Generalized Bernoulli Process with Long-Range Dependence And

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Depend. Model. 2021; 9:1–12

Research Article Open Access

Jeonghwa Lee* Generalized Bernoulli process with long-range dependence and fractional binomial distribution https://doi.org/10.1515/demo-2021-0100 Received October 23, 2020; accepted January 22, 2021

Abstract: Bernoulli process is a nite or innite sequence of independent binary variables, Xi , i = 1, 2, ··· , whose outcome is either 1 or 0 with probability P(Xi = 1) = p, P(Xi = 0) = 1 − p, for a xed constant p ∈ (0, 1). We will relax the independence condition of Bernoulli variables, and develop a generalized Bernoulli process that is stationary and has auto-covariance function that obeys power law with exponent 2H − 2, H ∈ (0, 1). Generalized Bernoulli process encompasses various forms of binary sequence from an independent binary sequence to a binary sequence that has long-range dependence. Fractional binomial random variable is dened as the sum of n consecutive variables in a generalized Bernoulli process, of particular interest is when its variance is proportional to n2H , if H ∈ (1/2, 1).

Keywords: Bernoulli process, Long-range dependence, Hurst exponent, over-dispersed binomial model

MSC: 60G10, 60G22

1 Introduction

Fractional process has been of interest due to its usefulness in capturing long lasting dependency in a stochastic process called long-range dependence, and has been rapidly developed for the last few decades. It has been applied to internet trac, queueing networks, hydrology data, etc (see [3, 7, 10]). Among the most well known models are fractional Gaussian noise, fractional Brownian motion, and fractional Poisson process. Fractional Brownian motion(fBm) BH(t) developed by Mandelbrot, B. and Van Ness, J. (1968) [11] is zero- mean, increment stationary Gaussian process that has self-similarity, i.e., for any a > 0, law H {BH(at), t ∈ R} = {a BH(t), t ∈ R} law where 0 < H < 1 is called Hurst parameter, and = means equality of all the nite distributions. FBm has auto-covariance function var(BH(1)) 2H 2H 2H cov(BH(s), BH(t)) = |t| + |s| − |t − s| . 2 2H Especially, the variance of BH(t) is var(BH(1))|t| . If H = 1/2, then fBm becomes standard Brownian motion which has independent increment. The one time dierence of fBm, Yj = BH(j) − BH(j − 1) for j ∈ Z, is called fractional Gaussian noise(fGn) which is mean-zero, stationary Gaussian process with auto-covariance function

var(Y0) 2H 2H 2H γ(j) := cov(Y , Yj) = |j + 1| − 2|j| + |j − 1| . 0 2 The auto-covariance function obeys power law with exponent 2H − 2 for large lag, 2H−2 γ(j) ∼ var(Y0)H(2H − 1)j as j → ∞. (1.1)

*Corresponding Author: Jeonghwa Lee: Department of Statistics, Truman State University, USA, E-mail: [email protected]

If 1/2 < H < 1, fGn has long-range dependence, X cov(Y0, Yj) = ∞. (1.2) j∈N

Note that a covariance function that decreases as slowly as the power law with exponent between 0 and -1 for large lag leads to long-range dependence of the process. The time fractional Poisson process (TFPP) developed by Laskin, N. (2003) is a fractional generalization of Poisson process. Poisson process N(t) is used for counting the number of events in time-interval [0, t] where inter-arrival times between events are independent and exponentially distributed with parameter λ. As a result, it has Markov property with expected number of events E(N(t)) = λt. In [9], TFPP, Nµ(t), 0 < µ ≤ 1, is dened with Mittag-Leer distributed waiting times where the inter-arrival distribution has heavy tail that decreases as slowly as 1/tµ+1, and this makes TFPP non-Markov process, possesses long-range dependence, µ and has expectation that is proportional to the fractional exponent of the time, E(Nµ(t))∝t . For other fractional generalizations of Poisson process, such as mixed fractional Poisson process and their properties, see [2, 4, 5, 8]. In this paper, we propose a generalized Bernoulli process(GBP) that is stationary and has long-range dependence, and fractional binomial distribution as the sum of consecutive variables in a generalized Bernoulli process. Bernoulli process is a nite or innite sequence of independent binary variables, Xi , i = 1, 2, ··· , whose outcome is either 1 or 0 with probability P(Xi = 1) = p, P(Xi = 0) = 1−p, for a xed constant p ∈ (0, 1). If we interpret Xi = 1 as a success in i-th trial, and Xi = 0 as a failure in i-th trial, then binomial variable that is the sum of n binary variables in Bernoulli process counts the number of successes among n trials, and the binomial variable has expectation np and variance np(1 − p). When applying binomial model in real data analysis, over-dispersion is frequently observed. Over- dispersion is a term referring to the phenomenon that larger variance is observed than the nominal variance under some presumed model. It occurs when assumptions in the presumed model are violated. In binomial model, it is assumed that trials are independent and each has the same probability of success. Due to dependence among trials or heterogeneous probability of success in trials, over-dispersed binomial model is widely used in application. See [6] for more account for over-dispersion in various statistical models. In this paper, we will relax the independence condition of Bernoulli variables so that binary variables have dependence with the auto-covariance which decreases with power law with exponent 2H − 2, for H ∈ (0, 1). If H ∈ (1/2, 1), the generalized Bernoulli process possesses long-range dependence, therefore, fractional binomial random variable, which is dened as the sum of n successive variables in a GBP,has larger variance than ordinary binomial random variable. Over-dispersion parameter ψn > 0 is dened and incorporated in binomial model as follows:

