Selected References Basic (or q − ) Hypergeometric Series for 0 < q < 1 r φ s ( a 1 , . . . , a r ; b 1 , . . . , b s ; q , z ) = ∞ ( a 1 , · · · , a r ; q ) n z n [( − 1 ) n q ( n 2 )] s − r + 1 � ( b 1 , · · · , b s ; q ) n ( q ; q ) n n = 0 Malvina Vamvakari A q -Random Walk Approximated by a q -Brownian Motion

Introduction Main Results Selected References q -Exponential functions Two kinds of q -exponential functions are the following ∞ ( 1 − q ) n z n 1 � e q ( z ) := = , | z | < 1 , ( q ; q ) n (( 1 − q ) z ; q ) ∞ n = 0 ∞ ( 1 − q ) n z n q n ( n − 1 ) / 2 = ( − ( 1 − q ) z ; q ) ∞ , � E q ( z ) := ( q ; q ) n n = 0 Malvina Vamvakari A q -Random Walk Approximated by a q -Brownian Motion

Introduction Main Results Selected References q -Exponential functions Two kinds of q -exponential functions are the following ∞ ( 1 − q ) n z n 1 � e q ( z ) := = , | z | < 1 , ( q ; q ) n (( 1 − q ) z ; q ) ∞ n = 0 ∞ ( 1 − q ) n z n q n ( n − 1 ) / 2 = ( − ( 1 − q ) z ; q ) ∞ , � E q ( z ) := ( q ; q ) n n = 0 The functions e q and E q satisfy e q ( z ) E q ( − z ) = 1 Malvina Vamvakari A q -Random Walk Approximated by a q -Brownian Motion

Introduction Main Results Selected References q -Differences, 0 < q < 1 A discrete analogue of the derivatives is the q -difference opera- tor ( D q f )( x ) = f ( x ) − f ( qx ) ( 1 − q ) x Malvina Vamvakari A q -Random Walk Approximated by a q -Brownian Motion

Introduction Main Results Selected References q -Differences, 0 < q < 1 A discrete analogue of the derivatives is the q -difference opera- tor ( D q f )( x ) = f ( x ) − f ( qx ) ( 1 − q ) x It is clear that ( D q x n )( x ) = 1 − q n 1 − q x n − 1 = [ n ] q x n − 1 Malvina Vamvakari A q -Random Walk Approximated by a q -Brownian Motion

Introduction Main Results Selected References q -Differences, 0 < q < 1 A discrete analogue of the derivatives is the q -difference opera- tor ( D q f )( x ) = f ( x ) − f ( qx ) ( 1 − q ) x It is clear that ( D q x n )( x ) = 1 − q n 1 − q x n − 1 = [ n ] q x n − 1 For differentiable functions q → 1 ( D q f )( x ) = f ′ ( x ) lim Malvina Vamvakari A q -Random Walk Approximated by a q -Brownian Motion

Introduction Main Results Selected References For finite a and b the q -integral is � a ∞ [ aq n − aq n + 1 ] f ( aq n ) , � f ( x ) d q x := 0 n = 0 � b � b � a f ( x ) d q x := f ( x ) d q x − f ( x ) d q a 0 0 Malvina Vamvakari A q -Random Walk Approximated by a q -Brownian Motion

Introduction Main Results Selected References For finite a and b the q -integral is � a ∞ [ aq n − aq n + 1 ] f ( aq n ) , � f ( x ) d q x := 0 n = 0 � b � b � a f ( x ) d q x := f ( x ) d q x − f ( x ) d q a 0 0 The q -intergal over [ 0 , ∞ ) uses the division points { q n : −∞ < n < ∞} and is � ∞ ∞ � q n f ( q n ) f ( x ) d q x := ( 1 − q ) 0 n = −∞ Malvina Vamvakari A q -Random Walk Approximated by a q -Brownian Motion

Introduction Main Results Selected References Consider a sequence of q - Bernoulli trials with the odds of suc- cess at the i th trial, θ i = θ q i − 1 , i = 1 , 2 , . . . , 0 < q < 1 , 0 < θ < ∞ or with probability of success at the i th trial, θ q i − 1 p i = 1 + θ q i − 1 , i = 1 , 2 , . . . , 0 < q < 1 , 0 < θ < ∞ . Malvina Vamvakari A q -Random Walk Approximated by a q -Brownian Motion

