Strongly consistent autoregressive predictors in abstract Banach spaces
M. D. Ruiz-Medina J. Álvarez-Liébana
math.ST stat.TH 62M10, 62M15, 62M20, 60G25, 46C05, 46C07, 62-02, 62J05, 65N25, 97K10












Strongly consistent autoregressive predictors in abstract Banach spaces
    M. Dolores Ruiz–Medina^1 and Javier Álvarez-Liébana^1
    December 30, 2023
=======================================================================



^1 Department of Statistics and O. R., University of Granada, Spain. 



E-mail: javialvaliebana@ugr.es




















This work derives new results on strong consistent estimation and prediction for autoregressive processes of order
1 in a separable Banach space B. The consistency results are obtained for the componentwise estimator of the autocorrelation
operator in the norm of the space ℒ(B) of bounded linear operators on B. The strong consistency of
the associated plug–in predictor then follows in the B-norm. A Gelfand triple is defined through the Hilbert space
constructed in Kuelbs' Lemma <cit.>. A Hilbert–Schmidt embedding introduces the Reproducing Kernel Hilbert space
(RKHS), generated by the autocovariance operator, into the Hilbert space conforming the Rigged Hilbert space structure.
This paper extends the work of <cit.> and <cit.>.



In press, manuscript accepted  in Journal of Multivariate Analysis















§ INTRODUCTION



In the last few decades, there exists a growing interest on the statistical analysis of high–dimensional data, from the  Functional Data Analysis (FDA) perspective.  The   book by  <cit.>  provides an  overview on FDA techniques, extended  from the multivariate data context, or specifically formulated for the FDA framework. 
 The   monograph by  <cit.> introduces  functional analytical tools usually  applied in    the estimation of  random elements in function spaces.  
The book by  <cit.> is mainly concerned with inference based on second order statistics. A central topic in this book is the analysis of functional data,  displaying  dependent  structures    in time and space. The methodological  survey paper by
  <cit.>, on 
the state of the art in FDA,  discusses central topics in FDA. 
Recent advances  in the statistical analysis of 
 high–dimensional data, from the parametric, semiparametric and nonparametric FDA frameworks,  are collected in the Special Issue  by  <cit.>.

Linear time series models traditionally arise  for processing temporal linear correlated data. In the FDA context, the monograph by  <cit.> introduces  linear functional time series theory. The RKHS, generated by the autocovariance operator,   plays a crucial role in the estimation approach presented in this monograph. In particular,  the eigenvectors of the autocovariance operator are considered for projection (see also <cit.>). Its  empirical version is computed, when they  are unknown. The resulting  plug–in predictor is obtained as a linear functional of the observations, based  on the empirical approximation of the autocorrelation operator. This approach exploits the Hilbert space structure, and its extension to the metric space context, and, in particular, to the Banach space context, requires to deriving a relationship (continuous embeddings) between the  Banach space norm, and the RKHS norm, induced by   the autocovariance operator, in contrast with the nonparametric  regression approach  for  functional prediction (see, for instance,  <cit.>, where asymptotic normality is derived). Specifically, in the nonparametric approach, a linear combination of the observed response values  is usually considered. That is the case of the nonparametric local–weighting–based  approach, involving  weights defined from an  isotropic  
kernel,  depending on the metric or semi–metric of the space, where the regressors take their values (see, for example, <cit.>; see also <cit.>, in the functional time series framework). The nonparametric  approach is then more flexible regarding the structure of the  space where the functional values of the regressors lie (usually a semi–metric space is considered). However, some computational drawbacks are present in its implementation,  requiring the resolution of several selection problems. For instance,  a choice of the  smoothing parameter, and  the kernel involved, in the definition of the weights, should be performed.
   Real–valued covariates were incorporated in the novel semiparametric kernel–based proposal by  <cit.>, involving  an extension to the functional partial linear time series framework (see also    <cit.>). <cit.> also adopt a semi–parametric approach in their formulation of a two–terms Partitioned Functional Single Index Model.  <cit.> exploits the alternative provided by semi–metrics to avoid the curse of infinite
dimensionality of some functional  estimators. 

On the other hand, in a parametric linear framework,    <cit.> introduced functional time series models in Banach spaces. In particular, strong mixing conditions and  the absolute regularity   of Banach–valued autoregressive processes  have  been studied  in  <cit.>. 
Empirical
estimators for Banach–valued autoregressive processes are  studied  in <cit.>, where,  under some regularity conditions, and for the case of  orthogonal innovations, 
the empirical mean is proved to be  asymptotically optimal, with respect to almost surely  (a.s.) convergence,  and convergence of
order two.  The empirical autocovariance operator
was also interpreted as a sample mean of an autoregressive process in  a suitable space of linear operators. The extension of these results to the case of weakly dependent innovations is obtained in  <cit.>.  
 A strongly–consistent  sieve estimator of the autocorrelation operator of a Banach–valued  autoregressive process is considered in <cit.>.  Limit theorems for a seasonality estimator, in the case of  Banach autoregressive perturbations, are formulated in 
 <cit.>. Confidence regions for the periodic seasonality function, in the Banach space
of continuous functions, is obtained as well.
An approximation of Parzen's optimal predictor,
in the RKHS framework, is applied in  <cit.>,
for prediction of temporal stochastic process in Banach spaces. The existence and uniqueness of an almost surely strictly periodically correlated solution, to the 
first order autoregressive  model in Banach spaces, is derived in  <cit.>. 
Under some regularity conditions, limit results are obtained 
for AR𝒟(1) processes in <cit.>, where 
𝒟=𝒟([0,1]) denotes  the Skorokhod space  of right–continuous      functions on [0,1], having limit to the left at each t∈ [0,1]. 
Conditions for the existence of  strictly stationary solutions of ARMA equations in Banach spaces, with independent and identically distributed noise innovations, are derived in 
 <cit.>.   

In the derivation of strong–consistency results for ARB(1) componentwise estimators and predictors,   <cit.> restricts his attention to the case  of the  Banach space   𝒞([0,1]) of continuous functions on [0,1], with the supremum norm. <cit.> considers  an ARB(1) context, for B being an arbitrary real separable Banach space,  under the construction of a Hilbert space  H, where B is 
 continuously embedded, as given in the Kuelbs's Lemma in <cit.>.
Under the existence of a continuous extension to
H of the  autocorrelation operator ρ∈ℒ(B),   <cit.> obtain the 
strong-consistency of the formulated  componentwise estimator of  
ρ ,   and of its associated   plug–in predictor, in the norms of  ℒ(H),  and   H,
respectively. 



 functional data in nuclear spaces, arising, for example, in the observation of the solution to stochastic fractional and multifractional linear pseudodifferential equations (see, for example, <cit.>). The scales of Banach spaces constituted by  fractional Sobolev and Besov spaces play a central role in the context of nuclear spaces. Continuous (nuclear)  embeddings usually connect the elements of these scales (see, for example,  <cit.>).
In this paper,  a 
Rigged–Hilbert–Space structure is defined, involving the separable Hilbert space 
  H, appearing in the construction of the Kuelbs's Lemma in <cit.>. A key assumption, here, is the existence of a   continuous (Hilbert–Schmidt) embedding  introducing the RKHS, associated with the autocovariance operator of the ARB(1) process, into the Hilbert space generating the Gelfand triple, equipped with a finer topology than the B–topology.  Under this scenario,    
  strong–consistency results are derived, in the space ℒ(B) of bounded linear operators on B,   considering an abstract separable Banach space framework.
    
  
  
The outline of this paper is as follows. Notation and preliminaries   are fixed in Appendix <ref>. Fundamental assumptions and some key lemmas are formulated in Appendix <ref>, and proved in Appendix <ref>. The main result of this paper on strong–consistency  is derived in Appendix <ref>. 
Appendix <ref> provides some examples. Final comments  on our approach can be found in Appendix <ref>. The Supplementary Material provides in Appendix <ref> illustrates numerically the results derived in Appendix <ref>, under  the scenario described in Appendix <ref>, in a simulation study.










§ PRELIMINARIES






Let (B, ·_B ) be a  real separable Banach space, with the norm ·_B, and let ℒ^2_B(Ω,𝒜, 𝒫), the space of zero-mean B–valued random variables X such that 
    √(∫_BX_B^2d𝒫)<∞.
 