E(Bn) = np, Var(Bn) = np(1 − p)(1 + ψn), (1.3) where Bn is over-dispersed binomial variable that has n number of trials and probability of success p. Among methods that incorporate over-dispersion in binomial distribution are beta-binomial model and generalized linear mixed model, [1, 6]. It turned out that the fractional binomial variable dened from a GBP shows over- 2H−1 dispersion (1.3) with various ψn . In particular, ψn is proportional to fractional exponent of n, ψn ∝ n , when H ∈ (1/2, 1). The contents of the paper are as follows. In section 2, we construct a generalized Bernoulli process whose auto-covariance function decreases with the power law of exponent 2H − 2. In section 3, some interesting properties of a GBP are investigated regrading conditional probability. In section 4, fractional binomial variable is dened from a GBP, and the connection of fractional binomial distribution to the over-dispersed binomial distribution is investigated. 0 0 0 Throughout this paper, {i, i0, i1, ··· }, {i , i0, i1, ··· } ⊂ N, and for any set A = {i0, i1, ··· , in}, |A| = n+1, the number of elements in the set A, with |A| = 0, if A = ∅. We also dene the maximum of empty set as 0, i.e., max ∅ = 0. Generalized Bernoulli process and fractional binomial distribution Ë 3

2 Generalized Bernoulli process

We will dene stationary process, {Xi , i ∈ N}, where each Xi takes one of two possible outcomes, 0 or 1, with P(Xi = 1) = p, P(Xi = 0) = 1 − p, and 0 2H−2 cov(Xi , Xj) = c |i − j| , i ≠ j, 0 for some constants c ∈ R+, H ∈ (0, 1). If H ∈ (1/2, 1), n 0 X c 2H−1 cov(X , Xi) ∼ |n| , 1 2H − 1 i=1 which diverges as n increases, and the process is said to have long-range dependence. Below is how we will proceed to dene such stationary process. Let Xi be a binary variable with

P(Xi = 1) = p, P(Xi = 0) = 1 − p, for a xed p ∈ (0, 1). (2.1) * Dene the following function P with constants c > 0, H ∈ (0, 1), and for any i0, i1, ··· , in , P* X p ( i0 = 1) := , P* X X p p c|i i |2H−2 ( i0 = 1, i1 = 1) := + 1 − 0 , P* X X X p p c|i i |2H−2 p c|i i |2H−2 ( i0 = 1, i1 = 1, i2 = 1) := + 1 − 0 + 2 − 1 , and in general, P* X X X p p c|i i |2H−2 p c|i i |2H−2 ( i0 = 1, i1 = 1, ··· , in = 1) := + 1 − 0 + 2 − 1

2H−2 × · · · × p + c|in − in−1| . (2.2) Furthermore, for any disjoint sets, A, B ⊂ N, A ≠ ∅, B ≠ ∅, dene |B| * * X X k * P (∩i0∈B{Xi0 = 0}) ∩ (∩i∈A{Xi = 1}) := P (∩i∈A{Xi = 1}) + (−1) P (∩i∈B0∪A{Xi = 1}) k=1 B0⊂B 0 |B |=k X X P* ∩ {X } P* ∩ {X } P* ∩ {X } = ( i∈A i = 1 ) − ( i∈A∪{j} i = 1 ) + ( i∈A∪{j1 ,j2} i = 1 ) j∈B j1

2.1 Dening generalized Bernoulli process

Denition 2.1. Dene the following operation on a set A = {i0, i1, ··· , in} ⊂ N with i0 < i1 < ··· < in . Y 2H−2 LH(A) := p + c|ij − ij−1| . j=1,··· ,n

If A = ∅, dene LH(A) := 1/p, and if |A| = 1, LH(A) := 1. 4 Ë Jeonghwa Lee

2H−2 2H−2 For example, if A = {1, 4, 9}, LH(A) = p + c|4 − 1| p + c|9 − 4| .

