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 (a; q)0 := 1, (a; q)n :=

n

• k=1

(1 − aqk−1), n = 1, 2, . . . , or ∞

Malvina Vamvakari A q-Random Walk Approximated by a q-Brownian Motion

Introduction Main Results Selected References

Basic Definitions

The q-shifted factorials are (a; q)0 := 1, (a; q)n :=

n

• k=1

(1 − aqk−1), n = 1, 2, . . . , or ∞ The multiple q-shifted factorials are defined by (a1, a2, . . . , ak; q)n :=

k

• j=1

(aj; q)n

Malvina Vamvakari A q-Random Walk Approximated by a q-Brownian Motion

Introduction Main Results Selected References

Basic Definitions

The q-shifted factorials are (a; q)0 := 1, (a; q)n :=

n

• k=1

(1 − aqk−1), n = 1, 2, . . . , or ∞ The multiple q-shifted factorials are defined by (a1, a2, . . . , ak; q)n :=

k

• j=1

(aj; q)n The q-binomial coefficient is n k

• q

:= (q; q)n (q; q)k(q; q)n−k

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

rφs(a1, . . . , ar; b1, . . . , bs; q, z) = ∞

• n=0

(a1, · · · , ar; q)n zn (b1, · · · , bs; q)n(q; q)n [(−1)nq(n

2)]s−r+1 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 eq(z) :=

∞

• n=0

(1 − q)nzn (q; q)n = 1 ((1 − q)z; q)∞ , |z| < 1, Eq(z) :=

∞

• n=0

(1 − q)nzn (q; q)n qn(n−1)/2 = (−(1 − q)z; q)∞,

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 eq(z) :=

∞

• n=0

(1 − q)nzn (q; q)n = 1 ((1 − q)z; q)∞ , |z| < 1, Eq(z) :=

∞

• n=0

(1 − q)nzn (q; q)n qn(n−1)/2 = (−(1 − q)z; q)∞, The functions eq and Eq satisfy eq(z)Eq(−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 (Dqf )(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 (Dqf )(x) = f (x) − f (qx) (1 − q)x It is clear that (Dqxn)(x) = 1 − qn 1 − q xn−1 = [n]qxn−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 (Dqf )(x) = f (x) − f (qx) (1 − q)x It is clear that (Dqxn)(x) = 1 − qn 1 − q xn−1 = [n]qxn−1 For differentiable functions lim

q→1(Dqf )(x) = f ′(x)

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 f (x)dqx :=

∞

• n=0

[aqn − aqn+1]f (aqn), b

a

f (x)dqx := b f (x)dqx − a f (x)dq

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 f (x)dqx :=

∞

• n=0

[aqn − aqn+1]f (aqn), b

a

f (x)dqx := b f (x)dqx − a f (x)dq The q-intergal over [0, ∞) uses the division points {qn : −∞ < n < ∞} and is ∞ f (x)dqx := (1 − q)

∞

• n=−∞

qnf (qn)

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 ith trial, θi = θqi−1, i = 1, 2, . . . , 0 < q < 1, 0 < θ < ∞

• r with probability of success at the ith trial,

pi = θqi−1 1 + θqi−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 ith trial, θi = θqi−1, i = 1, 2, . . . , 0 < q < 1, 0 < θ < ∞

• r with probability of success at the ith trial,

pi = θqi−1 1 + θqi−1 , i = 1, 2, . . . , 0 < q < 1, 0 < θ < ∞. Then the probability function of the number Xn of successes at n such trials is given by fXn(x) = P(Xn = x) = n x

• q

q(x

2)θx

n

j=1(1 + θqj−1), x = 0, 1, . . . , n,

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 µq, of the random variable Y = [Xn]1/q is µq = [n]q θ 1 + θqn−1

Malvina Vamvakari A q-Random Walk Approximated by a q-Brownian Motion

Introduction Main Results Selected References

The mean value, say µq, of the random variable Y = [Xn]1/q is µq = [n]q θ 1 + θqn−1 The variance, say σ2

q, of the r.v. Y is

σ2

q

= θ2[n]q[n − 1]q q(1 + θqn−1)(1 + θqn−2) + θ[n]q (1 + θqn−1) − θ2[n]2

q

(1 + θqn−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 lim

n→∞

n x

• q

q(x

2)θx

n

• j=1

(1 + θqj−1)−1 = eq(−λ)q(x

2)λx

[x]q! , x = 0, 1, 2, . . . , λ = θ/(1 − q), (1) for 0 < q < 1, 0 < λ < ∞, where eq(λ) = ∞

i=1(1 −

λ(1 − q)qi−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 δi(n) = (n)−1

q qi−1t, 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 δi(n) = (n)−1

q qi−1t, 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 δi(n), i = 1, 2, . . . , n, n ≥ 1, with the odds of success arrival at the ith trial, θi(n; t) = θ(n)−1

q qi−1t, 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 ith trial is pi(n; t) = θ(n)−1

q qi−1t

1 + θ(n)−1

q qi−1t , 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 ith trial is pi(n; t) = θ(n)−1

q qi−1t

1 + θ(n)−1

q qi−1t , i = 1, 2, . . . , n, 0 < q < 1, 0 < θ < ∞

Also, by (1) the probability function of the number of arrivals Xq,n during the time interval (0, t] is given by P(Xq,n = x) = n x