 Consider  X = { X_n,  n ∈ℤ} to be a zero–mean B–valued stochastic process  on the basic probability space ( Ω, 𝒜, 𝒫) satisfying  (see <cit.>):

    X_n = ρ( X_n-1) + ε_n,   n ∈ℤ,   ρ∈ℒ(B),

where ρ denotes  the autocorrelation operator of X. In equation (<ref>), the B–valued innovation  process ε = {ε_n,  n ∈ℤ} on ( Ω, 𝒜, 𝒫) is assumed to be strong white noise,  uncorrelated with the random initial condition.    Thus, ε is a zero–mean Banach–valued stationary process, with independent and identically distributed components, and with σ_ε^2 =  E{ε_n _B^2} < ∞ ,  for each n∈ℤ. Assume that there exists an integer j_0 ≥ 1 such that 

    ρ^j_0_ℒ(B ) < 1.


Then, 
equation (<ref>) admits an unique strictly stationary solution with σ_X^2= E{ X_n _B^2} < ∞; i.e., belonging to ℒ^2_B(Ω,𝒜, 𝒫),    given by
X_n = ∑_j=0^∞ρ^j ( ε_n-j),  for each n ∈ℤ (see  <cit.>).    
Under (<ref>), the autocovariance operator C of an ARB(1) process X is defined from the autocovariance operator of X_0∈ℒ_B^2(Ω, 𝒜, 𝒫),  
  as
 
    C ( x^∗)=   E{ x^∗(X_0)X_0},    x^∗∈ B^∗.


 The cross–covariance operator  D is given by

    D ( x^∗)  =  E{
    x^∗(X_0)X_1},     x^∗∈ B^∗.
 


Since C is assumed to be a nuclear operator,  there exists a sequence { x_j,  j ≥ 1 }⊂ B such that,  for every x^∗∈ B^∗ (see <cit.>):


    C( x^∗) = ∑_j=1^∞  x^∗( x_j ) x_j,   ∑_j=1^∞ x_j _B^2 < ∞.



 D is also assumed to be a nuclear operator.  Then, there exist  sequences { y_j,  j ≥ 1 }⊂ B and { x_j^∗∗,  j ≥ 1 }⊂ B^∗∗ 
such that, for every x^∗∈ B^∗,

    D( x^∗) = ∑_j=1^∞x_j^∗∗
    (x^∗)y_j,  ∑_j=1^∞ x_j^∗∗_B^∗∗ y_j < ∞,

(see  <cit.>). Empirical estimators of C and D are respectively 
 given by (see <cit.>),
  for n≥ 2, 
    C_n ( x^∗) = 1/n∑_i=0^n-1  x^∗( X_i ) (X_i ),    D_n ( x^∗) = 1/n-1∑_i=0^n-2  x^∗( X_i ) (X_i+1),    x^∗∈ B^∗.
 


 <cit.>, now formulated,  plays a key role in our approach.
 
 


If B is a real separable Banach space with norm ·_B,
then, there exists an inner product ⟨·,·⟩_H on B such that the norm ·_H, generated
by ⟨·, ·⟩_H , is weaker than ·_B. The completion of B under the   norm ·_H defines  the Hilbert space H,   where B is continuously embedded.




 Denote by 
{ x_n,  n ∈ℕ}⊂ B, a dense sequence in B, and by   { F_n,  n ∈ℕ}⊂ B^∗ a sequence of bounded linear functionals on B, satisfying

    F_n ( x_n ) =  x_n _B,    F_n  =  1,


such that  

    x _B  = sup_n ∈ℕ| F_n (x) 
    |,   x ∈ B.


The inner product  ⟨·,·⟩_H, and its associated norm,  in Lemma <ref>, is defined by 

    ⟨ x,y ⟩_H   =   ∑_n=1^∞ t_n F_n (x) F_n (y),    x,y ∈H,
    
     x _H^2   =   ∑_n=1^∞ t_n {F_n (x) }^2 ≤ x _B^2,    x ∈ B,
 
where  { t_n,  n ∈ℕ} is a sequence of positive numbers  such that  ∑_n=1^∞ t_n = 1. 






§ MAIN ASSUMPTIONS AND PRELIMINARY RESULTS




In view of Lemma <ref>, for every n∈ℤ, X_n∈ B↪H satisfies a.s. 


    X_nH=∑_j=1^∞⟨ X_n, v_j⟩_H v_j,    n∈ℤ,

for any orthonormal basis {v_j, j≥ 1} of H.  The 
 trace autocovariance operator 
    C= E{(∑_j=1^∞⟨ X_n, v_j⟩_H v_j)⊗(∑_j=1^∞⟨ X_n, v_j⟩_H v_j)}
 of the extended ARB(1) process is a trace operator in H, admitting a diagonal spectral representation, in terms of its eigenvalues 
 {C_j, j≥ 1} and eigenvectors {ϕ_j, j≥ 1}, that provide an orthonormal system in H.  
Summarizing, in the subsequent developments, the following identities in H will be considered, for the extended version of ARB(1) process X. For each f,h∈H, 


    C(f)H=      ∑_j=1^∞C_j⟨ f,ϕ_j⟩_Hϕ_j
    
    D(h)H=      ∑_j=1^∞∑_k=1^∞⟨ 
    D(ϕ_j),ϕ_k⟩_H⟨ h,ϕ_j⟩_Hϕ_k
    
    C_n(f)H a.s.=      ∑_j=1^n C_n,j⟨ f,ϕ_n,j⟩_Hϕ_n,j
    
    C_n,ja.s.=      1/n∑_i=0^n-1 X_i,n,j^2,  X_i,n,j = ⟨ X_i, ϕ_n,j⟩_H,  C_n(ϕ_n,j)H a.s.=C_n,jϕ_n,j
    
    D_n(h)H a.s.=      ∑_j=1^∞∑_k=1^∞⟨ D_n(ϕ_n,j), ϕ_n,k⟩_H⟨ h,ϕ_n,j⟩_Hϕ_n,k,

where, for n ≥ 2, {ϕ_n,j,  j ≥ 1 }  is a complete orthonormal  system in H,
and 
    C_n,1≥ C_n,2≥…≥ C_n,n≥ 0= C_n,n+1=
    C_n,n+2=….




The following assumption plays a crucial role in the derivation of the main results in this paper.






Assumption A1. X_0_B is a.s. bounded, and    the eigenspace V_j, associated with 
C_j>0 in (<ref>) is 
one-dimensional for every j≥ 1.




Under Assumption A1, we can define the following quantities:

    a_1 = 2 √(2)1/C_1 - C_2,    a_j = 2 √(2)max(1/C_j-1 - C_j, 1/C_j - C_j+1),    j ≥ 2.




This assumption can be relaxed to  considering multidimensional eigenspaces by redefining the quantities a_j, for each j≥ 1, as the quantities c_j, for each j≥ 1, given in <cit.>.


 Assumption A2. Let k_n such that

    C_n,k_n>0,      k_n→∞,  k_n/n→ 0, n→∞.






Consider

    Λ_k_n=sup_1≤ j≤ k_n(C_j-C_j+1)^-1.

For n sufficiently 
large,

    k_n<  C_k_n^-1 < 1/C_k_n - C_k_n + 1 < a_k_n < Λ_k_n<∑_j=1^k_n
     a_j.

 


Assumption A3.  The following limit holds:

    sup_x∈ B;  x_B≤ 1ρ(x)-∑_j=1^k⟨ρ(x),ϕ_j⟩_Hϕ_j_B→ 0,   k→∞.




Assumption A4. { C_j, j≥ 1 } are such that the inclusion of ℋ(X) into  H^∗ is continuous;  i.e.,

    ℋ(X) ↪H^∗,

where ↪ denotes, as usual,  the continuous embedding, H^∗ the dual space of H and ℋ(X) the Reproducing Kernel Hilbert Space associated with C.




Let us consider the closed subspace H of B with the norm induced by
the inner product ⟨·,·⟩_H  defined as follows:

    H    =   {x∈ B;  ∑_n=1^∞{F_n(x)}^2<∞},   ⟨ f,g⟩_H=∑_n=1^∞ F_n(f)F_n(g),     f,g∈ H.


Then, H is continuously embedded into B, and the following remark provides the isometric isomorphism   established by the Riesz Representation Theorem between the spaces H and its dual H^∗.  





Let f^∗,g^∗∈H^∗, and f,g∈H, such that, for every n≥ 1, consider F_n(f^∗)= 
    √(t_n)F_n(f), F_n(g^∗)= 
    √(t_n)F_n(g),  and F_n(f)=√(t_n)F_n(f), F_n(g)=√(t_n)F_n(g), for certain f,g∈ H.
Then, the following identities hold:

    ⟨ f^∗, g^∗⟩_H^∗   =   ∑_n=1^∞1/t_nF_n(f^∗)F_n(g^∗)=∑_n=1^∞1/t_n√(t_n)√(t_n)F_n(f)F_n(g)=⟨f,g⟩_H
       =   ∑_n=1^∞t_nF_n(f)F_n(g)=⟨ f,g⟩_H.
   