Denition 2.2. Dene for disjoint sets, A, B ⊂ N with |B| = m > 0, X 0 X 0 00 DH(A, B) :=LH(A) − LH(A ∪ {i }) + LH(A ∪ {i , i }) 0 0 00 i ∈B i

If B = ∅, DH(A, B) := LH(A).

Note that (2.2-2.4) can be expressed as

* P (∩i∈A{Xi = 1}) ∩ (∩i0∈B{Xi0 = 0}) = pDH(A, B), (2.5) for disjoint sets, A, B ⊂ N, A ∪ B ≠ ∅. Now we give an assumption on constants, p, H, c, so that (2.5) is always positive.

ASSUMPTION (A1) i) 0 < p, H < 1. ii) 1 H p 0 ≤ c < min{1 − p, (−2p + 22 −2 + 4p − p22H + 24H−4)}. (2.6) 2 Throughout this paper, it is assumed that p, H ∈ (0, 1) and (2.6) holds.

Lemma 2.1. For any i0 < i1 < i2, 2H−2 2H−2 2H−2 p + c|i1 − i0| p + c|i2 − i1| < p + c|i2 − i0| . (2.7)

See proof on page 7. The Lemma 2.1 states that

DH({i0, i2}, {i1}) = LH({i0, i2}) − LH({i0, i1, i2}) > 0, which is extended in the following Proposition.

Proposition 2.2. For any disjoint sets, A0, A1 ⊂ N, A0 ∪ A1 ≠ ∅,

DH(A1, A0) > 0. (2.8)

See proof on page 9. * By (2.5) and Proposition 2.2, it was proved that P is non-negative function, and therefore, {Xi , i ∈ N} is a stationary process with probability P = P* under the assumption (A1).

* Theorem 2.3. For binary variables {Xi , i ∈ N} that satisfy (2.1), if the joint probability is dened with P = P as (2.2-2.4) under the assumption (A1), then {Xi , i ∈ N} is a stationary process with covariance 2H−2 Cov(Xi , Xj) = pc|i − j| , i ≠ j. (2.9)

* Proof. By proposition 2.2, (2.3-2.4) are always positive, therefore, {Xi , i ∈ N} with probability P = P is well dened. It is a stationary process with P(Xi = 1) = p, P(Xi = 0) = 1 − p, and 2 2H−2 P(Xi = 1, Xj = 1) = p + pc|i − j| , from which (2.9) is derived.

We call this stationary process with probability P = P* a generalized Bernoulli process(GBP), and for the rest of the paper {Xi , i ∈ N} is assumed to be a GBP. Generalized Bernoulli process and fractional binomial distribution Ë 5

2.2 Properties of GBP

Next, we consider the conditional probability in a GBP. Following Theorem shows the result concerning the conditional probability of next observation given past observations when the last observation was 1. Recall that we dene the maximum of empty set as zero, i.e., max ∅ = 0.

* Theorem 2.4. Let A0, A1 be disjoint subsets of N such that max A1 > max A0. Then, for i > max A1, the conditional probability satises the following:

 * 2H−2 P {X }| ∩ 0 {X 0 } ∩ ∩ {X } p c|i A | i* = 1 ( i ∈A0 i = 0 ) ( i∈A1 i = 1 ) = + − max 1 . (2.10)

If it is called “success" at time i for Xi = 1, and “failure" at time i for Xi = 0, (2.10) implies that the conditional probability of success in next observation given past observations only depends on the time dierence between last observation and next observation, provided that the last observation was success. Furthermore, the conditional probability decreases as the time dierence increase by power law with exponent 2H − 2.

See the proof on page 10. Next we show the results for the conditional probability when the last observation was 0.