• q

q(x

2)θxtx(n)−x

q

n

j=1(1 + θ(n)−1 q qi−1t), x = 0, 1, . . . , n, (3)

for 0 < θ < ∞, 0 < q < 1.

Malvina Vamvakari A q-Random Walk Approximated by a q-Brownian Motion

Introduction Main Results Selected References

The limit of the probability function (3), as n → ∞ , is lim

n→∞ P(Xq,n = x)

= eq(−λt)q(x

2)(λt)x

[x]q! , x = 0, 1, 2, . . . , λ = θ, (4) for 0 < q < 1, 0 < λ < ∞.

Malvina Vamvakari A q-Random Walk Approximated by a q-Brownian Motion

Introduction Main Results Selected References

The limit of the probability function (3), as n → ∞ , is lim

n→∞ P(Xq,n = x)

= eq(−λt)q(x

2)(λt)x

[x]q! , x = 0, 1, 2, . . . , λ = θ, (4) for 0 < q < 1, 0 < λ < ∞. The limit distribution (4), as n → ∞, introduces a q-Poisson process, {Xq(t), t > 0}, of arrivals in the time interval (0, t], partitioned suitably by the time differences δi = (1 − q)qi−1t, i = 1, 2, . . . . (5)

Malvina Vamvakari A q-Random Walk Approximated by a q-Brownian Motion

Introduction Main Results Selected References

The limit of the probability function (3), as n → ∞ , is lim

n→∞ P(Xq,n = x)

= eq(−λt)q(x

2)(λt)x

[x]q! , x = 0, 1, 2, . . . , λ = θ, (4) for 0 < q < 1, 0 < λ < ∞. The limit distribution (4), as n → ∞, introduces a q-Poisson process, {Xq(t), t > 0}, of arrivals in the time interval (0, t], partitioned suitably by the time differences δi = (1 − q)qi−1t, i = 1, 2, . . . . (5)

Malvina Vamvakari A q-Random Walk Approximated by a q-Brownian Motion

Introduction Main Results Selected References

Definition of q-Poisson Process, {Xq(t), t > 0}

Malvina Vamvakari A q-Random Walk Approximated by a q-Brownian Motion

Introduction Main Results Selected References

Definition of q-Poisson Process, {Xq(t), t > 0}

1 In each of the consecutive mutually disjoint time interval of

length δk = (1 − q)qk−1t, k = 1, 2, . . . , t > 0, 0 < q < 1, at most one event (arrival) occurs with α1,k(δk) = P

• Xq(qk−1t) − Xq(qkt) = 1
• =

λ(1 − q)qk−1t 1 + λ(1 − q)qk−1t , α0,k(δk) = P

• Xq(qk−1t) − Xq(qkt) = 0
• =

1 1 + λ(1 − q)qk−1t , k = 1, 2, . . . , λ > 0

Malvina Vamvakari A q-Random Walk Approximated by a q-Brownian Motion

Introduction Main Results Selected References

Definition of q-Poisson Process, {Xq(t), t > 0}

1 In each of the consecutive mutually disjoint time interval of

length δk = (1 − q)qk−1t, k = 1, 2, . . . , t > 0, 0 < q < 1, at most one event (arrival) occurs with α1,k(δk) = P

• Xq(qk−1t) − Xq(qkt) = 1
• =

λ(1 − q)qk−1t 1 + λ(1 − q)qk−1t , α0,k(δk) = P

• Xq(qk−1t) − Xq(qkt) = 0
• =

1 1 + λ(1 − q)qk−1t , k = 1, 2, . . . , λ > 0

2 In the consecutive mutually disjoint time intervals of length

δk = (1 − q)qk−1t, k = 1, 2, . . . , ν, t > 0, ν ≥ 2, correspond ν independent events (arrivals).

