scholarly journals Generalized Bernoulli process with long-range dependence and fractional binomial distribution

2021 ◽  
Vol 9 (1) ◽  
pp. 1-12
Author(s):  
Jeonghwa Lee
Keyword(s):  
Long Range ◽  
Infinite Sequence ◽  
Binary Sequence ◽  
Random Variable ◽  
Long Range Dependence ◽  
Bernoulli Process ◽  
Binomial Random Variable ◽  
Fixed Constant ◽  
Bernoulli Variables ◽  
Auto Covariance

Abstract Bernoulli process is a finite or infinite sequence of independent binary variables, X i , i = 1, 2, · · ·, whose outcome is either 1 or 0 with probability P(X i = 1) = p, P(X i = 0) = 1 – p, for a fixed 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 defined as the sum of n consecutive variables in a generalized Bernoulli process, of particular interest is when its variance is proportional to n 2 H , if H ∈ (1/2, 1).

scholarly journals Long-Range Dependence of Markov Chains in Discrete Time on Countable State Space

2007 ◽  
Vol 44 (04) ◽  
pp. 1047-1055 ◽  
Author(s):  
K. J. E. Carpio ◽  
D. J. Daley
Keyword(s):  
State Space ◽  
Long Range ◽  
Transition Probability ◽  
Random Variable ◽  
Return Time ◽  
Long Range Dependence ◽  
Countable State Space ◽  
Countable State ◽  
Number Of Visits ◽  
Step Transition

When {X n } is an irreducible, stationary, aperiodic Markov chain on the countable state space X = {i, j,…}, the study of long-range dependence of any square integrable functional {Y n } := {y X n } of the chain, for any real-valued function {y i : i ∈ X }, involves in an essential manner the functions Q ij n = ∑ r=1 n (p ij r − π j ), where p ij r = P{X r = j | X 0 = i} is the r-step transition probability for the chain and {π i : i ∈ X } = P{X n = i} is the stationary distribution for {X n }. The simplest functional arises when Y n is the indicator sequence for visits to some particular state i, I ni = I {X n =i} say, in which case limsup n→∞ n −1var(Y 1 + ∙ ∙ ∙ + Y n ) = limsup n→∞ n −1 var(N i (0, n]) = ∞ if and only if the generic return time random variable T ii for the chain to return to state i starting from i has infinite second moment (here, N i (0, n] denotes the number of visits of X r to state i in the time epochs {1,…,n}). This condition is equivalent to Q ji n → ∞ for one (and then every) state j, or to E(T jj 2) = ∞ for one (and then every) state j, and when it holds, (Q ij n / π j ) / (Q kk n / π k ) → 1 for n → ∞ for any triplet of states i, j k.

scholarly journals Long-Range Dependence of Markov Chains in Discrete Time on Countable State Space

2007 ◽  
Vol 44 (4) ◽  
pp. 1047-1055 ◽  
Author(s):  
K. J. E. Carpio ◽  
D. J. Daley
Keyword(s):  
State Space ◽  
Long Range ◽  
Transition Probability ◽  
Random Variable ◽  
Return Time ◽  
Long Range Dependence ◽  
Countable State Space ◽  
Countable State ◽  
Number Of Visits ◽  
Step Transition

When {Xn} is an irreducible, stationary, aperiodic Markov chain on the countable state space X = {i, j,…}, the study of long-range dependence of any square integrable functional {Yn} := {yXn} of the chain, for any real-valued function {yi: i ∈ X}, involves in an essential manner the functions Qijn = ∑r=1n(pijr − πj), where pijr = P{Xr = j | X0 = i} is the r-step transition probability for the chain and {πi: i ∈ X} = P{Xn = i} is the stationary distribution for {Xn}. The simplest functional arises when Yn is the indicator sequence for visits to some particular state i, Ini = I{Xn=i} say, in which case limsupn→∞n−1var(Y1 + ∙ ∙ ∙ + Yn) = limsupn→∞n−1 var(Ni(0, n]) = ∞ if and only if the generic return time random variable Tii for the chain to return to state i starting from i has infinite second moment (here, Ni(0, n] denotes the number of visits of Xr to state i in the time epochs {1,…,n}). This condition is equivalent to Qjin → ∞ for one (and then every) state j, or to E(Tjj2) = ∞ for one (and then every) state j, and when it holds, (Qijn / πj) / (Qkkn / πk) → 1 for n → ∞ for any triplet of states i, jk.