* ** Theorem 2.5. Let A0, A1 be disjoint subsets of N such that max A1 < max A0. Then, for i > i > max A0, the conditional probability satises the following: i) * 2H−2 p c|i A | P {X }| ∩ 0 X 0 ∩ ∩ X + − max 1 > i* = 1 ( i ∈A0 i = 0) ( i∈A1 i = 1) . (2.11) ii) P {X * = 1}|(∩i0∈A {Xi0 = 0}) ∩ (∩i∈A {Xi = 1}) * 2H−2 i 0 1 p + |i − max A | > 1 . (2.12) p |i** A |2H−2 P {X }| ∩ 0 {X 0 } ∩ ∩ {X } + − max 1 i** = 1 ( i ∈A0 i = 0 ) ( i∈A1 i = 1 )

See proof on Page 10. Remark 1. Theorem 2.4 and 2.5 imply that a GBP has what we will call “conditioned Markov property", which means that the conditional probability depends only on the time dierence between the last observation and the next time of prediction, regardless of the past observations, when the last observation was success, and this is not true when the latest observation was failure. Remark 2. Note that from (2.9) both the parameters c and H determine the strength of the dependence in the 2H−2 process. The larger the parameters, c, H, are, the stronger the correlation is, Corr(Xi , Xj) = c|i−j| /(1−p), for i ≠ j. Especially, if c = 0 in (2.6), then a GBP becomes identical to Bernoulli process that has no dependence between any two variables in the process. Therefore, GBP can be considered as generalization of Bernoulli process that possesses long-range dependence if c ≠ 0 and H ∈ (1/2, 1). In fact, GBP is the unique binary stationary process that has conditioned Markov property and covariance function (2.9).

Theorem 2.6. {Xi , i ∈ N} is the unique stationary process that satises i)

P(Xi = 1) = p, P(Xi = 0) = 1 − p, p ∈ (0, 1), ii) 2H−2 Cov(Xi , Xj) = c|i − j| , i ≠ j, for some c > 0, 6 Ë Jeonghwa Lee

iii) there is a function h(·) such that

 * P {X }| ∩ 0 X 0 ∩ ∩ X h i A i* = 1 ( i ∈A0 i = 0) ( i∈A1 i = 1) = ( − max 1),

* * for any i ∈ N, and any disjoint subsets, A0, A1 ⊂ N, such that i > max A1 > A0.

Proof. Let {Xi , i ∈ N} is a stationary process that satises i)-iii). Note that by i), ii), P X X Cov X X p2 c|i i |2H−2 p2 ( i0 = 1, i1 = 1) = ( i0 , i1 ) + = 0 − 1 + , P X |X p c p|i i |2H−2 iii h i i p c p|i i |2H−2 which results in ( i1 = 1 i0 = 1) = + / 0 − 1 . Therefore, by ), ( 0 − 1) = + / 0 − 1 . By applying iii) repeatedly, we obtain

P ∩ {X } P X P X |X P X |X X ( i∈A1 i = 1 ) = ( i0 = 1) × ( i1 = 1 i0 = 1) × ( i2 = 1 i1 = 1, i0 = 1) P X |X X × · · · × ( in = 1 in−1 = 1, ··· i0 = 1) Y = p h(ik − ik−1), k=1,··· ,n which is equivalent to (2.2).

3 Fractional binomial distribution

A GBP {Xi , i ∈ N} dened in Section 2 is stationary process where each Xi has two outcomes, 0 or 1, with 0 2H−2 0 P(Xi = 1) = p, P(Xi = 0) = 1 − p, and cov(Xi , Xj) = c |i − j| , i ≠ j, for some constants c ∈ R+, H ∈ (0, 1). 0 Here, c = cp replaces cp in previous sections. If H ∈ (1/2, 1), the process has long-range dependence. Let S Pn X n E S p n → n = i=1 i / . Then ( n) = , and as ∞,  c0  −1 ∈  p(1 − p) + n H (0, 1/2),  2H − 1  0 ln n Var(Sn) ∼ c H = 1/2,  n  0  c H  |n|2 −2 H ∈ (1/2, 1). 2H − 1 B Pn X E B np n → Dene n = i=1 i . It follows that ( n) = , and as ∞,  c0  p(1 − p) + n H ∈ (0, 1/2),  2H − 1  0 Var(Bn) ∼ c nln n H = 1/2,  0  c H  |n|2 , H ∈ (1/2, 1). 2H − 1 0 We call Bn fractional binomial random variable. It becomes binomial variable with parameter n, p, if c = 0. 0 If c ≠ 0, Bn is over-dispersed binomial distribution with over-dispersion parameter,  c0   H ∈ (0, 1/2),  p(1 − p)(2H − 1)  0  c ln n ψn ∼ − 1 H = 1/2, p(1 − p)   0 2H−1  c n  − 1 H ∈ (1/2, 1), p(1 − p)2H − 1 as n → ∞, by following the conventional notation of over-dispersion parameter,

