Time series are ubiquitous in many applications and it is often the case that the time separating successive observations is itself random. We approach the study of such times series by using a continuous time stationary process and a renewal process which reflects the sampling scheme applied to . We assume that is strongly mixing or weakly dependent as defined, respectively, in R56 and DD08 and that is a process independent of with inter-arrival time sequence . In this general model set-up, we show under which assumptions the renewal sampled process defined as inherits strong mixing or weak dependence.
In the literature, the statistical inference methodologies based on renewal sampled data seldom make use of a strongly mixing or weakly dependent process . To the best of our knowledge, the only existing example of this approach can be found in ASM04 where it is shown that is
-strongly mixing and this property is used to study the consistency of maximum likelihood estimators for continuous time diffusion processes. On the contrary, there exist several statistical estimators whose asymptotic properties heavily rely on ad-hoc tailor made arguments in specific modeling set-ups. Examples of the kind are the estimators defined byLM92, M78a, M78b, and M83. In these works, the authors study non-parametric and parametric estimators of the spectral density of by means of an aliasing-free sampling scheme defined trough a renewal process, see LM92 for a general definition of this set-up. Such schemes like the Poisson one allow to overcome the aliasing problem which is typically observed with a not band limited process sampled in
. Moreover, working with an aliasing-free sampling allows to show that the spectral density estimators are consistent and asymptotic normal distributed once assumed thatM88, non-parametric estimators of volatility and drift for scalar diffusion in CT16, and parametric estimators of the covariance function of as in MW79 and BC18. In the latter, an estimator of the covariance function of a Gauss-Markov process and a continuous-time Lévy driven moving average are respectively analyzed. In BC18, in particular, the asymptotic properties of the estimator are obtained by an opportune truncation of a Lévy driven moving average process that is strongly mixing.
Determining conditions under which the process inherits the asymptotic dependence of can enlarge the field of applicability of renewal sample data beyond the literature above. Just as indicative examples, our analysis could enable the use of renewal sampled data to study spectral estimators as in R84, Whittle estimators as in BD08, and generalized method of moments estimators as in CS19 and SS19. Moreover, the knowledge of the asymptotic dependence of allows to apply well-established asymptotic results for -mixing processes like the ones in (B07, Chapter 10), DR00 and K17. The latter are respectively functional and triangular array central limit theorems. The same argument can be applied to central limit theorems for weakly dependent processes like the ones presented in BS05, DD03, and DW07. To this end, we give results on the decay rate of the weak dependence or strong-mixing coefficients, cf. (DD08, Section 2.2) and (B07, Definiton 3.5), of the process which can be used to determine under which conditions central limit theorems for strongly mixing or weak dependent processes can be applied to renewal sampled data.
More specifically, we show the inheritance of , , , , -weak dependence and -mixing which are extensively analyzed in the monographs DD08, B07 and D94, respectively. Moreover, under the additional condition that admits exponential or power decaying coefficients, we show that inherits strong mixing or weak dependence and its related coefficients preserve the exponential or power decay (at least asymptotically).
We present in Section 1 a unified formulation of weak dependence and -strong mixing conditions. We achieve this by introducing -weak dependence which encompasses both. In Section 2, we explicitly compute the weak dependence or strong mixing coefficients of the process . Finally, in Section 3, we show that if the underlying process admits exponential or power decaying coefficients then the process is -weakly dependent and has coefficients with (at least asymptotically) the same decay rate. Therefore, the process inherits the asymptotic dependence structure of . The last section includes several examples of renewal sampling in particular the Poisson one.
1 Weak dependence and strong mixing conditions
In the following, we will refer by to the set of positive integers, by to the set of the non-negative integers, by to the set of all integers and by to the set of the non-negative real numbers. We denote the Euclidean norm by . However, due to the equivalence of all norms, none of our results depends on the choice of the norm.
Although the theory developed below is probably most relevant for sampling processes defined in continuous time, we work with a general index set as this makes no difference and covers also other cases, like a sampling of discrete-time processes or random fields sampled along a walk, e.g. a self-avoiding walk that moves in positive coordinate directions. We refer the reader to CSS20 and the references therein for an overview of strongly mixing and weakly dependent random fields. Even if our theory extends to sampling of random fields, we always refer to as being a process to lighten the reading.
We assume throughout
The index set is denoting either , , or . Given and , we define .
Moreover, we consider
where and are respectively two classes of measurable functions from to and to that we specify individually later on. Finally, for a function that is unbounded or not Lipschitz, we set its norm or Lipschitz constant to infinity.
Let be an index set as in Definition 1.1, be a process with values in and a function from to non-decreasing in all arguments. The process is called -weakly dependent if there exists a sequence of coefficients converging to and satisfying the following inequality
where is a constant independent of .
W.l.o.g. we always choose the sequence of coefficients non-increasing.
Definition 1.2 encompasses the weak dependence conditions as described in DD08. For several choices of the function and , the coefficients are already well-known.
Let and be the class of bounded Lipschitz functions from to with respect to the distance on defined by
where and and for all . Then, . For
corresponds to the -coefficients as defined by DL99. If instead
corresponds to the -coefficients as defined by DW07. Moreover, for
corresponds to the -coefficients, and, for
corresponds to the -coefficients as defined by DL99.
Let be the class of bounded measurable functions from to and be the class of bounded Lipschitz functions from to with respect to the distance defined in (3). Then, for
corresponds to the -coefficients as defined by DD03.
The weak dependence conditions can all be alternatively formulated by further assuming that and are bounded by one. For more details on this issue see DL99 and DD03. Therefore, an alternative definition of -weak dependence exists where the function in Definition (2) does not depend on and . In this case, and are always bounded by one and therefore omitted in the notation.