Malvina Vamvakari A q-Random Walk Approximated by a q-Brownian Motion

Introduction Main Results Selected References

Definition of q-Poisson Process, {Xq(t), t > 0}

1 In each of the consecutive mutually disjoint time interval of

length δk = (1 − q)qk−1t, k = 1, 2, . . . , t > 0, 0 < q < 1, at most one event (arrival) occurs with α1,k(δk) = P

• Xq(qk−1t) − Xq(qkt) = 1
• =

λ(1 − q)qk−1t 1 + λ(1 − q)qk−1t , α0,k(δk) = P

• Xq(qk−1t) − Xq(qkt) = 0
• =

1 1 + λ(1 − q)qk−1t , k = 1, 2, . . . , λ > 0

2 In the consecutive mutually disjoint time intervals of length

δk = (1 − q)qk−1t, k = 1, 2, . . . , ν, t > 0, ν ≥ 2, correspond ν independent events (arrivals).

3 The process starts at epoch 0 with Xq(0) = 0. Malvina Vamvakari A q-Random Walk Approximated by a q-Brownian Motion

Introduction Main Results Selected References

Definition 2.1. A process {Xq(t), t > 0} satisfying the assumptions (1)-(3) is called q-Poisson process with parameters q and λ > 0.

Malvina Vamvakari A q-Random Walk Approximated by a q-Brownian Motion

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Definition 2.1. A process {Xq(t), t > 0} satisfying the assumptions (1)-(3) is called q-Poisson process with parameters q and λ > 0. Theorem 2.1. The q-Poisson process has the q-Poisson ditribution: Pk(t) = P(Xq(t) = k) = eq(−λt)q(k

2)(λt)k

[k]q! , k = 0, 1, 2, . . . , (6) for 0 < q < 1, 0 < λ < ∞.

Malvina Vamvakari A q-Random Walk Approximated by a q-Brownian Motion

Introduction Main Results Selected References

Distributions of the Interarrival and Waiting Times

Malvina Vamvakari A q-Random Walk Approximated by a q-Brownian Motion

Introduction Main Results Selected References

Distributions of the Interarrival and Waiting Times

Let Wν,q be the waiting time up to the νth arrival (event), ν = 1, 2, . . ., of the q-Poisson process. The interarrival times, say Tν,q, ν = 1, 2, . . ., can be defined as follows: Tν,q = Wν,q − Wν−1,q, ν = 1, 2, . . . , W0,q = 0.

Malvina Vamvakari A q-Random Walk Approximated by a q-Brownian Motion

Introduction Main Results Selected References

Distributions of the Interarrival and Waiting Times

Let Wν,q be the waiting time up to the νth arrival (event), ν = 1, 2, . . ., of the q-Poisson process. The interarrival times, say Tν,q, ν = 1, 2, . . ., can be defined as follows: Tν,q = Wν,q − Wν−1,q, ν = 1, 2, . . . , W0,q = 0. Equivalently, the waiting times Wν,q, ν = 1, 2, . . . , can be written as Wν,q = T1,q + T2,q + . . . + Tν,q, ν = 1, 2, . . . .

Malvina Vamvakari A q-Random Walk Approximated by a q-Brownian Motion

Introduction Main Results Selected References

Distributions of the Interarrival and Waiting Times

Let Wν,q be the waiting time up to the νth arrival (event), ν = 1, 2, . . ., of the q-Poisson process. The interarrival times, say Tν,q, ν = 1, 2, . . ., can be defined as follows: Tν,q = Wν,q − Wν−1,q, ν = 1, 2, . . . , W0,q = 0. Equivalently, the waiting times Wν,q, ν = 1, 2, . . . , can be written as Wν,q = T1,q + T2,q + . . . + Tν,q, ν = 1, 2, . . . . For establishing the distributions of the waiting time Wν,q and the interrarival time Tk,q, k = 1, 2, . . . , ν, we first need to give the following definitions of univariate, bivariate and multivariate q-continuous random variables respectively.

Malvina Vamvakari A q-Random Walk Approximated by a q-Brownian Motion

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Definition 2.2. A random variable Xq is called q-continuous if their exists a non-negative function fq(x) ≥ 0, x > 0 with ∞ fq(x)dqx = 1, where for every real number α and β with α < β, P (α < Xq ≤ β) = β

α

fq(x)dqx. The function fq(x), x > 0, is called q-density function of the random variable Xq.

Malvina Vamvakari A q-Random Walk Approximated by a q-Brownian Motion

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For the corresponding q-distribution function Fq(x) = P (Xq ≤ x) with P (α < Xq ≤ β) = Fq(β) − Fq(α) it holds that Fq(x) = x fq(t)dqt, x ≥ 0.