E(Bn) = np, Var(Bn) = np(1 − p)(1 + ψn). Generalized Bernoulli process and fractional binomial distribution Ë 7

If H ∈ (0, 1/2), the over-dispersion parameter ψn is asymptotically constant as n increases, whereas for H ∈ (1/2, 1), ψn increases with the rate of fractional exponent of n. Therefore, fractional binomial distribution can model various over-dispersed binomial random variable.

4 Conclusion

Over-dispersion in various models including binomial model is well observed in real data, and one of the reasons for over-dispersion in binomial model is explained by dependence among trials. In this paper, the independence assumption on binary sequence in Bernoulli process was relaxed, and GBP was dened as a binary sequence whose auto-covariance function decreases by power law with exponent 2H − 2, H ∈ (0, 1). If H ∈ (1/2, 1), the process has long-range dependence. There could be many other stationary models that feature binary sequences with dependence structure. However, GBP in this paper is the only stationary binary sequence that has the covariance function of the form,

2H−2 Cov(Xi , Xj) = c|i − j| , i ≠ j, for some c > 0, H ∈ (0, 1), and has conditioned Markov property where the Markov property holds only when the last observation was in certain state. Also, from the conditioned Markov property and (2.2-2.4), it can be derived that the number of failures (number of 0’s) between any two consecutive successes is independent of the number of failures between any other two consecutive successes and they are identically distributed. Fractional binomial random variable was dened as the sum of n successive variables in GBP, and it can be used for various over-dispersed binomial model whose over-dispersion parameter ranges from asymptotic constant to fractional exponent of n. As over-dispersed binomial data is widely observed, the hope is that the fractional binomial model provides another useful tool in those circumstances.

5 Proofs

* 2H−2 Proof of Lemma 2.1. Let c = c|i2 − i0| . Then, (2.7) becomes

 H H |i − i | p + c*|x|2 −2 p + c*|1 − x|2 −2 < p + c*, where 0 < x = 1 0 < 1. |i2 − i0|

Since |x(1 − x)|2H−2 and x2H−2 + (1 − x)2H−2 are both maximized when x has the smallest fraction possible (or the largest fraction possible), (2.7) holds if

 2H−2 2H−2 p + c p + c|i2 − 1| < p + c|i2| .

Note that 2H−2 2H−2 p + c|i2| p + c|2| min 2H−2 = , i2∈N/{1} p + c|i2 − 1| p + c the minimum occurs when i2 = 2. Therefore, (2.7) holds if

p + c|2|2H−2 p + c < , p + c that is,

1 H p 1 H p 22 −2 − 2p − (22H−2 − 2p)2 + 4(p − p2) < c < 22 −2 − 2p + (22H−2 − 2p)2 + 4(p − p2) . 2 2 8 Ë Jeonghwa Lee

Lemma 5.1. For any {i0, i1, ··· , in} ⊂ N,

DH({i0, in}, {i1, i2, ··· , in−1}) =

DH({i0, in}, {i1, i2, ··· , in−2}) − DH({i0, in−1, in}, {i1, i2, ··· , in−2}).

In general, for any {a0, a1, ··· , an , b0, b1, ··· , bm} ⊂ N,

DH({a0, a1, ··· , an}, {b0, b1, ··· , bm}) = DH({a0, a1, ··· , an}, {b0})

− DH({a0, a1, ··· , an , b1}, {b0}) − DH({a0, a1, ··· , an , b2}, {b0, b1})

· · · − DH({a0, a1, ··· , an , bm}, {b0, b1, ··· , bm−1}). (5.1)

Proof. Let A = {a0, a1, ··· , an}, B = {b0, b1, ··· , bm}, then by the denition of DH, X DH({a0, a1, ··· , an}, {b0, b1, ··· , bm}) = LH(A) − LH(A ∪ {b}) b∈B/{bm} X 0 00 m + LH(A ∪ {b , b }) + ··· (−1) LH(A ∪ B/{bm}) 0 00 b

b∈B/{bm} X 0 00 m + LH(A ∪ {bm} ∪ {b , b }) + ··· (−1) LH(A ∪ B) 0 00 b

= DH({a0, a1, ··· , an}, {b0, b1, ··· , bm−1}) − DH({a0, a1, ··· , an , bm}, {b0, b1, ··· , bm−1}).