Under Assumption A4,   the following continuous embeddings hold:

    ℋ(X) ↪H^∗↪  B^∗↪ H ↪ B↪H↪ [ℋ(X)]^∗,

where

    H   =   {x∈ B;  ∑_n=1^∞
    t_n{F_n(x)}^2<∞},   ⟨ f,g⟩_H=∑_n=1^∞t_n F_n(f)F_n(g), f,g∈H
    
    H    =   {x∈ B;  ∑_n=1^∞{F_n(x)}^2<∞},   ⟨ f,g⟩_H = ∑_n=1^∞ F_n(f)F_n(g), f,g∈ H
    H^∗   =   {x∈ B;  ∑_n=1^∞1/t_n{F_n(x)}^2<∞},   ⟨ f,g⟩_H^∗=∑_n=1^∞1/t_n F_n(f)F_n(g), f,g∈H^∗
    ℋ(X)   =   {x∈H;  ⟨ C^-1(x),x⟩_H<∞},  
    ⟨ f,g⟩_ℋ(X)   =   ⟨ C^-1(f),g⟩_H, f,g∈ C^1/2(H)
    
    [ℋ(X)]^∗   =   {x∈H;  ⟨ C(x),x⟩_H<∞}
    ⟨ f,g⟩_[ℋ(X)]^∗   =   ⟨ C(f),g⟩_H f,g∈ C^-1/2(H).





 Let us consider  the following inequalitites, for each x ∈ B,:


    x_H   =   √(∑_j=1^∞t_n{F_n(x)}^2)≤x_B=sup_n≥ 1|F_n(x)|,
    x_B   =   sup_n≥ 1|F_n(x)|≤√(∑_n=1^∞{F_n(x)}^2)=x_H≤∑_n=1^∞|F_n(x)|=x_B^∗,
    
    x_B^∗   =   ∑_n=1^∞|F_n(x)|≤√(∑_n=1^∞1/t_n{F_n(x)}^2)=x_H^∗.
 
 
Under Assumption A4 (see also Remark <ref>), for every  f∈ C^1/2(H)=ℋ(X),   

    f_ℋ(X)=√(⟨ C^-1(f),f⟩_H)≥f_H^∗=√(∑_n=1^∞1/t_n{F_n(x)}^2).

From equations (<ref>)–(<ref>), the  inclusions in  (<ref>) are continuous. 

▪




It is well–known that {ϕ_j, j≥ 1} is also an orthogonal system in ℋ(X).
Futhermore, under Assumption A4, from Lemma <ref>,

    {ϕ_j, j≥ 1}⊂ℋ(X)↪H^∗↪  B^∗↪ H.


Therefore, from  equation (<ref>), for every j≥ 1,


    ϕ_j_H^2=∑_m=1^∞{F_m(ϕ_j)}^2<∞.


 The following assumption is now considered on the norm (<ref>):



Assumption A5.
The continuous embedding i_ℋ(X),H: ℋ(X)↪ H belongs to the trace class. That is, 
   
    ∑_j=1^∞ϕ_j_H^2<∞.

  



 Let { F_m, m≥ 1} be defined as in Lemma  <ref>. Assumption A5 leads to 

    ∑_j=1^∞⟨ i_ℋ(X),H(ϕ_j),ϕ_j⟩_H=∑_j=1^∞∑_m=1^∞{F_m(ϕ_j)}^2=∑_m=1^∞N_m<∞ ,

where, in particular, from  equation (<ref>),

    N_m   =   ∑_j=1^∞{F_m(ϕ_j)}^2
    <∞,   sup_m≥ 1N_m=N<∞
    
    V    =   sup_j ≥ 1ϕ_j _B ≤∑_j=1^∞∑_m=1^∞{F_m(ϕ_j)}^2 <∞.


The following preliminary results are considered from <cit.>).





 
Under Assumption A1,  the following identities hold, for any standard 
ARH(1) process (e.g., the extension to H of ARB(1) process X satisfying equation (<ref>)), 

    C_n - C _𝒮(H) = 𝒪((ln(n) /n)^1/2) a.s.,    D_n - D _𝒮(H) = 𝒪((ln(n) /n)^1/2) a.s.,

where  ·_𝒮(H) is the  norm in the Hilbert space 𝒮(H) of  Hilbert–Schmidt operators on H; i.e., the subspace of compact operators 𝒜 such that 
    ∑_j=1^∞⟨𝒜^∗𝒜(φ_j),φ_j⟩_H<∞,
 for any orthonormal basis {φ_j, j≥ 1} of  H.  






Under Assumption A1, let  {C_j,  j≥ 1} and  {C_n,j,  j≥ 1}  in (<ref>)– (<ref>), respectively. Then,

    (n/ln(n))^1/2sup_j ≥ 1| C_n,j- C_j | ⟶ 0 a.s.,    n→∞.






 
(See details in <cit.>) Under Assumption A1,  
consider Λ_k_n in equation (<ref>) satisfying 
 
    Λ_k_n=o((n/ln(n))^1/2),    n→∞.
   Then, 

    sup_1≤ j≤ k_nϕ_n,j^'-ϕ_n,j_H⟶ 0 a.s.,   n→∞,

 where, for j ≥ 1, and  n≥ 2, 
 
    ϕ_n,j^'=  sgn⟨ϕ_n,j , ϕ_j⟩_Hϕ_j,    sgn⟨ϕ_n,j, ϕ_j ⟩_H= 1_⟨ϕ_n,j, ϕ_j ⟩_H≥ 0-1_⟨ϕ_n,j, ϕ_j ⟩_H< 0,
  with   1_· being the indicator function.




An upper bound for c_B× B=∑_j=1^∞C_jϕ_j⊗ϕ_j_B× B is now obtained.





Under Assumption A5, the following inequality holds: 

    c  _B × B
     =
     sup_n,m ≥ 1| C ( F_n ) (F_m ) |
     ≤ N C _ℒ(H),
 where N  has been introduced in equation (<ref>), ℒ(H) denotes the space of bounded linear operators on H, and ·_ℒ(H) the usual uniform norm on such a space.





Let us consider the following notation.


    cH⊗H=      ∑_j=1^∞C_jϕ_n,j^'⊗ϕ_n,j^'H⊗H=∑_j=1^∞C_jϕ_j⊗ϕ_j,   c_nH⊗H=∑_j=1^∞C_n,jϕ_n,j⊗ϕ_n,j.
    
    c-c_nH⊗H=      ∑_j=1^∞C_jϕ^'_n,j⊗ϕ^'_n,j- ∑_j=1^∞C_n,jϕ_n,j⊗ϕ_n,j







From Lemma <ref>, for n sufficiently large, 
there exist positive constants K_1 and K_2 such that 

    K_1⟨ C(φ ),φ⟩_H≤⟨ C_n(φ ),φ⟩_H≤ K_2⟨ C(φ ),φ⟩_H,  ∀φ∈H.

In particular, for every x∈ℋ(X)=C^1/2(H), considering  n sufficiently large,

    1/K_1⟨ C^-1(x),x⟩_H≥⟨ C_n^-1(x),x⟩_H≥1/K_2⟨ C^-1(x),x⟩_H
          ⇔ 1/K_1x_ℋ(X)^2≥⟨ C_n^-1(x),x⟩_H≥1/K_2x_ℋ(X)^2.

Equation (<ref>) means that,   for n sufficiently large, the norm of the RKHS ℋ(X) of X is equivalent to the norm of the RKHS generated by C_n, with spectral kernel c_n given in (<ref>). 






Under Assumptions A1 and  A4–A5, let us consider Λ_k_n in (<ref>) satisfying 

    √(k_n)Λ_k_n=o(√(n/ln(n))),   n→∞,
 
where k_n has been introduced in Assumption A2.
  The following a.s.  inequality then holds:

    c-c_n_B × B≤max(N, √(N))[
    C-C_n_ℒ(H).
          .+2max(√(C_ℒ(H)),√(C_n_ℒ(H)))[sup_l≥ 1sup_m≥ 1|F_l(ϕ^'_n,m)|].
          ×√(k_n8Λ_k_n^2C_n-C_ℒ(H)^2+∑_m=k_n+1^∞ϕ_n,m-
    ϕ_n,m^'_H^2).

Therefore, c-c_n_B × B→_a.s. 0, as n→∞.