Malvina Vamvakari A q-Random Walk Approximated by a q-Brownian Motion

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For the corresponding q-distribution function Fq(x) = P (Xq ≤ x) with P (α < Xq ≤ β) = Fq(β) − Fq(α) it holds that Fq(x) = x fq(t)dqt, x ≥ 0. Taking the q-derivative of the above relation we have that dFq(x) dqx = fq(x) and by the definition of the q-derivative we obtain fq(x) = Fq(x) − Fq(qx) (1 − q)x = P (qx < Xq ≤ x) (1 − q)x .

Malvina Vamvakari A q-Random Walk Approximated by a q-Brownian Motion

Introduction Main Results Selected References

Definition 2.3. A bivariate random variable (Xq, Yq) is called q-continuous if their exists a non-negative function fq(x, y) ≥ 0, x, y > 0 with ∞ ∞ fq(x, y)dqxdqy = 1, where for every real numbers α, β, γ and δ with α < β and γ < δ, P (α < Xq ≤ β, γ < Yq ≤ δ) = δ

γ

β

α

fq(x, y)dqxdqy. The function fq(x, y), x > 0, y > 0, is called q-density function

• f the bivariate random variable (Xq, Yq) or joint q-density function
• f the random variables Xq and Yq.

Malvina Vamvakari A q-Random Walk Approximated by a q-Brownian Motion

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For the corresponding joint q-distribution function Fq(x, y) = P (Xq ≤ x, Yq ≤ y) with P (α < Xq ≤ β, γ < Yq ≤ δ) = Fq(β, δ)−Fq(α, δ)−Fq(β, γ)+Fq(α, γ) it holds that Fq(x, y) = y x fq(t, u)dqtdqu, x, y ≥ 0.

Malvina Vamvakari A q-Random Walk Approximated by a q-Brownian Motion

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For the corresponding joint q-distribution function Fq(x, y) = P (Xq ≤ x, Yq ≤ y) with P (α < Xq ≤ β, γ < Yq ≤ δ) = Fq(β, δ)−Fq(α, δ)−Fq(β, γ)+Fq(α, γ) it holds that Fq(x, y) = y x fq(t, u)dqtdqu, x, y ≥ 0. Taking the q-derivative of the above relation we have that ∂Fq(x, y) ∂qy∂qx = fq(x, y), x, y ≥ 0 and by the definition of the partial q-derivative we obtain fq(x, y) = Fq(x, y) − Fq(qx, y) − Fq(x, qy) + Fq(qx, qy) (1 − q)x(1 − q)y = P (qx < Xq ≤ x, qy < Yq ≤ y) (1 − q)x(1 − q)y .

Malvina Vamvakari A q-Random Walk Approximated by a q-Brownian Motion

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Definition 2.4. A kth-variate random variable X = (X1,q, X2,q, . . . , Xk,q) is called q-continuous if their exists a non-negative function fq(x1, x2, . . . , xk) ≥ 0, xi > 0, i = 1, 2, . . . k with ∞ · · · ∞ ∞ fq(x1, x2, . . . , xk)dqx1dqx2 · · · dqxk = 1, where for every real numbers αi, βi with αi < βi, i = 1, 2, . . . , k P

• α1 < X1,q ≤ β1, α2 < X2,q ≤ β2, . . . , αk < Xk,q ≤ βk
• =

βk

αk

· · · β2

α2

β1

α1

fq(x1, x2, . . . , xk)dqx1dqx2 · · · dqxk. (7) The function fq(x1, x2, . . . , xk) ≥ 0, xi > 0, i = 1, 2, . . . k, is called q-density function of the kth-variate random variable X = (X1,q, X2,q, . . . , Xk,q) or joint q-density function of the random variables X1,q, X2,q, . . . , Xk,q.

Malvina Vamvakari A q-Random Walk Approximated by a q-Brownian Motion

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For the corresponding joint q-distribution function Fq(x1, x2, . . . , xk) = P

• X1,q ≤ x1, X2,q ≤ x2, . . . , Xk,q ≤ xk
• with xi > 0, i = 1, 2, . . . k, it holds that

Fq(x1, x2, . . . , xk) = xk · · · x2 x1 fq(t1, t2, . . . , tk)dqt1dqt2 · · · dqtk. Taking the q-derivative of the above relation we have that ∂Fq(x1, x2, . . . , xk) ∂qxk · · · ∂qx2∂qx1 = fq(x1, x2, . . . , xk), xi > 0, i = 1, 2, . . . k and by the definition of the partial q-derivative we obtain fq(x1, x2, . . . , xk) = P

• qx1 < X1,q ≤ x1, qx2 < X2,q ≤ x2, . . . , qxk < Xk,q ≤ xk
• (1 − q)x1(1 − q)x2 · · · (1 − q)xk

.