Applying the above result inductively leads to (5.1).

Lemma 5.2. For any {a a a a0 a0 a0 } ⊂ such that a Pj a a0 Pj a0 for 0, 1, ··· , n , 0, 1, ··· , n R+ 0 − i=1 i > 0, 0 − i=1 i > 0, j = 1, 2, ··· , n, i) if a0 a1 an a0 > a0 > ··· > a0 , 0 1 n then a0 − a1 − a2 − · · · − an a0 a0 a0 a0 a0 > a0 . 0 − 1 − 2 − · · · − n 0 ii) If a0 a1 an a0 < a0 < ··· < a0 , 0 1 n then a0 − a1 − a2 − · · · − an a0 a0 a0 a0 a0 < a0 . 0 − 1 − 2 − · · · − n 0 Proof. We will prove i) by mathematical induction. ii) follows in the same way. a0 a1 a0 0 0 Let n = 1. Since a0 > a0 , a0 − a1 > a0 (a0 − a1), which leads to 0 1 0

a0 − a1 a0 a0 a0 > a0 . 0 − 1 0 Assume i) holds for n = k. Let n = k + 1. Since

a0 − a1 − · · · − ak a0 ak+1 a0 a0 a0 > a0 > a0 , 0 − 1 − · · · − k 0 k+1

a0 0 0 0 0 (a0 − a1 − · · · − ak) − ak+1 > a0 {(a0 − a1 − · · · − ak) − ak }, therefore, i) holds for n = k + 1. By mathematical 0 +1 induction, i) follows. Generalized Bernoulli process and fractional binomial distribution Ë 9

Proof of Proposition 2.2. If A0 = ∅, DH(A1, A0) = LH(A1) > 0. Assume A0 ≠ ∅. We will prove (2.8) by mathematical induction. If |A0| = 1, DH(A1, A0) = LH(A1) − LH(A1 ∪ {A0}) > 0, by Lemma 2.1. Assume (2.8) holds with |A0| ≤ m. We will show that (2.8) holds when |A0| = m + 1. 0 0 0 0 0 0 0 0 Let A0 = {i0, i1, i2, ··· , im} with i0 < i1 < i2 < ··· < im . By Lemma 5.1, it is enough to show 0 0 0 DH(A1, A0/{im}) > DH(A1 ∪ {im}, A0/{im}).

By (5.1), 0 DH(A , A /{im}) a − a − a − a · · · − am 1 0 = 0 1 2 3 −1 , (5.2) ∪ { 0 } { 0 } a0 a0 a0 a0 a0 DH(A1 im , A0/ im ) 0 − 1 − 2 − 3 · · · − m−1 where ( 0 0 0 0 DH(A1 ∪ {ij}, {i0, i1, ··· , ij−1}) for j = 1, ··· , m − 1, aj = 0 DH(A1, {i0}) for j = 0, and ( 0 0 0 0 0 0 DH(A1 ∪ {ij , im}, {i0, i1, ··· , ij−1}) for j = 1, ··· , m − 1, aj = 0 0 DH(A1 ∪ {im}, {i0}), for j = 0.

0 0 0 We will apply Lemma 5.2 i) to show that (5.2) > 1. Note that a0 − a1 − · · · − aj > 0 and a0 − a1 − · · · − aj > 0 for j = 0, 1, ··· , m − 1, by the earlier assumption that (2.8) holds for |A0| ≤ m. * 0 0 * 0 0 * Dene ij = max{ij , i : i ∈ A1, i < im}, and i = min{i : i ∈ A1, i > im} if max A1 > im . Note that ij is non-decreasing as j increases from 1 to m − 1. For j = 1, ··· , m − 1, 0 0 0 0 aj DH(A1 ∪ {ij}, {i0, i1, ··· , ij }) = −1 a0 D A ∪ {i0 i0 } {i0 i0 i0 } j H( 1 j , m , 0, 1, ··· , j−1 )  p c|i* i*|2H−2  + − j 0  if max A > im ,  p c|i0 i*|2H−2 p c|i0 i*|2H−2 1 ( + m − j )( + m − )   1 0 = A im (5.3) 0 * 2H−2 if max 1 < ,  p + c|im − ij |    1 A ∅  0 0 2H−2 if 1 = , p + c|im − ij| which non-increases as j goes from 1 to m − 1. Also, 0 0 a DH(A , {i }) LH(A ) − LH(A ∪ {i }) 0 = 1 0 = 1 1 0 a0 D A ∪ {i0 } {i0 } L A ∪ {i0 } L A ∪ {i0 i0 } 0 H( 1 m , 0 ) H( 1 m ) − H( 1 0, m ) with  1 0 if min A > im ,  0 * 2H−2 1  p + c|im − i |   1 0  if max A1 < im , LH(A )  p c|i0 i |2H−2 1 = + m − * L A ∪ {i0 } H( 1 m )  p c|i* i |2H−2  + − * A i0 A  0 H 0 H if min 1 < m < max 1, (p + c|im − i*|2 −2)(p + c|im − i |2 −2)  *  1/p if A1 = ∅, 0 0 where i* = max{i : i ∈ A1, i < im} if min A1 < im , and

