CONVOLVED SUBSAMPLING ESTIMATION WITH APPLICATIONS TO BLOCK BOOTSTRAP

Transcription

1 Submitted to the Annals of Statistics arxiv: arxiv: CONVOLVED SUBSAMPLING ESTIMATION WITH APPLICATIONS TO BLOCK BOOTSTRAP By Johannes Tewes, Dimitris N. Politis and Daniel J. Nordman Ruhr-Universität Bochum, University of California-San Diego, Iowa State University The block bootstra aroximates samling distributions from deendent data by resamling data blocks. A fundamental roblem is establishing its consistency for the distribution of a samle mean, as a rototyical statistic. We use a structural relationshi with subsamling to characterize the bootstra in a new and general manner. While subsamling and block bootstra differ, the block bootstra distribution of a samle mean equals that of a k-fold self-convolution of a subsamling distribution. Motivated by this, we rovide simle necessary and sufficient conditions for a convolved subsamling estimator to roduce a normal limit that matches the target of bootstra estimation. These conditions may be linked to consistency roerties of an original subsamling distribution, which are often obtainable under minimal assumtions. Through several examles, the results are shown to validate the block bootstra for means under significantly weakened assumtions in many existing (and some new) deendence settings, which also addresses a standing conjecture of Politis, Romano and Wolf (1999). Beyond samle means, convolved subsamling may not match the block bootstra, but instead rovides an alternative resamling estimator that may be of interest. Under minimal deendence conditions, results also broadly establish convolved subsamling for general statistics having normal limits. 1. Introduction. Subsamling and block bootstra are two common nonarametric tools for statistical inference under deendence; see Politis, Romano and Wolf [29] and Lahiri [19], resectively, for monograhs on these. Both aim to aroximate distributions of statistics with correlated data, and both are data resamling methods that use blocks of neighboring observations to cature deendence. The subsamling aroach of Politis and Romano [28] treats data blocks as small scale renditions of the original data, which rovides relication of a statistic for estimating a samling distribution. The block bootstra differs hilosohically by using data blocks as building material to re-create the original data. Essentially, data blocks are randomly selected and asted together to reroduce a full-scale set of bootstra data, as roosed by Künsch [17] and Liu and Singh [25] for extending Efron [11] s bootstra to time series. As noted in Politis, Romano and Wolf [29] (cf. sec. 3.9), subsamling is often valid under weak assumtions about the deendent rocess, basically requiring that a non-degenerate (ossibly non-normal) limit exist for the samling distribution being aroximated. In contrast, the block bootstra alies to mean-like statistics with normal limits and tyically requires comaratively much stronger assumtions for its validity. Case-by-case treatments are commonly needed to validate the bootstra across differing deendence conditions. Research suorted by by the Collaborative Research Center Statistical modeling of nonlinear dynamic rocesses (SFB 823, Teilrojekt C2) of the German Research Foundation (DFG). Research artially suorted by NSF DMS AMS 2000 subject classifications: Primary 62G09; secondary 62G20, 62J05, 62M10 Keywords and hrases: Convolution, Mixing, Moving blocks, Non-stationary 1

2 2 TEWES, POLITIS, NORDMAN However, while erhas not widely recognized, subsamling can in fact be used to verify the block bootstra in some cases, which is a theme of this work. We investigate estimators defined by the k-fold self-convolution of a subsamling distribution, and establish a new and general theory for their consistency to normal limits. There are two basic motivations for considering such convolved subsamling. The first is that, in the fundamental case of samle means, the block bootstra estimator is a k-fold self-convolution of a subsamling distribution (centered and normalized), where the level k of convolution corresonds to the number of resamled blocks. This observation was originally noted by Politis, Romano and Wolf [29], who suggested this asect as a otential technique for showing the validity of the bootstra. Secifically, they conjectured that convolved subsamling might rovide a route for establishing the block bootstra under minimal conditions for non-stationary, strongly mixing rocesses, in analogy to bootstra results existing for stationary, mixing series due to Radulovic [30, 31]. For the bootstra under deendence, the findings for the samle mean in [30, 31] have stood out as an excetion, verifying the