Malvina Vamvakari A q-Random Walk Approximated by a q-Brownian Motion

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Theorem 2.2. The waiting time Wν,q, ν = 1, 2, . . . up to the νth arrival of the q-Poisson process has q-distribution function Fν,q(w) = P(Wν,q ≤ w) = 1 −

ν−1

• k=0

eq(−λw)q(k

2)(λw)k

[k]q! (8) and q-density function given by fν,q(w) = λeq(−λw)q(ν

2)(λw)ν−1

[ν − 1]q! , w > 0. Its moments are given by µr = E

• W r

ν,q

• = λ−rq−(ν−1)r−(r+1

2 )(qν; q)r(1−q)−r, r = 1, 2, . . . .

(9)

Malvina Vamvakari A q-Random Walk Approximated by a q-Brownian Motion

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Definition 2.5. The distribution of the q-continuous random variable Wν,q, ν = 1, 2, . . . with q-density function fν,q(w) is called q-Gamma distribution with parameters q, λ and ν.

Malvina Vamvakari A q-Random Walk Approximated by a q-Brownian Motion

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Definition 2.5. The distribution of the q-continuous random variable Wν,q, ν = 1, 2, . . . with q-density function fν,q(w) is called q-Gamma distribution with parameters q, λ and ν. Note that the limiting distribution of the q-Gamma distribution as q tends to 1 is the classical Gamma distribution (Erlang), i.e. fν,q(w) → exp(−λw)λνwν−1 (ν − 1)!, w > 0, as q → 1.

Malvina Vamvakari A q-Random Walk Approximated by a q-Brownian Motion

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Corollary 2.1. The first interarrival T1,q of the q-Poisson process has q-distribution function FT1,q(t) = F1,q(t) = P(T1,q ≤ t) = 1 − eq(−λt) (10) and q-density function given by fT1,q(t) = f1,q(t) = λeq(−λt) t > 0. Its moments are given by µr = E

• T r

1,q

• = λ−rq−(r+1

2 )(q; q)r(1 − q)−r, r = 1, 2, . . . .

(11)

Malvina Vamvakari A q-Random Walk Approximated by a q-Brownian Motion

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Theorem 2.3. The interarrival times Tk,q, k = 1, 2, . . . , ν, ν ≥ 1,

• f the q-Poisson process are independent, equidistributed

q-continuous random variables of the q-Exponential distribution with q-density function given by fTk,q(tk) = λeq(−λtk), tk > 0, k = 1, 2, . . . , ν, ν ≥ 1.

Malvina Vamvakari A q-Random Walk Approximated by a q-Brownian Motion

Introduction Main Results Selected References

Theorem 2.3. The interarrival times Tk,q, k = 1, 2, . . . , ν, ν ≥ 1,

• f the q-Poisson process are independent, equidistributed

q-continuous random variables of the q-Exponential distribution with q-density function given by fTk,q(tk) = λeq(−λtk), tk > 0, k = 1, 2, . . . , ν, ν ≥ 1. Note that the joint q-density function of the random variables of the interrarival times Tk,q, k = 1, 2, . . . , ν, is given by fT (t1, t2, . . . , tν) = λν

ν

• i=1

eq(−λti). (12)

Malvina Vamvakari A q-Random Walk Approximated by a q-Brownian Motion

Introduction Main Results Selected References

Corollary 2.2. Let the νth-variate random variable W = (W1,q, W2,q, . . . , Wν,q) , where Wk,q, k = 1, 2, . . . , ν the random variables of the waiting times of the q-Poisson process. Then the joint q-density function of the random variables Wk,q, k = 1, 2, . . . , ν, is given by fW(w1, w2, . . . , wν) = λν

ν

• j=1

eq (−λ(wj − wj−1)) , w0 = 0. (13)

Malvina Vamvakari A q-Random Walk Approximated by a q-Brownian Motion

Introduction Main Results Selected References

Conjecture

The conditions leading to the q-Poisson process likely to be fulfilled by several phenomena in Graphs and Networks

Malvina Vamvakari A q-Random Walk Approximated by a q-Brownian Motion

Introduction Main Results Selected References

Conjecture

The conditions leading to the q-Poisson process likely to be fulfilled by several phenomena in Graphs and Networks In Graph or Network processes, the time steps might be geome- trically decreasing with rate q and at each time step a vertex or a pageview would be added according to q-Bernoulli trials.