For a standard ARB(1) process satisfying equation  (<ref>),  under Assumptions A1 and A3–A5,    for n sufficiently large,

    sup_1≤ j≤ k_nϕ_n,j - ϕ_n,j^'_B
          ≤2/C_k_n[max(N, √(N))[
    C-C_n_ℒ(H)..
          ..+2max(√(C_ℒ(H)),√(C_n_ℒ(H)))(sup_l≥ 1sup_m≥ 1|F_l(ϕ^'_n,m)|)..
          ..×√(k_n8Λ_k_n^2C_n-C_ℒ(H)^2+∑_m=k_n+1^∞ϕ_n,m-
    ϕ_n,m^'_H^2)].
          .+sup_1≤ j≤ k_nϕ_n,j - ϕ_n,j^'_HNC_𝒮(H)+ 
    VC-C_n_𝒮(H)]

Under (<ref>),  
    sup_1≤ j≤ k_nϕ_n,j - ϕ_n,j^'_B ⟶ 0 a.s.,    n→∞.
 






Under Assumption A3, if 
    ∑_j=1^k_nϕ_n,j-ϕ_n,j^'_B→_a.s. 0,,    n→∞,
 then 

    sup_x∈ B; x_B≤ 1ρ(x)-∑_j=1^k_n⟨ρ(x),ϕ_n,j⟩_Hϕ_n,j_B⟶ 0 a.s.,   n→∞ .







Under the conditions of Lemma <ref>,  if 
    k_n^3/2Λ_k_n=o(√(n/ln(n))),  ∑_m=k_n+1^∞ϕ_n,m-
    ϕ_n,m^'_H^2=o(1/k_n),n→∞,

then, equation (<ref>) holds.




Let us know consider the projection operators

    Π^k_n( x )   =   ∑_j=1^k_n⟨ x, ϕ_n,j⟩_Hϕ_n,j,  Π^k_n(x)=∑_j=1^k_n⟨ x, ϕ_n,j^'⟩_Hϕ_n,j^',   x∈ B⊂H.




 

Under the conditions of Remark <ref>,   let 
    Π^k_nρΠ^k_n=∑_j=1^k_n∑_p=1^k_n⟨ρ(ϕ_n,j),ϕ_n,p⟩_Hϕ_n,j⊗ϕ_n,p,
 then 
 
    sup_x∈ B; x_B≤ 1ρ(x)-∑_j=1^k_n∑_p=1^k_n⟨ x,ϕ_n,j⟩_H⟨ρ(ϕ_n,j),ϕ_n,p⟩_Hϕ_n,p_B⟶ 0 a.s., n→∞.






§ PROOFS OF LEMMAS





 §.§ Proof of Lemma <ref>



Applying the Cauchy–Schwarz's inequality, for every k,l≥ 1, 

    |C(F_k,F_l)|   =   |∑_j=1^∞C_jF_k(ϕ_j)F_l(ϕ_j)| ≤√(∑_j=1^∞C_j[F_k(ϕ_j)]^2∑_p=1^∞C_p[F_l(ϕ_p)]^2)
       ≤   sup_j ≥ 1|C_j|
    √(∑_j=1^∞[F_k(ϕ_j)]^2∑_p=1^∞[F_l(ϕ_p)]^2) = sup_j ≥ 1|C_j|√(N_kN_l),

where {F_n, n≥ 1} have been introduced in equation (<ref>), and satisfy (<ref>)–(<ref>).
Under Assumption A5, from equation  (<ref>), 
 
    c_B× B= sup_k,l ≥ 1|C(F_k,F_l)|≤sup_k,l ≥ 1sup_j ≥ 1|C_j|√(N_kN_l)
    =Nsup_j ≥ 1|C_j|=NC_ℒ(H).

 

▪








 §.§ Proof of Lemma <ref>




Let us first consider the following identities and inequalities:

    |C-C_n(F_k)(F_l)|
       =   | ∑_j=1^∞C_jF_k(ϕ_n,j^')F_l(ϕ_n,j^')-C_n,jF_k(ϕ_n,j)F_l(ϕ_n,j)|
       ≤   ∑_j=1^∞|C_j||F_k(ϕ_n,j^')||F_l(ϕ_n,j^')-F_l(ϕ_n,j)|
    
          
    +sup_j|C_j-C_n,j|
    |F_k(ϕ_n,j^')F_l(ϕ_n,j)|
          +|C_n,jF_l(ϕ_n,j)||F_k(ϕ_n,j^')-F_k(ϕ_n,j)|
       ≤   √(∑_j=1^∞C_j{F_k(ϕ_n,j^')}^2∑_j=1^∞C_j{F_l(ϕ_n,j^')-F_l(ϕ_n,j)}^2)
       +   sup_j≥ 1|C_j-C_n,j|√(∑_j=1^∞{F_k(ϕ_n,j^')}^2∑_j=1^∞{F_l(ϕ_n,j)}^2)
       +   √(∑_j=1^∞C_n,j{F_l(ϕ_n,j)}^2∑_j=1^∞C_n,j{F_k(ϕ_n,j^')-F_k(ϕ_n,j)}^2)
       ≤   √(N_k)√(∑_j=1^∞C_j{F_l(ϕ_n,j^')-F_l(ϕ_n,j)}^2)
       +   sup_j≥ 1|C_j-C_n,j|√(N_k)√(N_l)
       +   √(N_l)√(∑_j=1^∞C_n,j{F_k(ϕ_n,j^')-F_k(ϕ_n,j)}^2)
       ≤   max(N, √(N))[√(C_ℒ(H)∑_j=1^∞{F_l(ϕ_n,j^'-ϕ_n,j)}^2).
          . +C-C_n_ℒ(H).
          .+√(C_n_ℒ(H)∑_j=1^∞{F_k(
    ϕ_n,j^'-ϕ_n,j)}^2)]


    ≤   max(N, √(N))[C-C_n_ℒ(H).
          .+√(C_ℒ(H)∑_j=1^∞∑_m=1^∞{F_l(ϕ^'_n,m)}^2{⟨ϕ_n,j^',ϕ_n,m^'⟩_H-⟨ϕ_n,j,ϕ_n,m^'⟩_H}^2).
          .+√(C_n_ℒ(H)∑_j=1^∞∑_m=1^∞{F_k(ϕ^'_n,m)}^2{⟨ϕ_n,j^',ϕ_n,m^'⟩_H-⟨ϕ_n,j,ϕ_n,m^'⟩_H}^2)]
       =   max(N, √(N))[C-C_n_ℒ(H).
          .+√(C_ℒ(H)∑_m=1^∞{F_l(ϕ^'_n,m)}^2∑_j=1^∞{⟨ϕ_n,j^',ϕ_n,m^'⟩_H-⟨ϕ_n,j,ϕ_n,m^'⟩_H}^2).
          .+√(C_n_ℒ(H)∑_m=1^∞{F_k(ϕ^'_n,m)}^2∑_j=1^∞{⟨ϕ_n,j^',ϕ_n,m^'⟩_H-⟨ϕ_n,j,ϕ_n,m^'⟩_H}^2)]
       =   max(N, √(N))[C-C_n_ℒ(H).
          .+√(C_ℒ(H)∑_m=1^∞{F_l(ϕ^'_n,m)}^2∑_j=1^∞{⟨ϕ_n,j,ϕ_n,m⟩_H-⟨ϕ_n,j,ϕ_n,m^'⟩_H}^2).
          .+√(C_n_ℒ(H)∑_m=1^∞{F_k(ϕ^'_n,m)}^2∑_j=1^∞{⟨ϕ_n,j,ϕ_n,m⟩_H-⟨ϕ_n,j,ϕ_n,m^'⟩_H}^2)]
       =   max(N, √(N))[
    C-C_n_ℒ(H).
          .+√(C_ℒ(H)∑_m=1^∞{F_l(ϕ^'_n,m)}^2ϕ_n,m-ϕ_n,m^'_H^2).
          .+√(C_n_ℒ(H)∑_m=1^∞{F_k(ϕ^'_n,m)}^2ϕ_n,m-ϕ_n,m^'_H^2)]
       ≤   max(N, √(N))[
    C-C_n_ℒ(H).
          .+sup_m≥ 1|F_l(ϕ^'_n,m)|√(C_ℒ(H)∑_m=1^∞ϕ_n,m-ϕ_n,m^'_H^2).
          .+sup_m≥ 1| F_k(ϕ^'_n,m)|√(C_n_ℒ(H)∑_m=1^∞ϕ_n,m-ϕ_n,m^'_H^2)]

    ≤   max(N, √(N))[
    C-C_n_ℒ(H).
          .+max(√(C_ℒ(H)),√(C_n_ℒ(H))).
          .[sup_m≥ 1|F_l(ϕ^'_n,m)|+sup_m≥ 1|F_k(ϕ^'_n,m)|]
    √(∑_m=1^∞ϕ_n,m
    -ϕ_n,m^'_H^2)].