method under the same weak assumtions as subsamling (i.e., conditions essentially needed for a normal limit to exist). By investigating the convolved subsamling aroach here, we can answer the above conjecture affirmatively. Moreover, we show convolved subsamling leads to a simle and unified rocedure for establishing the block bootstra for samle means under further tyes of rocesses and much weaker conditions than reviously considered, such as linear time rocesses, long-memory sequences, (non-stationary) almost eriodic time series, and satial fields. Hence, convolved subsamling estimation allows for bootstra consistency under deendence to be generally extended under the same weak assumtions used by subsamling, containing the conclusions of Radulovic [30, 31] for stationary time series as a secial case. While connections to the bootstra are useful, our study of convolved subsamling estimation is intended to be broad, alying also to general statistics with normal limits and with arbitrary levels of convolution. Consistency results often do not require articular assumtions about the underlying deendent rocess, but are rather formulated in terms of mild convergence roerties of the original subsamling distribution and its variance. Furthermore, we show that a consistent subsamling variance is not only sufficient, but essentially necessary, for the consistency of convolved subsamling (and the block bootstra in some cases). Due to its imortance, we also rovide tools for verifying the consistency of subsamling variance estimators. For general statistics beyond the samle mean, the convolved subsamling distribution may differ from the block bootstra, which relates to a second motivation for our develoment. That is, a general theory for convolved subsamling may be of interest in its own right, as the aroach can be comutationally less demanding than the block bootstra while also otentially enhancing ordinary subsamling for aroximating samling distributions with normal limits. In fact, there has been recent interest in establishing generalized tyes of subsamling estimation for comlicated statistics under various deendence structures, where numerical studies suggest such methods can exhibit better finite samle erformance than standard subsamling when the target distribution is normal; for examle, see Lenart [22] and Shariov, Tewes and Wendler [32] for sectral estimates and U-statistics, resectively, with time series. While not formally recognized as such, however, these roosed methods are exactly convolved subsamling estimators. By exloiting this realization, our results can facilitate future work and allow such revious findings with generalized subsamling to be demonstrated in an alternative, simler manner with weaker assumtions; see Section 5 for illustrations of the examles mentioned above. Section 2 describes convolved subsamling estimation and its connection to block bootstra. General distributional results for convolved subsamling are given in Section 3, while Section 4 resents some alications with differing deendence structures. Section 4.1 rovides a broad result for convolved subsamling estimation with statistics from mixing time series. Under weak

3 CONVOLVED SUBSAMPLING 3 conditions, Sections aly convolved subsamling for demonstrating the block bootstra for samle means with non-stationary time series (Section 4.2 and the conjecture of Politis, Romano and Wolf [29]), linear time rocesses (Section 4.3), long-range deendence (Section 4.4), and satial data (Section 4.5). Section 5 describes relationshis to other recent work with generalized subsamling, and Section 6 rovides a short treatment of indeendent data. A numerical study of subsamling, block bootstra and convolved subsamlng aears in Section 7, while Section 8 contains concluding remarks. The roofs of main results are given in a sulement [15]. Finally, to be clear, we stress that a central advantage of classical subsamling is its validity for non-normal limits (cf. Section 4.4), which convolved subsamling does not share. The convolution of a subsamling distribution essentially induces a sum of indeendently resamled terms so that, like the block bootstra, reroducing a non-normal limit is imossible. However, for aroximating normal targets, convolved subsamling does inherit the alicability of subsamling under weak conditions with general statistics. 