Malvina Vamvakari A q-Random Walk Approximated by a q-Brownian Motion

Introduction Main Results Selected References

Conjecture

The conditions leading to the q-Poisson process likely to be fulfilled by several phenomena in Graphs and Networks In Graph or Network processes, the time steps might be geome- trically decreasing with rate q and at each time step a vertex or a pageview would be added according to q-Bernoulli trials. In Networks, the number of pageviews on web sites might fit much better to the q-Poisson process than to the Poisson pro- cess

Malvina Vamvakari A q-Random Walk Approximated by a q-Brownian Motion

Introduction Main Results Selected References

Fitting Interarrival Times of Pageviews on Harokopio University’s Web to the q-Exponential Distribution

Proposition 3.1. The moment method’s estimates of the parameters q and λ of a random sample X1, X2, . . . , Xn of the q-Exponential distribution are given by ˆ q = (m1

′)2

m2 − (m1

′)2

and ˆ λ = 1 ˆ q m1

′ ,

(14) where mr

′ = 1

n

n

• i=1

X r

i , r = 1, 2, . . .

the rth order sample moment. ⋄

Malvina Vamvakari A q-Random Walk Approximated by a q-Brownian Motion

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Data set: Interarrival times of pageviews on Harokopio University’s web in minutes at 27th of last April between 1.00pm and 2.00pm.

Malvina Vamvakari A q-Random Walk Approximated by a q-Brownian Motion

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Data set: Interarrival times of pageviews on Harokopio University’s web in minutes at 27th of last April between 1.00pm and 2.00pm. Table I: Interarrival times of pageviews in minutes in ascending

• rder

0.01 0.01 0.01 0.01 0.01 0.01 0.02 0.02 0.02 0.02 0.04 0.04 0.04 0.05 0.05 0.06 0.06 0.06 0.07 0.07 0.07 0.07 0.08 0.09 0.11 0.12 0.12 0.14 0.15 0.16 0.23 0.26 0.38 0.41 0.42 0.44 0.45 0.46 0.46 0.46 0.47 0.47 0.47 0.47 0.48 0.54 0.55 0.59 1.11 1.25 1.32 1.40 2.50 2.57 4.44

Malvina Vamvakari A q-Random Walk Approximated by a q-Brownian Motion

Introduction Main Results Selected References

Underdispersed data with mean ¯ x = 0.4434545455 and variance s2 = 0.5838480661

Malvina Vamvakari A q-Random Walk Approximated by a q-Brownian Motion

Introduction Main Results Selected References

Underdispersed data with mean ¯ x = 0.4434545455 and variance s2 = 0.5838480661 By (14), the moment method’s estimates for the q-Exponential model are equal with ˆ q = 0.3368203910 and ˆ λ = 6.695029785

Malvina Vamvakari A q-Random Walk Approximated by a q-Brownian Motion

Introduction Main Results Selected References

Underdispersed data with mean ¯ x = 0.4434545455 and variance s2 = 0.5838480661 By (14), the moment method’s estimates for the q-Exponential model are equal with ˆ q = 0.3368203910 and ˆ λ = 6.695029785 Moment method’s estimate of the ordinary Exponential model with probability density function f (t) = θexp(−θt), θ > 0, is ˆ θ = 2.255022550

Malvina Vamvakari A q-Random Walk Approximated by a q-Brownian Motion

Introduction Main Results Selected References

Table II: Empirical and fitted commulative distribution functions for the interarrival times of pageviews and their differences (ˆ θ = 2.296022960, ˆ q = 0.3288279561, ˆ λ = 6.982444520) t Empirical Fitted Differences

• Expon. model

q-Expon. model

• Emp. vs Exp.
• Emp. vs q-Exp.