 
 Under Assumption A5, from equation (<ref>), 
    ∑_m=1^∞ϕ_n,m-ϕ_n,m^'_H^2<∞,   sup_m≥ 1|F_k(ϕ^'_n,m)|<∞ ,     k≥ 1.


 Thus, considering k_n, as given in Assumption A2,
 
    ∑_m=1^∞ϕ_n,m-
    ϕ_n,m^'_H^2
    =∑_m=1^k_nϕ_n,m-
    ϕ_n,m^'_H^2+∑_m=k_n+1^∞ϕ_n,m-
    ϕ_n,m^'_H^2
          ≤ k_nsup_1≤ m≤ k_nϕ_n,m-
    ϕ_n,m^'_H^2+∑_m=k_n+1^∞ϕ_n,m-
    ϕ_n,m^'_H^2
          ≤ k_n8Λ_k_n^2C_n-C_ℒ(H)^2+∑_m=k_n+1^∞ϕ_n,m-
    ϕ_n,m^'_H^2


From equation (<ref>), under 
Λ_k_n=o(√(n/ln(n))),

    k_n8Λ_k_n^2C_n-C_ℒ(H)^2≤ 
    k_n8Λ_k_n^2C_n-C_𝒮(H)^2→_a.s. 0,n→∞.


Under Assumption A5, 

    ∑_m=k_n+1^∞ϕ_n,m-
    ϕ_n,m^'_H^2→_a.s. 0,  
    n→∞.


From equations (<ref>)–(<ref>),
since, under Assumption A5, 
    sup_k≥ 1sup_m≥ 1|F_k(ϕ^'_n,m)|<∞,

we have c-c_n_B× B=sup_k,l|(C-C_n)(F_k)(F_l)|→_a.s. 0, as n→∞.

  ▪
 
 
 

 §.§ Proof of Lemma <ref>

 
 

Let us first consider the following a.s. equalities

    C_n,j( ϕ_n,j - ϕ_n,j^')    =    C_n ( ϕ_n,j) - C_n,jϕ_n,j^' = ( C_n - C ) ( ϕ_n,j) 
        +     C ( ϕ_n,j - ϕ_n,j^') + ( C_j - C_n,j) ϕ_n,j^'.


 From equation (<ref>), keeping in mind Assumption A2,

    ϕ_n,j - ϕ_n,j^'_B    ≤   1/C_n,j( C_n - C ) ( ϕ_n,j)_B + 1/C_n,jC ( ϕ_n,j - ϕ_n,j^')
    _B 
        +    1/C_n,j( C_j - C_n,j) ϕ_n,j^'_B =  1/C_n,j[S_1+S_2+S_3],  .


For n sufficiently large,  from Lemmas <ref> and <ref>, applying the Cauchy–Schwarz's inequality, for every j≥ 1,

    S_1   =   ( C_n - C ) ( ϕ_n,j)
    _B
       =   sup_m|∑_k=1^∞C_n,kF_m(ϕ_n,k)⟨ϕ_n,k,ϕ_n,j⟩_H-∑_k=1^∞C_kF_m(ϕ_n,k^')⟨ϕ_n,k^',ϕ_n,j⟩_H|
       =   sup_m|∑_k=1^∞∑_l=1^∞t_lF_l(ϕ_n,j){C_n,kF_m(ϕ_n,k)F_l(ϕ_n,k)-C_kF_m(ϕ_n,k^')F_l(ϕ_n,k^')}|
       =   sup_m|∑_l=1^∞t_lF_l(ϕ_n,j)∑_k=1^∞C_n,kF_m(ϕ_n,k)F_l(ϕ_n,k)-C_kF_m(ϕ_n,k^')F_l(ϕ_n,k^')|
       ≤   sup_m√(∑_l=1^∞t_l[F_l(ϕ_n,j)]^2)
          ×√(∑_l=1^∞t_l{∑_k=1^∞C_n,kF_m(ϕ_n,k)F_l(ϕ_n,k)-C_kF_m(ϕ_n,k^')F_l(ϕ_n,k^')}^2)
       ≤   ϕ_n,j_H√(∑_l=1^∞t_l)sup_m,l|∑_k=1^∞C_n,kF_m(ϕ_n,k)F_l(ϕ_n,k)-C_kF_m(ϕ_n,k^')F_l(ϕ_n,k^')|
    
          =c_n-c_B× B
          ≤max(N, √(N))[
    C-C_n_ℒ(H).
          .+2max(√(C_ℒ(H)),√(C_n_ℒ(H)))[sup_l≥ 1sup_m≥ 1|F_l(ϕ^'_n,m)|].

    ×√(k_n8Λ_k_n^2C_n-C_ℒ(H)^2+∑_m=k_n+1^∞ϕ_n,m-
    ϕ_n,m^'_H^2)


    S_2   =   C ( ϕ_n,j - ϕ_n,j^')
    _B=sup_m|∑_k=1^∞∑_l=1^∞
    t_lC_kF_m(ϕ_n,k^')F_l(ϕ_n,k^')
    F_l( ϕ_n,j - ϕ_n,j^')|
       ≤   sup_m√(∑_l=1^∞t_l{F_l( 
    ϕ_n,j - ϕ_n,j^')}^2)√(∑_l=1^∞t_l{∑_k=1^∞ C_kF_m(ϕ_n,k^')F_l(ϕ_n,k^')}^2)
       ≤   ϕ_n,j - ϕ_n,j^'_Hsup_m,l|∑_k=1^∞ C_kF_m(ϕ_n,k^')F_l(ϕ_n,k^')|
       =   ϕ_n,j - ϕ_n,j^'_Hc_B× B≤ϕ_n,j - ϕ_n,j^'_HNC_𝒮(H),

Under Assumption A3,

    S_3≤sup_j≥ 1|C_j - C_n,j| ϕ_n,j^'_B
    ≤  VC-C_n_ℒ(H)≤  VC-C_n_𝒮(H),


 In addition,  from Lemma <ref>, 

    C_n-C_𝒮(H)→_a.s. 0,    n→∞,
  and 
    C_n,j→_a.s. C_j,    n→∞.
  For ε=C_k_n/2, we can  find n_0 such that for n≥ n_0,


    C_n-C_ℒ(H)≤ε=C_k_n/2,  
          | C_n,k_n-C_k_n|≤ε≤C_n-C_ℒ(H)
           C_n,k_n≥ C_k_n-ε≥ C_k_n-C_n-C_ℒ(H)≥ C_k_n-C_k_n/2≥ C_k_n/2.




From equations (<ref>)–(<ref>), for n large enough such that equation (<ref>) holds,  the following almost surely inequalities are satisfied. For 1≤ j≤ k_n,

    sup_1≤ j≤ k_nϕ_n,j - ϕ_n,j^'_B 
          ≤1/C_n,k_n[max(N, √(N))[
    C-C_n_ℒ(H)..
          ..+2max(√(C_ℒ(H)),√(C_n_ℒ(H))){sup_l≥ 1sup_m≥ 1|F_l(ϕ^'_n,m)|}..
          ..×√(k_n8Λ_k_n^2C_n-C_ℒ(H)^2+∑_m=k_n+1^∞ϕ_n,m-
    ϕ_n,m^'_H^2)].
          .+sup_1≤ j≤ k_nϕ_n,j - ϕ_n,j^'_HNC_𝒮(H)+ 
    VC-C_n_𝒮(H)]
          ≤2/C_k_n[max(N, √(N))[
    C-C_n_ℒ(H)..
          ..+2max(√(C_ℒ(H)),√(C_n_ℒ(H))){sup_l≥ 1sup_m≥ 1|F_l(ϕ^'_n,m)|}..
          ..×√(k_n8Λ_k_n^2C_n-C_ℒ(H)^2+∑_m=k_n+1^∞ϕ_n,m-
    ϕ_n,m^'_H^2)].
          .+sup_1≤ j≤ k_nϕ_n,j - ϕ_n,j^'_HNC_𝒮(H)+ 
    VC-C_n_𝒮(H)]    a.s.


Hence, equation (<ref>) holds. The a.s. convergence to zero directly follows from Lemma <ref>, under (<ref>).

▪







 §.§ Proof of Lemma <ref>



The following  identities  are considered:

    ∑_j=1^k_n⟨ρ(x),ϕ_n,j⟩_Hϕ_n,j-∑_j=1^k_n⟨ρ(x),ϕ_n,j^'⟩_Hϕ_n,j^'
          =∑_j=1^k_n⟨ρ(x),ϕ_n,j⟩_H(ϕ_n,j-ϕ_n,j^')+
    ∑_j=1^k_n⟨ρ(x),
    ϕ_n,j-ϕ_n,j^'⟩_Hϕ_n,j^'.