2. Descrition of convolved subsamling estimators Problem background and original subsamling estimation. Consider data X 1,..., X n from a real-valued rocess equied with a robability structure P. For concreteness, we may view such observations as arising from a time series rocess {X t }, though satial and other data schemes may be treated as well. Based on X 1,..., X n, consider the roblem of aroximating the distribution of T n τ n (t n (X 1,..., X n ) t(p )), involving an estimator t n t n (X 1,..., X n ) of a arameter t(p ) and a sequence of ositive scaling factors τ n yielding a distributional limit for T n. For examle, if t n (X 1,..., X n ) X n = n i=1 X i/n is the samle mean, then t(p ) may corresond to a common rocess mean µ and T n may be defined with usual scaling τ n = n under weak time deendence. Denote the samling distribution function of T n as F n (x) = P (T n x), x R. We next define the subsamling estimator of F n ; see [28]. For a ositive integer b b n < n, let {(X i,..., X i+b 1 ) : i = 1,..., N n } denote the set of N n n b + 1 overlaing data blocks, or subsamles, of length b. To kee blocks relatively small, the block size is often assumed to satisfy b 1 + b/n + τ b /τ n 0 as n. For each subsamle, we comute the statistic as t n,b,i = t b (X i,..., X i b+1 ) and define a scale b version of T n τ n (t n (X 1,..., X n ) t(p )) as τ b [t n,b,i t n ] for i = 1,..., N n. Letting I( ) denote the indicator function, the subsamling estimator of F n is given by (2.1) S (x) = 1 N n N n I ( τ b [t n,b,i t n ] x ), x R, i=1 or the emirical distribution of subsamle analogs {τ b [t n,b,i t n ]} Nn i=1 (cf. [29]). Suose that S is consistent for the distribution of T n, which has an asymtotically normal N(0, σ 2 ) limit for some σ > 0, that is, as n, (2.2) T n d N(0, σ 2 ), (2.3) su S (x) Φ(x/σ) 0, where Φ( ) is the standard normal distribution function. We wish to consider estimators of the distribution F n of T n formed by self-convolutions of the subsamling estimator S. This rovides

4 4 TEWES, POLITIS, NORDMAN a general class of block resamling estimators in its own right, but also has exlicit connections to block bootstra estimators in the imortant case that the statistic of interest t n (X 1,..., X n ) = X n is a samle mean, as described next Convolved subsamling and connections to block bootstra. Let k n be a sequence of ositive integers and define a triangular array {Yn,1,..., Y n,k n } n 1, where, for each n, {Yn,j }kn j=1 are iid variables following the subsamling distribution S, as determined by (2.1) from data X 1,..., X n. For n 1, define a centered and scaled sum (2.4) Z n 1 kn k n j=1 (Y n,j m ) where m xds (x) = Nn 1 Nn i=1 τ b[t n,b,i t n ] is the mean of the subsamling distribution S, and let C n,kn (x) P (Zn x), x R, denote the induced resamling distribution P of Zn. Then, C n,kn reresents the k n -fold selfconvolution of the subsamling distribution S, with aroriate centering/scaling adjustments. That is, C n,kn (x) = S S S (x k n + k n m ), x R. }{{} k n times We consider C n,kn as an estimator of the distribution F n of T n and formulate general conditions under which such convolved subsamling is consistent. As suggested earlier, such results have direct imlications for block bootstra estimation as well, because the convolved subsamling estimator C n,kn exactly matches a block bootstra estimator in the basic samle mean case t n (X 1,..., X n ) = X n. To illustrate, consider aroximating the distribution of T n = n( X n µ) where t(p ) µ = E X n and τ n = n. In this setting, the block bootstra uses an analog (2.5) T n = n 1 ( X n 1 E X n1 ) based on the average X n 1 n 1 n1 1 i=1 X i from a block bootstra samle X1,..., X n 1 of size n 1 k n b, which is defined by drawing k n blocks of length b, indeendently and with relacement, from the subsamle collection {(X i,..., X i+b 1 ) : i = 1,..., N n } and asting these together (where above E X n1 = Nn 1 Nn i=1 b 1 i+b 1 j=i X j denotes the bootstra exectation of X n1 ); see ch. 2, Lahiri [19]. Most tyically, the number of resamled blocks is taken as k n = n/b so that the bootstra samle re-creates the aroximate length n/b b n of the original samle. The bootstra distribution of Tn here is then equivalent to the convolved subsamling distribution C n,kn. This is because Tn has the same resamling distribution as Zn in (2.4) as a sum of k n iid block averages (Yn,i m )/ k n, with each Yn,i drawn from S in (2.1) where t n = X n and τ b [t n,b,i t n ] = b[b 1 i+b 1 j=i X j X n ], 1 i N n, for the samle mean case. Consequently, if convolved subsamling estimators C n,kn are shown to be valid under weak conditions, such results entail that block bootstra estimation is as well. In the following, we make comrehensive use of the fact that C n,kn is always and exactly a block bootstra estimator whenever the underlying statistic t n (X 1,..., X n ) = X n is a samle mean; this holds true across all the various deendent data structures considered here, including cases where the usual block bootstra form (2.5) requires modification for samle means (cf. long-range deendence in Section 4.3).