0.01 0.1091 0.0222978696 0.0637454325 0.0868021304 0.0453545675 0.02 0.1818 0.0440985442 0.1216246712 0.1377014558 0.0601753288 0.04 0.2364 0.0862524069 0.2225473787 0.1501475931 0.0138526213 0.05 0.2727 0.1066270316 0.2667175893 0.1660729684 0.0059824107 0.06 0.3273 0.1265473456 0.3073110137 0.2007526544 0.0199889863 0.07 0.4000 0.1460234790 0.3447107729 0.2539765210 0.0552892271 0.08 0.4182 0.1650653361 0.3792487421 0.2531346639 0.0389512579 0.09 0.4364 0.1836826004 0.4112137571 0.2527173996 0.0251862429 0.11 0.4545 0.2196810093 0.4684038966 0.2348189907 0.0139038966 0.12 0.4909 0.2370804604 0.4940457951 0.2538195396 0.0031457951 0.14 0.5091 0.2707241015 0.5402893264 0.2383758985 0.0311893264 0.15 0.5273 0.2869854004 0.5611803234 0.2403145996 0.0338803234 0.16 0.5455 0.3028841070 0.5807511867 0.2426158930 0.0352511867 0.23 0.5636 0.4046793949 0.6886802765 0.1589206051 0.1250802765 0.26 0.5818 0.4436211673 0.7232081417 0.1381788327 0.1414081417 0.38 0.6000 0.5755277171 0.8188770737 0.0244722829 0.2188770737 0.41 0.6182 0.6032937694 0.8355387603 0.0149062306 0.2173387603 0.42 0.6364 0.6121394732 0.8406298831 0.0242605268 0.2042298831 0.44 0.6545 0.6292435578 0.8501906784 0.0252564422 0.1956906784 0.45 0.6727 0.6375106368 0.8546818593 0.0351893632 0.1819818593 0.46 0.7273 0.6455933772 0.8589934290 0.0817066228 0.1316934290 0.47 0.8000 0.6534958897 0.8631345089 0.1465041103 0.0631345089 0.48 0.8182 0.6612221933 0.8671136552 0.1569778067 0.0489136552 0.54 0.8364 0.7040936255 0.8880054953 0.1323063745 0.0516054953 0.55 0.8545 0.7106917071 0.8910510280 0.1438082929 0.0365510280 0.59 0.8727 0.7356452437 0.9021861585 0.1370547563 0.0294861585 1.11 0.8909 0.9181670275 0.9688269714 0.0272670275 0.0779269714 1.25 0.9091 0.9403211855 0.9757465767 0.0312211855 0.0666465767 1.32 0.9273 0.9490356936 0.9784691366 0.0217356936 0.0511691366 1.40 0.9455 0.9574481340 0.9811199110 0.0119481340 0.0356199110 2.50 0.9636 0.9964384391 0.9955835362 0.0328384391 0.0319835362 2.57 0.9818 0.9969585106 0.9959086227 0.0151585106 0.0141086227 4.44 1.0000 0.9999551551 0.9992160976 0.0000448449 0.0007839024 KS-crit.val. 0.2195194752

Malvina Vamvakari A q-Random Walk Approximated by a q-Brownian Motion

Introduction Main Results Selected References Figure 1. Empirical C.D.F (Red Curve), Fitted Exponential C.D.F. (Yellow Curve), Fitted q-Exponential C.D.F (Green Curve) Malvina Vamvakari A q-Random Walk Approximated by a q-Brownian Motion

Introduction Main Results Selected References

A q-Random Walk

Let the time interval (0, t], t > 0 and a suitable partition by considering the geometrically decreasing with rate q time diffe- rences δi(n) = (n)−1

q qi−1t, i = 1, 2, . . . , n, n ≥ 1.

(15)

Malvina Vamvakari A q-Random Walk Approximated by a q-Brownian Motion

Introduction Main Results Selected References

A q-Random Walk

Let the time interval (0, t], t > 0 and a suitable partition by considering the geometrically decreasing with rate q time diffe- rences δi(n) = (n)−1

q qi−1t, i = 1, 2, . . . , n, n ≥ 1.

(15) Consider the process generated by making a step of length s to the right and a step s to the left at every time period (n)−1

q qi−1t,

i = 1, 2, . . . , n, with probability of success (right step) P (Xi = s) = θ(n)−1

q qi−1t

1 + θ(n)−1

q qi−1t ,

(16) and probability of failure (left step) P (Xi = −s) = 1 1 + θ(n)−1

q qi−1t , 0 < q < 1, 0 < θ < ∞. (17)

Malvina Vamvakari A q-Random Walk Approximated by a q-Brownian Motion

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At time n

i=1 δi(n) = t the position of the process is

X(t) =

n

• i=1

Xi = ks −(n −k)s = (2k −n)s = rs, k = (r +n)/2 with k right steps and n − k left steps, which implies that P(X(t) = rs) =

• n

(r + n)/2

• q

q((r+n)/2

2

)θ(r+n)/2t(r+n)/2(n)−(r+n)/2

q

n

j=1(1 + θ(n)−1 q qi−1t)

, −n ≤ r ≤ n (18) for 0 < θ < ∞, 0 < q < 1.

Malvina Vamvakari A q-Random Walk Approximated by a q-Brownian Motion

Introduction Main Results Selected References

Definition 2.6 The discrete process {X(t), t > 0}. satisfying (15)-(18) is called q-Random Walk.