0 LH(A ∪ {i }) 1 0 = (5.3) L A ∪ {i0 i0 } H( 1 0, m ) * * * * with j = 0. Since i* ≤ i0 ≤ i1 ≤ i2 ≤ · · · ≤ im−1, 0 0 0 LH(A ) LH(A ∪ {i }) DH(A ∪ {i }, {i }) 1 ≥ 1 0 ≥ 1 1 0 L A ∪ {i0 } L A ∪ {i0 i0 } D A ∪ {i0 i0 } {i0 } H( 1 m ) H( 1 0, m ) H( 1 1, m , 0 ) 10 Ë Jeonghwa Lee

0 0 0 0 0 0 0 DH(A ∪ {i }, {i , i }) DH(A ∪ {i }, {i , i , ··· , i }) ≥ 1 2 0 1 ≥ · · · ≥ 1 m−1 0 1 m−2 , D A ∪ {i0 i0 } {i0 i0 } D A ∪ {i0 i0 } {i0 i0 i0 } H( 1 2, m , 0, 1 ) H( 1 m−1, m , 0, 1, ··· , m−2 ) therefore, by Lemma 5.2 i), LH(A1) (5.2) ≥ 0 . LH(A1 ∪ {im})

LH (A1) The result follows as 0 > 1 by Lemma 2.1. LH (A1∪{im})

Proof of Theorem 2.4. (2.10) is derived from the fact that P ∩ 0 {X 0 } ∩ ∩ * {X } ( i ∈A0 i = 0 ) ( i∈A1∪{i } i = 1 ) =

 * 2H−2 P ∩ 0 {X 0 } ∩ ∩ {X } p c|i A | ( i ∈A0 i = 0 ) ( i∈A1 i = 1 ) × + − max 1 ,

0 0 0 * since there is no element i such that i ∈ A0 and max A1 < i < i .

0 0 0 0 Proof of Theorem 2.5. Let A0 = {i0, i1, ··· , im}, A1 = {i0, i1, ··· , in} with max A1 = in < max A0 = im . i) Note P (∩i0∈A {Xi0 = 0}) ∩ (∩i∈A ∪{i*}{Xi = 1}) * 0 1 DH(A ∪ {i }, A ) = 1 0 , (5.4) D A A P ∩ 0 {X 0 } ∩ ∩ {X } H( 1, 0) ( i ∈A0 i = 0 ) ( i∈A1 i = 1 ) and by (5.1), * DH(A1 ∪ {i }, A0) a0 − a1 − a2 − a3 · · · − am = a0 a0 a0 a0 a0 , (5.5) DH(A1, A0) 0 − 1 − 2 − 3 · · · − m where  0 * 0 0 0 DH(A1 ∪ {ij , i }, {i0, i1, ··· , ij })  −1 for j = 1, ··· , m,  D A ∪ {i0} {i0 i0 i0 } aj H( 1 j , 0, 1, ··· , j−1 ) 0 = a * 0 j  DH(A ∪ {i }, {i })  1 0 for j = 0.  D A {i0 } H( 1, 0 ) We obtain

* 0 * * 0 a0 DH(A1 ∪ {i }, {i0}) LH(A1 ∪ {i }) − LH(A1 ∪ {i , i0}) * 2H−2 = = < p + c|i − in| , a0 D A {i0 } L A L A ∪ {i0 } 0 H( 1, 0 ) H( 1) − H( 1 0 ) by