5 CONVOLVED SUBSAMPLING 5 3. Fundamental results for convolved subsamling. From (2.1) and the subsamling mean m xds (x) = Nn 1 Nn j=1 τ b[t n,b,i t n ], we have the variance of the original subsamling distribution S as ˆσ 2 (x m ) 2 ds (x) = 1 N n N n (τ b [t n,b,i t n ] m ) 2, which estimates the asymtotic variance σ 2 of T n as in (2.2) (cf. [29]). Note that ˆσ 2 is also the variance of the convolved subsamling distribution C n,kn (i.e., the variance of the iid sum from (2.4)). Corresondingly, ˆσ 2 is then a block bootstra variance estimator when alied to samle means. Sections rovide basic distributional results for convolved subsamling estimators, describing when and how these have normal limits. These findings do not involve articular assumtions about the rocess {X t }, but are instead exressed through roerties of the original subsamling distribution S and, secifically, convergence of the subsamling variance ˆσ. 2 Such subsamling roerties can often be verified under weak assumtions about a rocess, allowing the limit behavior of convolved estimators C n,kn, and the block bootstra, to be established under minimal conditions. Results in Section 3.1 address the imortant case where the original subsamling d distribution S has a normal limit (2.3), as is often natural when the statistic T n N(0, σ 2 ) is asymtotically normal. These findings are exected to be the most ractical for establishing convolved subsamling C n,kn estimation with normal targets (2.2). Droing the condition that S converges to a normal law but assuming convolved estimators C n,kn are based on increasing convolution k n of S, Section 3.2 characterizes the convergence of C n,kn to normal limits through the subsamling variance ˆσ. 2 In many roblems involving the block bootstra for samle means (cf. Section 4), where T n has a normal limit (2.2), these results rovide both necessary and sufficient conditions for the validity of the block bootstra as well as convolved subsamling generally. Finally, because convergence ˆσ 2 σ 2 of the subsamling variance emerges as central to the behavior of convolved estimators C n,kn, Section 3.3 develos basic results for establishing this feature Convolution of subsamling distributions with normal limits. Theorem 1 rovides a sufficient condition for the general validity of the convolved estimator C n,kn via fundamental subsamling quantities, S and ˆσ. 2 Theorem 1. Suose (2.3) holds (i.e., su S (x) Φ(x/σ) as n. Then, su C n,kn (x) Φ(x/σ) 0 as n for any ositive integer sequence k n. j=1 0) and ˆσ 2 σ 2 > 0 d Furthermore, when (2.2) holds additionally (i.e., T n N(0, σ 2 )), then C n,kn is consistent for the distribution F n of T n, su C n,kn (x) F n (x) 0 as n. To re-iterate, the integer sequence k n, n 1, need not even be convergent in Theorem 1. The consistency of the subsamling variance estimator ˆσ 2 automatically guarantees that, for any amount k n of convolution of S, the convolved subsamling estimator C n,kn will have a normal limit if the subsamling distribution S does. In other words, if (2.2)-(2.3) hold so that S is

