Preservation of strong mixing and weak dependence under renewal - PowerPoint PPT Presentation

Introduction Ψ -Weak Dependence Preservation Preservation of strong mixing and weak dependence under renewal sampling Imma Valentina Curato based on a joint work with D. Brandes and R. Stelzer Institute of Mathematical Finance, University of Ulm 2nd ISM-UUlm Joint Workshop, 10th October 2019

Introduction Ψ -Weak Dependence Preservation Sampling Schemes Given a strictly-stationary data generating process X = ( X t ) t ∈ R

Introduction Ψ -Weak Dependence Preservation Sampling Schemes Given a strictly-stationary data generating process X = ( X t ) t ∈ R Equidistant sampling → General asymptotic theory for sample moment statistics, i.e. when X is strong mixing (Bradley, 2007).

Introduction Ψ -Weak Dependence Preservation Sampling Schemes Given a strictly-stationary data generating process X = ( X t ) t ∈ R Equidistant sampling → General asymptotic theory for sample moment statistics, i.e. when X is strong mixing (Bradley, 2007). Random Sampling → ?

Introduction Ψ -Weak Dependence Preservation Renewal Sampling Definition Let τ = ( τ i ) i ∈ Z \{ 0 } be a non-negative sequence of i.i.d. random variable with distribution function µ such that µ ( { 0 } ) < 1 . For i ∈ Z , we define ( T i ) i ∈ Z as  i  � i ∈ N ∗ ,  τ j ,     j =1 T 0 := 0 T i := and (1) − 1  � − i ∈ N ∗ . − τ j ,      j = i The sequence ( T i ) i ∈ Z is called a renewal sampling sequence.

Introduction Ψ -Weak Dependence Preservation Renewal Sampling Let X = ( X t ) t ∈ R a stationary process with values in R d -valued and let ( T i ) i ∈ Z be a sequence of random times as defined in (1) and independent of X , we define the sequence Y = ( Y i ) i ∈ Z as a stochastic process with values in R d +1 given by � X T i � Y i = . τ i

Introduction Ψ -Weak Dependence Preservation Renewal Sampling Target We show that if

Introduction Ψ -Weak Dependence Preservation Renewal Sampling Target We show that if X is strictly-stationary and satisfies a weak dependent property

Introduction Ψ -Weak Dependence Preservation Renewal Sampling Target We show that if X is strictly-stationary and satisfies a weak dependent property X admits exponential or power decaying weak dependent coefficients Then, we can apply to Y the existing asymptotic theory for equidistant sampling.

Introduction Ψ -Weak Dependence Preservation Definition Let T a non empty index set equipped with a distance d and X = ( X t ) t ∈ T a process with values in R d . The process is called a Ψ -weak dependent process if there exists a function Ψ and a sequence of coefficients ι = ( ι ( r )) r ∈ R + converging to zero satisfying | Cov ( F ( X i 1 , . . . , X i u ) , G ( X j 1 , . . . , X j v )) | ≤ c Ψ( F, G, u, v ) ι ( r ) (2) for all ( u, v ) ∈ N ∗ × N ∗ ;   r ∈ R + ;    ( i 1 , . . . , i u ) ∈ T u and ( j 1 , . . . , j v ) ∈ T v ,  such that r = min { d ( i l , j m ) : 1 ≤ l ≤ u, 1 ≤ m ≤ v }    for functions F : ( R d ) u → R and G : ( R d ) v → R   and where c is a constant independent of r . ι is called the sequence of the weak dependent coefficients.

Introduction Ψ -Weak Dependence Preservation η -weak dependence Let F u = G u be classes of bounded and Lipschitz functions with Ψ( F, G, u, v ) = uLip ( F ) � G � ∞ + vLip ( G ) � F � ∞ , then ι corresponds to the η -coefficients defined in Doukhan and Louhichi, (1999).

Introduction Ψ -Weak Dependence Preservation η -weak dependence Let F u = G u be classes of bounded and Lipschitz functions with Ψ( F, G, u, v ) = uLip ( F ) � G � ∞ + vLip ( G ) � F � ∞ , then ι corresponds to the η -coefficients defined in Doukhan and Louhichi, (1999). Also λ -weak dependence and κ -weak dependence , as defined in Doukhan and Wintenberger (2007), are encompassed by (2).

Introduction Ψ -Weak Dependence Preservation BL-dependence If, instead, Ψ( F, G, u, v ) = min ( u, v ) Lip ( F ) Lip ( G ) , then ι corresponds to the BL-weak dependent coefficients defined in Bulinski and Sashkin (2005).

Introduction Ψ -Weak Dependence Preservation θ -weak dependence Let F u be the class of bounded functions and G v the class of bounded and Lipschitz functions with Ψ( F, G, u, v ) = v � F � ∞ Lip ( G ) , then ι corresponds to the θ -coefficients defined in Dedecker and Doukhan, (2003).

Introduction Ψ -Weak Dependence Preservation Strong mixing Proposition (Brandes, C., Stelzer) Let X = ( X t ) t ∈ T be a process with values in R d and F u = G u are classes of bounded functions. X is α -mixing (Rosenblatt, 1956) if and only if there exists a sequence ( ι ( r )) r ∈ R + converging to zero such that (2) is satisfied for Ψ( F, G, u, v ) = � F � ∞ � G � ∞ .

Introduction Ψ -Weak Dependence Preservation Weak dependent coefficients of the renewal sampled process Theorem (Brandes, C., Stelzer) Let Y = ( Y i ) i ∈ Z be a R d +1 -valued process with X = ( X t ) t ∈ R being strictly-stationary and Ψ -weak dependent with coefficients ι = ( ι ( r )) r ∈ R + . Then, it exists a sequence ( I ( n )) n ∈ N ∗ satisfying | Cov ( F ( Y i 1 , . . . , Y i u ) , G ( Y j 1 , . . . , Y j v )) | ≤ C Ψ( F, G, u, v ) I ( n ) where C is a constant independent of n and Ψ satisfies the same weak dependence conditions of the data generating process X . Moreover, � R + ι ( r ) µ ∗ n ( dr ) , I ( n ) = with µ ∗ n the n-fold convolution of µ .

Introduction Ψ -Weak Dependence Preservation Ψ -weak dependence of the renewal sampled process Exponential decay If X is a Ψ -weak dependent process with coefficients ι ( r ) = c e − γr with γ > 0 and µ a distribution function in R + , then Y is Ψ -weak dependent with coefficients 1 � − n � �� I ( n ) = C , L µ ( γ ) � R + e − γr µ ( dr ) is the Laplace transform of the where L µ ( γ ) = distribution function µ .

Introduction Ψ -Weak Dependence Preservation Ψ -weak dependence of the renewal sampled process Power decay If X is a Ψ -weak dependent process with power decaying coefficients such that ι ( r ) = cr − γ for γ > 0 . Then, the process Y is Ψ -weak dependent with coefficients I ( n ) ≤ Cn − γ for large n .

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