Malvina Vamvakari A q-Random Walk Approximated by a q-Brownian Motion

Introduction Main Results Selected References

The mean value and the variance of the process X(t) are given respectively by E(X(t)) =

n

• i=1

E(Xi) = s

n

• i=1

θqi−1 − 1 1 + θqi−1 E(X(t)) =

n

• i=1

E(Xi) = s

n

• i=1

θ(n)−1

q qi−1t − 1

1 + θ(n)−1

q qi−1t

(19) and V (X(t)) =

n

• i=1
• E(X 2

i ) − E(Xi)2

= s2

n

• i=1

θ(n)−1

q qi−1t(2 − θ(n)−1 q qi−1t)

1 + θ(n)−1

q qi−1t

(20)

Malvina Vamvakari A q-Random Walk Approximated by a q-Brownian Motion

Introduction Main Results Selected References

Consider θ = θn, n = 0, 1, 2, . . . such that θn = q−αn with 1/2 < α < 1 constant. Then, for n → ∞, by a q-analogue of the De-Moivre Laplace theorem: P(X(t) = rs) ∼ = q1/8(log q−1)1/2 (2π)1/2 ·

• q−3/2(1 − q)1/2 [r/2]1/q − µq

σq + q−1 1/2 · exp

• 1

2 log q log2

• q−3/2(1 − q)1/2 [r/2]1/q − µq

σq + q−1

• ,

r ≥ 0. (21)

Malvina Vamvakari A q-Random Walk Approximated by a q-Brownian Motion

Introduction Main Results Selected References

with µq = E

• (X(t))1/q
• =

θt 1 + θ(n)−1

q qn−1t

and (22) σ2

q

= V

• (X(t))1/q
• = 1 − q

q (θt)2 (1 + θ(n)−1

q qn−1t)2(1 + θ(n)−1 q qn−2t)

+ θt (1 + θ(n)−1

q qn−1t)(1 + θ(n)−1 q qn−2t).

Malvina Vamvakari A q-Random Walk Approximated by a q-Brownian Motion

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A q-Brownian Motion

Theorem 2.4 Let the q-Random Walk with steps of length s = c/θ, c positive constant, where θ = θn = q−αn with 1/2 < α < 1 . The q-Random walk is approximated by a continuous analogue one, as n → ∞, with probability density function f (y, t) = q1/8(log q−1)1/2 (2π)1/2

• q−3/2(1 − q)1/2 (y − µ)

σ + q−1 1/2 · exp

• 1

2 log q log2

• q−3/2(1 − q)1/2 (y − µ)

σ + q−1

• ,

(23) where y the position after time t ≥ 0 and µ = E (Y (t)) = ct and σ2 = V (Y (t)) = 1 − q q (ct)2 + ct. (24)

Malvina Vamvakari A q-Random Walk Approximated by a q-Brownian Motion

Introduction Main Results Selected References

Definition 2.7 The limiting process {Y (t), t ≥ 0} with p.d.f. (23), being the continuous analogue of the q-Random walk, is called q-Brownian motion.

Malvina Vamvakari A q-Random Walk Approximated by a q-Brownian Motion

Introduction Main Results Selected References

• B. Bollob´

as and M. Riordan, Mathematical results on scale-free random graphs, In Handbook of Graphs and Networks: From the Genome to the Internet, S. Bornholdt, H.G. Schuster (Eds), Wiley-Vch, 2003.

• B. Bollob´

as, S. Janson and O. Riordan, The phase transition in inhomogeneous random graphs, Random Structures and Algo- rithms, 31 (2007), 3-122. Ch.A. Charalambides, Discrete q-distributions on Bernoulli tri- als with geometricall varying success probability, J. Stat. Plan, Infer., 140 (2010), 2355-2383. A.W. Kemp and J. Newton, Certain state-dependent processes for dichotomized parasite populations, J.Appl. Probab., 27, 251- 258, 1990. A.W. Kemp, Heine Euler Extensions of the Poisson ditribution,

• Comm. Statist.-Theory Meth., 21 (1992), 571-588.

Malvina Vamvakari A q-Random Walk Approximated by a q-Brownian Motion

Introduction Main Results Selected References

• A. Kyriakoussis and M.G. Vamvakari, On a q-analogue of the

Stirling formula and a continuous limiting behaviour of the q- Binomial distribution-Numerical calculations, Methodology and Computing in Applied Probability, 15(1), 187-213, 2013.

• A. Kyriakoussis and M.G. Vamvakari, A q-Poisson Process Ap-

proximated by a Stieltjes-Wigert Process, In Proc. 9th Edition

• f the Conference GASCom on Random Generation of Combi-

natorial Structures, GaSCom 2014 (Bertinoro, 2014), pp. 1-5

• A. Kyriakoussis and M. Vamvakari, A q-Poisson Process–Waiting

and Interrarival Times .Accepted for Publication in Commun. in

• Stat. Theory and Methods.

Malvina Vamvakari A q-Random Walk Approximated by a q-Brownian Motion

Introduction Main Results Selected References

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Malvina Vamvakari A q-Random Walk Approximated by a q-Brownian Motion