From equation (<ref>), applying the Cauchy–Schwarz's inequality, under Assumption A3,
    sup_x∈ B; x_B≤ 1∑_j=1^k_n⟨ρ(x),ϕ_n,j⟩_Hϕ_n,j-∑_j=1^∞⟨ρ(x),ϕ_n,j^'⟩_Hϕ_n,j^'_B
          ≤sup_x∈ B; x_B≤ 1∑_j=1^k_nρ(x)_Hϕ_n,j_Hϕ_n,j-ϕ_n,j^'_B
          
    +ρ(x)_Hϕ_n,j-ϕ_n,j^'_Hϕ_n,j^'_B
          +sup_x∈ B; x_B≤ 1∑_j=k_n+1^∞⟨ρ(x),ϕ_n,j^'⟩_Hϕ_n,j^'_B
          ≤sup_x∈ B; x_B≤ 1ρ(x)_H(∑_j=1^k_nϕ_n,j-ϕ_n,j^'_B+ϕ_n,j-ϕ_n,j^'_Bsup_jϕ_n,j^'_B)
          +sup_x∈ B; x_B≤ 1∑_j=k_n+1^∞⟨ρ(x),ϕ_n,j^'⟩_Hϕ_n,j^'_B
          ≤sup_x∈ B; x_B≤ 1ρ_ℒ(H)x_H
    (1+V)∑_j=1^k_nϕ_n,j-ϕ_n,j^'_B
          +sup_x∈ B; x_B≤ 1∑_j=k_n+1^∞⟨ρ(x),ϕ_n,j^'⟩_Hϕ_n,j^'_B
          ≤ρ_ℒ(H)(1+V)∑_j=1^k_nϕ_n,j-ϕ_n,j^'_B
          +sup_x∈ B; x_B≤ 1∑_j=k_n+1^∞⟨ρ(x),ϕ_n,j^'⟩_Hϕ_n,j^'_B→ 0,    n→_a.s.∞.


▪


 







§ ARB(1) ESTIMATION AND PREDICTION. STRONG CONSISTENCY RESULTS





For every x∈ B⊂H, the following componentwise estimator ρ_k_n of ρ will be considered: 

    ρ_k_n (x) = ( Π^k_n D_n C_n^-1Π^k_n) (x) =  ( ∑_j=1^k_n1/C_n,j⟨ x, ϕ_n,j⟩_HΠ^k_n D_n(ϕ_n,j) ),

where Π^k_n has been introduced in equation (<ref>), and  C_n, C_n,j, ϕ_n,j and D_n  have been defined in equations  
(<ref>)–(<ref>), respectively.



 


Let X be, as before, a standard ARB(1) process. Under the conditions of Lemmas <ref> and <ref> (see Remark <ref>), for all η >0, 

    𝒫(ρ_k_n-ρ_ℒ(B)≥η)≤𝒦exp( -nη^2/Q_n),

where 
    Q_n=𝒪({C_k_n^-1k_n∑_j=1^k_na_j}^2),   n→∞.
 Therefore, if 

    k_nC_k_n^-1∑_j=1^k_na_j=o(√(n/ln(n))),   n→∞,

then, 

    ρ_k_n-ρ_ℒ(B)→_a.s 0,   n→∞.






For every x∈ B, such that x_B≤ 1, applying the triangle inequality, under Assumptions A1–A2,

    Π^k_nD_nC_n^-1Π^k_n(x)-Π^k_nρΠ^k_n(x)_B   ≤   Π^k_n(D_n-D)C_n^-1Π^k_n(x)_B
       +   Π^k_n
    (DC_n^-1-ρ)Π^k_n(x)_B
       =    S_1(x)+S_2(x).


Under Assumption A3,   considering inequality (<ref>),

    S_1(x)   =   Π^k_n(D_n-D)C_n^-1Π^k_n(x)_B
          ≤C_n,k_n^-1∑_j=1^k_n∑_p=1^k_n⟨ x,ϕ_n,j⟩_H⟨ (D_n-D)(ϕ_n,j),ϕ_n,p⟩_Hϕ_n,p_B
          ≤|C_n,k_n^-1|∑_j=1^k_n∑_p=1^k_n|⟨ x,ϕ_n,j⟩_H||⟨ (D_n-D)(ϕ_n,j),ϕ_n,p⟩_H|ϕ_n,p_B
          ≤ 2C_k_n^-1 k_nD_n-D_ℒ(H)∑_p=1^k_nϕ_n,p_B
          ≤ 2VC_k_n^-1k_n^2D_n-D_𝒮(H).


Furthermore, applying the triangle inequality,

    S_2(x)   =   Π^k_n
    (DC_n^-1-ρ)Π^k_n(x)_B
       ≤   Π^k_n
    DC_n^-1Π^k_n(x)-Π^k_nDC^-1Π^k_n(x)_B
       +   Π^k_nDC^-1Π^k_n(x)-
    Π^k_nρΠ^k_n(x)_B=S_21(x)+S_22(x).


Under Assumptions A1–A2, C^-1 and C_n^-1 are bounded on the subspaces generated by  {ϕ_j, j=1,…,k_n} and {ϕ_n,j, j=1,…,k_n}, respectively.      Consider now

    S_21(x)   =   Π^k_n
    DC_n^-1Π^k_n(x)-Π^k_nDC^-1Π^k_n(x)_B
       =   ∑_j=1^k_n∑_p=1^k_n1/C_n,j⟨ x,ϕ_n,j-ϕ_n,j^'⟩_H⟨ D(ϕ_n,j),ϕ_n,p⟩_Hϕ_n,p.
          .+∑_j=1^k_n∑_p=1^k_n(1/C_n,j-1/C_j)⟨ x,ϕ_n,j^'⟩_H⟨ D(ϕ_n,j),ϕ_n,p⟩_Hϕ_n,p.
          .+∑_j=1^k_n∑_p=1^k_n1/C_j⟨ x,
    ϕ_n,j^'⟩_H⟨ D(ϕ_n,j-ϕ_n,j^'),ϕ_n,p⟩_Hϕ_n,p_B
       ≤   ∑_j=1^k_n∑_p=1^k_n|1/C_n,k_n|ϕ_n,j-ϕ_n,j^'_HD_ℒ(H)ϕ_n,p_B
          +|1/C_n,j-1/C_j|D_ℒ(H)ϕ_n,p_B
          +|1/C_j|D_ℒ(H)ϕ_n,j-ϕ_n,j^'_Hϕ_n,p_B.


From <cit.>, for every j≥ 1, under  Assumption A1,

    ϕ_n,j-ϕ_n,j^'_H≤ 
    a_jC_n-C_ℒ(H),

where  { a_j,  j ≥ 1 } has been introduced in (<ref>), for j≥ 1.
Then, in equation (<ref>), considering again  inequality (<ref>),  keeping in mind that C_j^-1≤ a_j,  we obtain

    S_21(x)   ≤   
    4C_k_n^-1∑_p=1^k_nϕ_n,p_BD_ℒ(H)C_n-C_ℒ(H)∑_j=1^k_na_j
       ≤   
    4Vk_nC_k_n^-1D_ℒ(H)C_n-C_𝒮(H)∑_j=1^k_na_j.


Applying again the triangle and  the Cauchy–Schwarz inequalities, from  (<ref>),

    S_22   =   Π^k_nDC^-1Π^k_n(x)-
    Π^k_nρΠ^k_n(x)_B
       =   ∑_j=1^k_n∑_p=1^k_n⟨ x,ϕ_n,j^'-ϕ_n,j⟩_H⟨ρ(ϕ_n,j^'),ϕ_n,p⟩_Hϕ_n,p.
          . +⟨ x,ϕ_n,j⟩_H⟨ρ(ϕ_n,j^'-ϕ_n,j),ϕ_n,p⟩_Hϕ_n,p
       ≤   ∑_j=1^k_n∑_p=1^k_nx_Hϕ_n,j^'-ϕ_n,j_Hρ_ℒ(H)ϕ_n,j^'_Hϕ_n,p_Hϕ_n,p_B
          +x_Hϕ_n,j_Hρ_ℒ(H)ϕ_n,j^'-ϕ_n,j_Hϕ_n,p_Hϕ_n,p_B
       ≤    2ρ_ℒ(H)C_n-C_𝒮(H)(∑_p=1^k_nϕ_n,p_B)
    (∑_j=1^k_na_j)
       ≤   2Vρ_ℒ(H)C_n-C_𝒮(H)
    k_n∑_j=1^k_na_j.