Introduction Main Results Selected References Consider a sequence of q - Bernoulli trials with the odds of suc- cess at the i th trial, θ i = θ q i − 1 , i = 1 , 2 , . . . , 0 < q < 1 , 0 < θ < ∞ or with probability of success at the i th trial, θ q i − 1 p i = 1 + θ q i − 1 , i = 1 , 2 , . . . , 0 < q < 1 , 0 < θ < ∞ . Then the probability function of the number X n of successes at n such trials is given by q ( x 2 ) θ x � n � f X n ( x ) = P ( X n = x ) = j = 1 ( 1 + θ q j − 1 ) , x = 0 , 1 , . . . , n , � n x q for θ > 0 , 0 < q < 1 (see Charalambides(2010)). Malvina Vamvakari A q -Random Walk Approximated by a q -Brownian Motion

Introduction Main Results Selected References The mean value, say µ q , of the random variable Y = [ X n ] 1 / q is θ µ q = [ n ] q 1 + θ q n − 1 Malvina Vamvakari A q -Random Walk Approximated by a q -Brownian Motion

Introduction Main Results Selected References The mean value, say µ q , of the random variable Y = [ X n ] 1 / q is θ µ q = [ n ] q 1 + θ q n − 1 The variance, say σ 2 q , of the r.v. Y is θ 2 [ n ] q [ n − 1 ] q θ [ n ] q σ 2 = q ( 1 + θ q n − 1 )( 1 + θ q n − 2 ) + q ( 1 + θ q n − 1 ) θ 2 [ n ] 2 q − ( 1 + θ q n − 1 ) 2 Malvina Vamvakari A q -Random Walk Approximated by a q -Brownian Motion

Introduction Main Results Selected References The limit of the probability function of the q -binomial distribu- tion as the number of trials tends to infinity is the probability function of the q -Poisson (Heine) distribution n ( 1 + θ q j − 1 ) − 1 = e q ( − λ ) q ( x 2 ) λ x � n � q ( x 2 ) θ x � lim [ x ] q ! , x n →∞ q j = 1 x = 0 , 1 , 2 , . . . , λ = θ/ ( 1 − q ) , (1) for 0 < q < 1, 0 < λ < ∞ , where e q ( λ ) = � ∞ i = 1 ( 1 − λ ( 1 − q ) q i − 1 ) − 1 , a q -exponential function (see Charalambi- des(2010)). Malvina Vamvakari A q -Random Walk Approximated by a q -Brownian Motion

Introduction Main Results Selected References Let the time interval ( 0 , t ] , t > 0 and a suitable partition by considering the geometrically decreasing with rate q time diffe- rences δ i ( n ) = ( n ) − 1 q q i − 1 t , i = 1 , 2 , . . . , n , n ≥ 1 . (2) Malvina Vamvakari A q -Random Walk Approximated by a q -Brownian Motion

Introduction Main Results Selected References Let the time interval ( 0 , t ] , t > 0 and a suitable partition by considering the geometrically decreasing with rate q time diffe- rences δ i ( n ) = ( n ) − 1 q q i − 1 t , i = 1 , 2 , . . . , n , n ≥ 1 . (2) Let also a sequence of independent q -Bernoulli trials in the n mutual disjoint consecutive time subintervals of length δ i ( n ) , i = 1 , 2 , . . . , n , n ≥ 1 , with the odds of success arrival at the i th trial, θ i ( n ; t ) = θ ( n ) − 1 q q i − 1 t , i = 1 , 2 , . . . , n , 0 < q < 1 , 0 < θ < ∞ . Malvina Vamvakari A q -Random Walk Approximated by a q -Brownian Motion

Introduction Main Results Selected References Then the probability of success (arrival) at the i th trial is θ ( n ) − 1 q q i − 1 t p i ( n ; t ) = q q i − 1 t , i = 1 , 2 , . . . , n , 0 < q < 1 , 0 < θ < ∞ 1 + θ ( n ) − 1 Malvina Vamvakari A q -Random Walk Approximated by a q -Brownian Motion

Introduction Main Results Selected References Then the probability of success (arrival) at the i th trial is θ ( n ) − 1 q q i − 1 t p i ( n ; t ) = q q i − 1 t , i = 1 , 2 , . . . , n , 0 < q < 1 , 0 < θ < ∞ 1 + θ ( n ) − 1 Also, by (1) the probability function of the number of arrivals X q , n during the time interval ( 0 , t ] is given by q ( x 2 ) θ x t x ( n ) − x � n � q P ( X q , n = x ) = q q i − 1 t ) , x = 0 , 1 , . . . , n , (3) � n j = 1 ( 1 + θ ( n ) − 1 x q for 0 < θ < ∞ , 0 < q < 1. Malvina Vamvakari A q -Random Walk Approximated by a q -Brownian Motion