We now show that Definition 1.2 also encompasses -mixing introduced by R56.
We suppose that and are sub--fields of and define
Let a set as in Definition 1.1, then a process with values in is said to be -mixing if
converges to zero as , where and . Throughout the paper, are called the -coefficients.
Let be a set as in Definition 1.1 and be a process with values in and where is the class of bounded measurable functions from to . is -mixing if and only if there exists a sequence converging to such that
and where is a constant independent of .
Set and . For arbitrary and , let and be arbitrary subsets of such that . Moreover, choose arbitrary and ,
By Theorem 4.4(a) in B07, it holds that
We assume now that the sequence is -weakly dependent with given by (6). By Theorem 4.4(a) and Remark 3.17(ii) in B07, we can rewrite the definition of the -coefficients as
Thus, if is -weakly dependent, is -mixing.
2 Strong mixing and weak dependence coefficients under renewal sampling
We assume given a strictly stationary -valued process , i.e. for all and all it holds
We want to investigate the asymptotic dependence of sampled at a renewal sequence. We first need the following definitions.
Let be a set as in Definition 1.1 and be an -valued sequence of non-negative (component-wise) i.i.d. random vectors with distribution function
-valued sequence of non-negative (component-wise) i.i.d. random vectors with distribution functionsuch that . For , we define an -valued stochastic process as
The sequence is called a renewal sampling sequence. When , we call the sequence of the inter-arrival times.
Let be a process with values in and let be a renewal sampling sequence independent of . We define the sequence as the stochastic process with values in given by
We call the underlying process and the renewal sampled process.
Definition 2.1 comes from H74 and determines the sampling of a random field through self-avoiding paths that move in positive coordinate directions. However, there are other interesting walks in that we could investigate by dropping the non-negativity of the sequence and using, for example, the definition of a renewal sequence as given in S69. This latter definition is also compatible with sampling a random field along a walk that moves in lexicographically increasing directions. The study of the asymptotic dependence of such samples is beyond the scope of the present paper but constitutes an interesting future research direction.
In the following theorem, we work with the class of functions defined in (1) and
where and are respectively two classes of measurable functions from to and to which can be either bounded or bounded Lipschitz.
Let be a renewal sampled process with the underlying process being strictly stationary and -weakly dependent with coefficients . Then, there exists a sequence such that
where is a constant independent of . Moreover,
where is the Dirac delta measure in zero, and, is the n-fold convolution of for .
is a strictly stationary process by Proposition 2.1 in BC18. Consider arbitrary fixed , , and with , and functions and . By conditioning with respect to the sequence of the inter-arrival times and using the law of total covariance (cf. Proposition A.1 in C19), we obtain that
Let us first discuss the summand (13). The term
because is independent of . On the other hand,
and, by stationarity of the process and the i.i.d property of , it is equal to
because of the independence between and
. Thus, the summand (13) is equal to zero, because
The summand (12) is less than or equal to
indicates the joint distribution of the inter-arrival times sequence. For a given , we have that and . is a -weakly dependent process, then the above inequality is less than or equal to
and, because the sequence is independent of the sequence , it is less than or equal to
We have that and that w.l.o.g. the coefficients are non increasing. Thus, we can conclude that the above integral is less than or equal to
Note that, if the coefficients (11) converge to zero as goes to infinity then inherits the asymptotic dependence structure of .
3 -weakly dependent renewal sampled processes
In this section, we consider renewal sampling of . Therefore, the inter-arrival times are a sequence of non-negative i.i.d random variables with values in .
We first show that if is -weakly dependent and admits exponential or power decaying coefficients then is in turn -weakly dependent and its coefficients preserve (at least asymptotically) the decay behavior of . This result directly enables the application of the limit theory for a vast class of -weakly dependent processes of which we present several examples throughout the section.
In fact, central limit theorems for a -weakly dependent process typically hold under sufficient conditions of the following type: for some and the coefficients satisfy a condition
where is a certain function of . If admits coefficients with exponential or sufficiently fast power decay then conditions of type (14) are satisfied. If in turn, is -weakly dependent with coefficients having exponential or sufficiently fast power decay, then conditions of type (14) are satisfied also under renewal sampling.
3.1 Exponential decay
In terms of the Laplace transform of the inter-arrival times, we can obtain a general formula for the coefficients .
Let , and be as in Theorem 2.4. Let us assume that for and denote the Laplace transform of the distribution function by
Then, the process admits coefficients
which converge to zero as goes to infinity.
We notice that for and that , cf. (S13, Proposition 2.6).
Using the result obtained in Theorem 2.4, we have that
To summarize, if is , , , , -weakly dependent or -mixing then inherits the same kind of asymptotic dependence structure under renewal sampling as long as the , , , , or -coefficients of are exponentially decaying.
If we have a renewal sampling with -distributed inter-arrival times for , then is the distribution function of a distributed random variable. By Proposition 3.1, we obtain the coefficients
A special case of the coefficients above is obtained in the case of Poisson sampling, i.e. with . In this instance, is the distribution function of a distributed random variable. We then obtain the coefficients
3.2 Power decay
We now assume that the underlying process is -weakly dependent with coefficients for .
We start with some concrete examples of inter-arrival time distributions which preserve the power decay of the coefficients .
Let us consider renewal sampling with distributed inter-arrival times for . Then, is a distribution. Thus,
by using Stirling’s series, see TE51, for to infinity.
In the special case of Poisson sampling, is a distribution and
We consider now the case where is an inverse Gaussian distribution with mean
is an inverse Gaussian distribution with meanand shape parameter (short ). We have that is a distribution and