From equations (<ref>)–(<ref>),

    sup_x∈ B; x_B≤ 1Π^k_nD_nC_n^-1Π^k_n(x)-Π^k_nρΠ^k_n(x)_B
          ≤ 2VC_k_n^-1k_n^2D_n-D_𝒮(H)
          +
    C_n-C_𝒮(H)2Vk_n∑_j=1^k_na_j( 2C_k_n^-1D_ℒ(H)+ρ_ℒ(H)).


 From equation (<ref>),  applying now <cit.>, one can get, for η>0,
 

    𝒫( sup_x∈ B; x_B≤ 1Π^k_nD_nC_n^-1Π^k_n(x)-Π^k_nρΠ^k_n(x)_B > η)
          ≤𝒫(sup_x∈ B; x_B≤ 1 S_1(x)> η)+𝒫(sup_x∈ B; x_B≤ 1 S_21(x)+S_22(x)> η)
          ≤𝒫(D_n-D_𝒮(H)>
    η/2VC_k_n^-1k_n^2)
          +𝒫(C_n-C_𝒮(H)>
    η/2Vk_n∑_j=1^k_na_j[ 2C_k_n^-1D_ℒ(H)+ρ_ℒ(H)])
          ≤  8exp( -nη^2/(2VC_k_n^-1k_n^2)^2(γ+δ(η/2VC_k_n^-1k_n^2)))+4exp( -nη^2/Q_n),

with    γ and δ   being positive numbers,  depending on ρ and 𝒫_ε_0,  respectively, introduced   in <cit.>. Here, 

    Q_n   =   
    4V^2k_n^2(∑_j=1^k_na_j)^2[ 2C_k_n^-1D_ℒ(H)+ρ_ℒ(H)]^2
          ×[α_1+β_1η/2Vk_n∑_j=1^k_na_j[ 2C_k_n^-1D_ℒ(H)+ρ_ℒ(H)]],

 where again α_1 and β_1 are positive constants depending on ρ and 𝒫_ε_0, respectively.
From equations  (<ref>) and  (<ref>), if
  
    k_nC_k_n^-1∑_j=1^k_na_j=o(√(n/ln(n))),    n→∞,
 then, the Borel–Cantelli lemma, and Lemma <ref> and Remarks <ref>– <ref> lead to the desired a.s. convergence to zero.
 
 ▪






Under the conditions of Theorem <ref>,  

    ρ_k_n(X_n)-ρ(X_n)_B→_a.s. 0,   n→∞ .





 The proof is straightforward from Theorem <ref>, since 

    ρ_k_n(X_n)-ρ(X_n)_B≤ρ_k_n-ρ_ℒ(B)X_0_B→_a.s 0,    n→∞,
 under   Assumption A1.







§ EXAMPLES: WAVELETS IN BESOV AND SOBOLEV SPACES



It is well–known that wavelets provide orthonormal bases of 
L^2(ℝ), and 
unconditional bases for several function spaces including Besov spaces, 
    { B_p,q^s,    s∈ℝ,    1≤ p,q≤∞}.
 

Sobolev or Hölder spaces constitute  interesting  particular cases of Besov spaces (see, for example,  <cit.>).
Consider now orthogonal wavelets on the interval [0,1]. Adapting wavelets
to a finite interval requires some modifications as described in <cit.>. 
Let s>0, for an [s]+1-regular Multiresolution Analysis  (MRA) of L^2([0,1]), where [·] stands for the integer part,  
 the father φ and the mother ψ wavelets are such that  φ ,ψ∈𝒞^[s]+1([0,1]). Also φ and  its derivatives, up to order [s]+1, have a fast decay  (see <cit.>).
Let 2^J ≥ 2([s]+1), the construction in <cit.> starts from a finite set of  2^J scaling functions {φ_J,k,  k=0,1,…,2^J - 1 }. For each j≥ J, a set 2^j wavelet functions  {ψ_j,k,  k=0,1,…,2^j - 1 } are also considered. The collection of these
functions,  
    {φ_J,k,  k=0,1,…,2^J - 1 },   {ψ_j,k,  k=0,1,…,2^j - 1 },    j≥ J,
 form a complete orthonormal system of L^2([0,1]). The associated reconstruction formula is given by:

    f(t) = ∑_k=0^2^J-1α_J,k^fφ_J,k (t) + ∑_j≥ J∑_k=0^2^j-1β_j,k^fψ_j,k(t),  ∀ t∈ [0,1],  ∀ f∈ L^2([0,1]),

where 

    α_J,k^f   =   ∫_0^1 f(t) φ_J,k(t)dt,    k=0,…,2^J - 1, 
    β_j,k^f   =   ∫_0^1 f(t) ψ_j,k(t)dt,   k=0,…, 2^j-1, j≥ J.


 The Besov spaces B_p,q^s([0,1])   can be characterized in terms of wavelets coefficients. Specifically,  denote by   𝒮^'  the dual of 𝒮, the Schwarz space, f∈𝒮^' belongs to B_p,q^s([0,1]), s∈ℝ, 1≤ p,q≤∞, if and only if

    f_p,q^s≡φ*f_p+( ∑_j=1^∞(2^jsψ_j*f_p)^q)^1/q<∞.


For β >1/2, consider  𝒯:H_2^-β([0,1])⟶ H_2^β([0,1])  be a self–adjoint positive  operator on L^2([0,1]), belonging to the unit ball of trace operators on L^2([0,1]). Assume that     
    𝒯:H_2^-β([0,1])⟶ H_2^β([0,1]),   𝒯^-1:H_2^β([0,1])⟶ 
    H_2^-β([0,1])
 are bounded linear operators. In particular, there exists an orthonormal basis { v_k,  k≥ 1 }   of L^2([0,1]) such that, for every l≥ 1,
𝒯(v_l)=t_lv_l, with ∑_l≥ 1t_l=1.
In what follows, consider { v_l,  l≥ 1 } to be  a wavelet basis, and define the kernel t of  𝒯
as, for s,t∈ [0,1],

    t(s,t)=1/2^J∑_k=0^2^J-1φ_J,k(s)φ_J,k(t)+
    2^2β-1/2^2β (1-J)∑_j≥ J∑_k=0^2^j-12^-2jβψ_j,k(s)ψ_j,k(t).

 
In Lemma <ref>,  
    {F_𝐦}= {F^φ_J,k, k=0,…,2^J-1}∪{F_j,k^ψ, k=0,…, 2^j-1,  j≥ J}
 are then defined as  follows:
  
    F^φ_J,k   =   φ_J,k,   k=0,…, 2^J-1
    
      F_j,k^ψ   =   ψ_j,k,   k=0,…, 2^j-1,   j≥ J.


Furthermore, the sequence 
    {t_𝐦}={ t^φ_J,k, k=0,…,2^J-1}∪{ t^ψ_j,k, k=0,…, 2^j-1,  j≥ J},
 involved in the definition of the inner product in H, is given by:

    t^φ_J,k   =   1/2^J,    k=0,…,2^J-1.
    
    t^ψ_j,k   =   2^2β-1/2^2β (1-J)2^-2jβ,    k=0,…,2^j-1,   j≥ J.


In view of <cit.>, the choice (<ref>)–(<ref>) of 
{F_𝐦} and {t_𝐦} leads to the definition of 
    H=
    [H_2^β([0,1])]^∗=H_2^-β([0,1]),
 constituted by the restriction to [0,1] of the tempered distributions  g∈𝒮^'(ℝ), such that (I-Δ )^-β /2g∈ L^2(ℝ), with (I-Δ )^-β /2 denoting the Bessel potential of order β (see <cit.>). 
 Let now define B=B_∞,∞^0([0,1],) and  B^∗=
 B^0_1,1([0,1]). From equation (<ref>), the corresponding norms, in term of the discrete wavelet transform introduced in equation (<ref>), are given by, for every f ∈ B,

    f _B   =   sup{|α_J,k^f|, k=0,…, 2^J-1;  |β_j,k^f|, k=0,…, 2^j-1;  j≥ J }


    g  _B^∗   =   ∑_k=0^2^J-1| α_J,k^ g|+  ∑_j=J^∞∑_k=0^2^j-1| β_j,k^ g|,   ∀ g ∈ B^∗.


 Therefore, 
 
    B^∗=B_1,1^0([0,1])↪ H=L^2([0,1]) ↪ B=B_∞,∞^0↪H= H_2^-β([0,1]).