6 6 TEWES, POLITIS, NORDMAN consistent, then C n,kn will be as well rovided ˆσ 2 σ 2. When the statistic t n (X 1,..., X n ) = X n is a samle mean, then C n,kn again denotes a block bootstra estimator based on k n resamled blocks, which is thereby consistent under Theorem 1 for any sequence k n, including the common choice k n = n/b. Proosition 1 next characterizes the convolved subsamling estimator C n,kn under bounded levels k n of convolution. In this case, a normal limit for the subsamling estimator S entails the same for the convolved estimator C n,kn, rovided the mean m xds (x) of the subsamling distribution converges to zero. But, if the subsamling mean m converges in this fashion, a normal limit for C n,kn with bounded {k n } is equivalent to a normal limit for the original subsamling distribution S. Proosition 1. Suose su n k n <. (i) If (2.3) holds (i.e., su S (x) Φ(x/σ) 0), then su C n,kn (x) Φ(x/σ) 0 as n if and only if m xds (x) 0. (ii) If m 0 as n, then (2.3) holds if and only if su C n,kn (x) Φ(x/σ) 0 as n. When the original subsamling estimator S is consistent for a distribution with a normal limit (i.e., (2.2)-(2.3)), both Theorem 1 and Proosition 1 show that the convolved subsamling estimator C n,kn is consistent under an additional subsamling moment condition. With bounded levels k n of convolution, the additional condition under Proosition 1 is that the subsamling mean converge m 0. But, for general and otentially unbounded kn, the additional condition from Theorem 1 for consistency of C n,kn is a convergent subsamling variance ˆσ 2 σ 2. With diverging amounts k n of convolution, which is often encountered in ractice and in connection to the block bootstra, it turns out that convergence ˆσ 2 σ 2 is also necessary for consistency of the convolved estimator C n,kn, as treated in the next section Unbounded convolution of subsamling distributions. We next consider the behavior of convolved subsamling estimators with unbounded convolution k n as n, which arises, for examle, with the block bootstra C n,kn for samle means with k n = n/b resamled blocks. Results here do not exlicitly require convergence of the original subsamling estimator S to a normal d limit (2.3). While a reasonable condition in roblems where the target quantity T n N(0, σ) 2 is asymtotically normal, limits for S are not directly necessary for convolved estimators C n,kn to yield normal limits from increasing convolution k n of S. However, convergence of the subsamling variance ˆσ 2 is crucial, as shown next. Theorem 2. Suose k n and x k nɛ x2 ds (x) 0 for each ɛ > 0 as n. (i) Then, su C n,kn (x) Φ(x/σ) 0 if and only if ˆσ 2 σ 2 > 0 as n. (ii) When ˆσ 2 σ 2 > 0 as n, then C n,kn is a consistent estimator of the distribution F n of

7 CONVOLVED SUBSAMPLING 7 T n if and only if T n 0). d N(0, σ 2 ) (i.e., a normal limit (2.2) for T n holds or su F n (x) Φ(x/σ) For an unbounded sequence k n of convolution (e.g., block bootstra with k n = n/b concatenated blocks), Theorem 2 imoses no direct assumtion on the convergence of the original subsamling distribution, but rather that S fulfills a mild truncated second moment roerty. From this, the convergence of the convolved subsamling estimator C n,kn to a normal limit is comletely determined by the subsamling variance ˆσ 2 under Theorem 2. Furthermore, when ˆσ 2 converges, the convolved estimator C n,kn will be valid for estimating the distribution F n of a target quantity T n having a normal limit (Theorem 2(ii)). In cases where T n fails to have a normal limit, the convolved estimator C n,kn does not aly. The following corollary of Theorem 2 shows that a convolved estimator C n,kn will quite generally have a normal limit, rovided that the subsamling variance converges ˆσ 2 σ 2 > 0 and that some other basic feature exists for the subsamling distribution S or for comosite statistics {τ b [t n,b,i t n ] τ b [t b (X i,..., X i b+1 ) t n (X 1,..., X n )]} Nn n b+1 i=1 defining S in (2.1). Essentially, Corollary 