* * 2H−2 LH(A1 ∪ {i }) p + c|i − in| = LH(A1) * 0 LH(A1 ∪ {i , i0}) * 0 2H−2 ≤ = p + c|i − max{in , i }| L A ∪ {i0 } 0 H( 1 0 ) and Lemma 5.2 ii). Note also that

* 0 * 0 0 DH(A1 ∪ {i }, {i0}) DH(A1 ∪ {i , i1}, {i0}) * 0 2H−2 ≤ = p + c|i − max{in , i }| . D A {i0 } D A ∪ {i0 } {i0 } 1 H( 1, 0 ) H( 1 1 , 0 ) 0 0 0 As i0 < i1 < ··· < ij , we have

D A ∪ {i* i0} {i0 i0 i0 } H( 1 , j , 0, 1, ··· , j−1 ) * 0 2H−2 = p + c|i − max{in , i }| , D A ∪ {i0} {i0 i0 i0 } j H( 1 j , 0, 1, ··· , j−1 ) which is non-decreasing as j increases from 1 to m. Applying Lemma 5.2 ii) leads to

* DH(A1 ∪ {i }, A0) * 2H−2 < p + c|i − in| . DH(A1, A0) Generalized Bernoulli process and fractional binomial distribution Ë 11 ii) The proof is similar to i). First, note that P {Xi* = 1}|(∩i0∈A {Xi0 = 0}) ∩ (∩i∈A {Xi = 1}) * 0 1 DH(A ∪ {i }, A ) = 1 0 , D A ∪ {i**} A P {X }| ∩ 0 {X 0 } ∩ ∩ {X } H( 1 , 0) i** = 1 ( i ∈A0 i = 0 ) ( i∈A1 i = 1 ) and by (5.1), * DH(A ∪ {i }, A ) a − a − a − a · · · − am 1 0 = 0 1 2 3 , (5.6) ∪ { **} a0 a0 a0 a0 a0 DH(A1 i , A0) 0 − 1 − 2 − 3 · · · − m where  0 * 0 0 0 DH(A1 ∪ {ij , i }, {i0, i1, ··· , ij })  −1 for j = 1, ··· , m,  D A ∪ {i0 i**} {i0 i0 i0 } aj H( 1 j , , 0, 1, ··· , j−1 ) 0 = a * 0 j  DH(A ∪ {i }, {i })  1 0 for j = 0.  D A ∪ {i**} {i0 } H( 1 , 0 ) It is derived that

* 0 * * 0 DH(A ∪ {i }, {i }) LH(A ∪ {i }) − LH(A ∪ {i , i }) 1 0 = 1 1 0 D A ∪ {i**} {i0 } L A ∪ {i**} L A ∪ {i** i0 } H( 1 , 0 ) H( 1 ) − H( 1 , 0 ) 2H−2 p + c|i* − in| > H , p + c|i** − in|2 −2 by Lemma 5.2 i) and the fact that

* 2H−2 * p + |i − in| LH(A1 ∪ {i }) ** 2H−2 = ** p + c|i − in| LH(A1 ∪ {i }) * 0 * 0 2H−2 LH(A ∪ {i , i }) p + c|i − max{in , i }| ≥ 1 0 = 0 . L A ∪ {i** i0 } p c|i** {i i0 }|2H−2 H( 1 , 0 ) + − max n , 0 Note also that

* 0 * 0 0 * 0 2H−2 DH(A ∪ {i }, {i }) DH(A ∪ {i , i }, {i }) p + c|i − max{in , i }| 1 0 ≥ 1 1 0 = 1 . D A ∪ {i**} {i0 } D A ∪ {i** i0 } {i0 } p c|i** {i i0 }|2H−2 H( 1 , 0 ) H( 1 , 1 , 0 ) + − max n , 1 0 0 0 As i0 < i1 < ··· < ij , we have

* 0 0 0 0 * 0 2H−2 DH(A1 ∪ {i , ij}, {i0, i1, ··· , ij }) p + c|i − max{in , ij}| −1 = , D A ∪ {i** i0} {i0 i0 i0 } p c|i** {i i0}|2H−2 H( 1 , j , 0, 1, ··· , j−1 ) + − max n , j which is non-increasing as j increases from 1 to m. Applying Lemma 5.2 i) leads to

* * 2H−2 DH(A1 ∪ {i }, A0) p + c|i − in| ** > ** 2H−2 . DH(A1 ∪ {i }, A0) p + c|i − in|

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