 
Also, for β >1/2, 
    H^∗=H^β([0,1])↪ B^∗=B_1,1^0([0,1]).
 

For γ>2β ,  consider the operator C=(I-Δ )^-γ;    i.e., given by the 2γ/β power of the Bessel potential of order β , restricted to L^2([0,1]).  From spectral theorems on spectral calculus (see <cit.>), for every g∈ C^1/2(H^-β([0,1])),

    g_ℋ(X)^2   =   ⟨ C^-1(f),f⟩_H^-β([0,1])
    = ⟨ (I-Δ )^-β /2(C^-1(f)),(I-Δ )^-β /2(f)⟩_L^2([0,1])
       =   ∑_j=1^∞f_j^2λ_j((I-Δ )^(γ -β ))≥∑_j=1^∞f_j^2λ_j((I-Δ )^β)
       =   f_H^β([0,1])^2=f_H^∗^2,

where 
    f_j=∫_0^1(I-Δ )^-β /2(f)(s)(I-Δ )^-β /2(ϕ_j)(s)ds,

with {ϕ_j, j≥ 1} denoting the eigenvectors of the Bessel potential (I-Δ )^-β/2 of order β , restricted to L^2([0,1]),  and {λ_j((I-Δ )^γ -β), j≥ 1} being the eigenvalues of (I-Δ )^-βC^-1 on L^2([0,1]). Thus, Assumption A4 holds.
 Furthermore, from embedding theorems between fractional Sobolev spaces
 (see <cit.>), Assumption A5 also holds, under the condition γ >2β > 1, considering  H=L^2([0,1]). 



    
 


§ FINAL COMMENTS
 
  
  
Appendix <ref> illustrates the motivation of the presented approach in relation to functional prediction in nuclear spaces.  Specifically, the current literature on ARB(1) prediction has been developed for B=𝒞[0,1], the space of continuous functions on [0,1], with the supremum norm (see, for instance, <cit.>), and  B=𝒟([0,1]), constituted by  the right–continuous  functions on [0,1], having limit to the left at each t∈ [0,1], with the  Skorokhod topology
  (see, for example, <cit.>). 
 This paper provides a more flexible framework, where functional prediction can be performed, in a consistent way, for instance, in nuclear spaces,  as follows from the continuous inclusions showed in   Appendix <ref>.

Note that the two above–referred usual  Banach spaces, 𝒞[0,1] and 𝒟([0,1]), are included in the Banach space B considered in  Appendix <ref> (see Supplementary Material in Appendix <ref> about the simulation study undertaken). 






§ SUPPLEMENTARY MATERIAL



This document provides the Supplementary Material to the current paper.
Specifically, a  simulation study is undertaken to illustrate the results derived, on  strong consistency of functional predictors, in abstract Banach spaces, from the ARB(1) framework. The results are also illustrated in the case of discretely observed functional  data. 
 
 
 
 
 
 
  





 §.§ Simulation study







    f _B   =   sup{|α_J,k^f|, k=0,…, 2^J-1;  |β_j,k^f|, k=0,…, 2^j-1;  j= J,…,M }

where 

    α_J,k^f   =   ∫_0^1 f(t) φ_J,k(t)dt,    k=0,…, 2^J-1, 
    β_j,k^f   =   ∫_0^1 f(t) ψ_j,k(t)dt,    k=0,…, 2^j-1,    j≥ J.


Thus, equation (<ref>) corresponds to the choice B=B_∞,∞^0([0,1]), when resolution level M is  fixed for truncation.  Therefore, B^∗=B^0_1,1([0,1]) is considered  with the truncated norm

    g  _B^∗   =   ∑_k=0^2^J-1| α_J,k^ g|+  ∑_j=J^M∑_k=0^2^j-1| β_j,k^ g|,      g ∈ B^∗,
 
where  {α_J,k^g} and {β_j,k^g} are the respective father and mother wavelet coefficients of function g. 
Furthermore, as given in Appendix <ref> of the manuscript, 
    H^∗=H_2^β([0,1])=B_2,2^β([0,1]),   H=H_2^-β([0,1])=B_2,2^-β([0,1]),
 for β >1/2.  Since Daubechies wavelets of order N=10 are selected as orthogonal wavelet basis,  with N=10 vanishing moments, according to 
 <cit.>, and 
 <cit.>, we have considered J=2, and M = ⌈log_2 (L/2) ⌉=10,  for L=2^11 nodes, in the  discrete wavelet transform applied. In addition, value β = 6/10>1/2 has been tested, with γ =2β +ϵ, ϵ= 0.01 (see definition above of the extended version of operator C on H=H^-β([0,1])).
 The covariance kernel is now displayed in Figure <ref> (see <cit.> and <cit.>).







 Under Assumption A3, operator ρ admits the following extended representation in H=H^-β([0,1]), and in B:


    ⟨ρ(ϕ_j),ϕ_h⟩_H^-β([0,1]) = ( 1 + j )^-1.5      j=h 
    
    e^- | j - h | / W      j ≠ h 
    ,


Operator C_ε also admits, in this case,  the following  extended version in H=H^-β([0,1]): 

    ⟨ C_ε(ϕ_j),ϕ_h⟩_H^-β([0,1]) = 
    C_j(1 - ρ_j,j^2)       j=h 
    
    e^- | j - h |^2 / W^2      j ≠ h 
    ,

 being W = 0.4.
 
 





  §.§.§ Large-sample behaviour of the ARB(1) plug-in predictor





The ARB(1) process is generated with   discretization step size  Δ h =  0.0372.
 The resulting functional values of ARB(1) process X are showed in Figure <ref> for sample sizes

    n_t =  [2500, 5000, 15000, 25000, 40000, 55000, 80000, 100000, 130000, 165000].

 In this section (but not in the next one), the generated discrete values  are interpolated and smoothed, applying the 'cubicspline' option in  'fit.m' MatLab function, with, as commented before,   the number of nodes L = 2^11 = 2048, then  M = 10, and  Δh = 0.0093. In the following computations,  N = 250 replications are generated for each functional sample size, and  k_n =  ln(n) has been tested. 



The random initial condition X_0 has been generated from a truncated zero–mean Gaussian distribution.  Figure <ref> illustrates the fact that   Assumption A1 holds, and Figure <ref> is displayed to  check    Assumption A2.





 















Condition (<ref>) in Theorem <ref>  has been checked as well (see Figure <ref>).




 
  
  



To illustrate Theorem <ref> and Corollary <ref>,   Table <ref> displays the proportion of values of the random variable  ρ(X_n_t) - X_n_t+1_B  that are larger than the upper bound  
 
    ξ_n_t =  exp( -n_t/C_k_n_t^-2 k_n_t^2( ∑_j=1^k_n_t a_j )^2),    t=1,…,10,

  from the 250 values generated, for each functional sample size n_t, t=1,…, 10,  reflected below.  
    

  




  Figure <ref> below illustrates the asymptotic efficiency. The curve  n^-1/4 is also displayed (red dotted line).














  §.§.§ Asymptotic behaviour of discretely observed ARB(1) processes




The results in  Theorem <ref> and Corollary <ref> are now tested for different discretization step sizes:  
    {Δ h_r = (2^8+r - 1 )^-1,  r = 1,…,7 },   Δ h_r ⟶^r →∞ 0,
 that is,

    Δ h_1    =    1.96 (10^-3),   Δ h_2 = 9.78 (10^-4) , 
    Δ h_3    =    4.89 (10^-4),   Δ h_4 = 2.44 (10^-4), 
    Δ h_5    =    1.22 (10^-4),   Δ h_6 = 6.10 (10^-5), 
    Δ h_7    =    3.06 (10^-5).


Due to computational limitations involved in the smallest discretization step sizes, we restrict our attention  here to the sample sizes 
    { n_t = 5000 + 10000 (t-1),  t=1,2,3 },
 and N = 120 realizations have been generated, for each functional sample size. The same nodes are considered as in the previous section, in  the implementation of the discrete wavelet transform,  without previous smoothing of the discretely generated data.



Table <ref> displays the results obtained on the proportion of values, from the 120 generated values,  
    ρ(X_n_t^h,r) - X_n_t+1^h,r_B,    h=1,…, 120,
  that are larger than the upper bound (<ref>),  considering different discretization step sizes, 
 for each sample size 
    { n_t = 5000 + 10000 (t-1),  t=1,2,3 },
 and for the corresponding  truncation orders { k_n_t =  ln(n_t),  t=1,2,3 }.




  



 


 



§ ACKNOWLEDGMENTS


This work has been supported in part by project MTM2015–71839–P (co-funded by Feder funds), of the DGI, MINECO, Spain.







dinat