THE MAXIMUM OF MULTIFRACTAL CASCADES: EXACT DISTRIBUTION AND APPROXIMATIONS

Fractals ◽  
2005 ◽  
Vol 13 (04) ◽  
pp. 311-324 ◽  
Author(s):  
DANIELE VENEZIANO ◽  
ANDREAS LANGOUSIS
Keyword(s):  
Long Range ◽  
Random Variable ◽  
Exact Distribution ◽  
Simple Approximation ◽  
Long Range Dependence ◽  
Distribution Of The Maximum ◽  
Cascade Generator ◽  
Distribution Type ◽  
Multifractal Measures

We study the distribution of the maximum M of multifractal measures using discrete cascade representations. For such discrete cascades, the exact distribution of M can be found numerically. We evaluate the sensitivity of the distribution of M to simplifying approximations, including independence of the measure among the cascade tiles and replacement of the dressing factor by a random variable with the same distribution type as the cascade generator. We also examine how the distribution of M varies with the dimensionality of the support and the multiplicity of the cascade. Of these factors, dependence of the measure among different cascade tiles has the highest effect on the distribution of M. This effect comes mainly from long-range dependence. We use these findings to propose a simple approximation to the distribution of M and give charts to implement the approximation for beta-lognormal cascades.

scholarly journals Representations of hermite processes using local time of intersecting stationary stable regenerative sets

2020 ◽  
Vol 57 (4) ◽  
pp. 1234-1251
Author(s):  
Shuyang Bai
Keyword(s):  
Long Range ◽  
Local Time ◽  
Limit Theorems ◽  
Local Times ◽  
Long Range Dependence ◽  
Random Interval ◽  
Stationary Increments ◽  
Self Similar ◽  
Regenerative Sets

AbstractHermite processes are a class of self-similar processes with stationary increments. They often arise in limit theorems under long-range dependence. We derive new representations of Hermite processes with multiple Wiener–Itô integrals, whose integrands involve the local time of intersecting stationary stable regenerative sets. The proof relies on an approximation of regenerative sets and local times based on a scheme of random interval covering.

scholarly journals On nonparametric regression for bivariate circular long-memory time series

2021 ◽  
Author(s):  
Jan Beran ◽  
Britta Steffens ◽  
Sucharita Ghosh
Keyword(s):  
Time Series ◽  
Nonparametric Regression ◽  
Long Range ◽  
Long Memory ◽  
Regression Function ◽  
Confidence Bands ◽  
Long Range Dependence ◽  
Asymptotic Results ◽  
Memory Time ◽  
Circular Time Series

AbstractWe consider nonparametric regression for bivariate circular time series with long-range dependence. Asymptotic results for circular Nadaraya–Watson estimators are derived. Due to long-range dependence, a range of asymptotically optimal bandwidths can be found where the asymptotic rate of convergence does not depend on the bandwidth. The result can be used for obtaining simple confidence bands for the regression function. The method is illustrated by an application to wind direction data.

Long range dependence in stock market returns

2006 ◽  
Vol 16 (18) ◽  
pp. 1331-1338 ◽  
Author(s):  
Christos Christodoulou-Volos ◽  
Fotios M. Siokis
Keyword(s):  
Stock Market ◽  
Long Range ◽  
Long Range Dependence ◽  
Market Returns ◽  
Stock Market Returns

scholarly journals Bootstrap testing for discontinuities under long-range dependence

2012 ◽  
Vol 105 (1) ◽  
pp. 322-347 ◽  
Author(s):  
Jan Beran ◽  
Yevgen Shumeyko
Keyword(s):  
Long Range ◽  
Long Range Dependence
Sign in / Sign up

Export Citation Format

Share Document