1 entails that the truncated second moment assumtion in Theorem 2 is mild in conjunction with ˆσ 2 σ 2. Corollary 1. Suose one of the following conditions (C.1)-(C.4) holds: (C.1) for some distribution J 0 with variance σ 2 > 0, S (x) J 0 (x) as n for any continuity oint x R of J 0 ; (C.2) for some ɛ 0 > 0, Nn 1 Nn [ i=1 τb (t n,b,i t n ) ] 2+ɛ 0 = O (1). (C.3) the subsamle-based sequence {Tb,i 2 τb 2[t n,b,i t(p )] 2 : i = 1,..., N n } n 1 is uniformly integrable and T n τ n (t n t(p )) = O (τ n /τ b ) (C.4) {X t } is stationary, {Tn 2 : n 1} is uniformly integrable, and τ b /τ n = O(1). Then, as n, su C n,kn (x) Φ(x/σ) 0 for any sequence k n with lim n k n = if and only if ˆσ 2 σ 2 > 0 Remark 1: For reference, note τ b /τ n 0 often holds with subsamle scaling as n so that conditions τ b /τ n = O(1) and T n = O (τ n /τ b ) are mild. Hence, if k n and ˆσ 2 σ 2, then the convolved estimator C n,kn will converge to a normal limit if the subsamling distribution S is convergent (C.1) or has an aroriate stochastically bounded moment (C.2), or if the subsamling statistics related to comuting S have uniformly integrable second moments (C.3)-(C.4). Condition (C.4) is a secial case of (C.3) under stationarity, and corresonds to an underlying assumtion of Radulovic [30, 31] for examining the block bootstra estimator C n,kn of a samle mean with stationary, mixing rocesses; see also Remark 2 to follow. When restricted to Condition (C.1), the art of Corollary 1 corresonds to an initial convolved subsamling result due to Politis, Romano and Wolf [29] (Proosition 4.4.1) for unbounded convolution k n, which was develoed for establishing the block bootstra estimator C n,kn for the samle mean of non-stationary data, as re-considered here in Section 4.2. Note that, for inference with T n having a normal N(0, σ 2 ) limit (2.2), Condition C.1 in Corollary 1 is erhas most natural and aroachable by verifying convergence S to a normal (2.3). In which case, the imlication of Corollary 1 (involving k n ) for guaranteeing that convolved subsamling and block bootstra estimators relicate normal limits when ˆσ 2 σ 2 also becomes a secial

8 8 TEWES, POLITIS, NORDMAN case of Theorem 1 (involving any k n ). Remark 2: For block bootstra estimation of the samle mean T n = n( X n EX 1 ) with strongly mixing, stationary rocesses, Radulovic [30, 31] rovides necessary and sufficient conditions for convergence of C n,kn (with k n = n/b ) to a normal limit, assuming {T 2 n : n 1} is uniformly integrable. Under such assumtions, the main result there is that normal limits for both C n,kn and T n are equivalent. In comarison, the necessary and sufficient conditions for normality of the block bootstra estimator C n,kn for a mean in Theorem 2 are erhas more basic in that the conclusions of [30, 31], under the additional assumtions made there, follow from Theorem 2 (cf. Corollary 1). In this sense, Theorem 2 broadly re-frames the findings in [30, 31], by not involving articular rocess assumtions (i.e., stationarity or mixing) and alying to convolved subsamling estimators C n,kn with general statistics and arbitrarily increasing convolution levels k n. Further connections to, and extensions of, the results of Radulovic [30, 31] are made in Section 4.1 for strongly mixing rocesses Consistency of subsamling variance estimators. Theorems 1-2 demonstrate that the subsamling variance ˆσ 2 lays a key role in the convergence of the convolved subsamling estimator C n,kn generally, and of the block bootstra for the samle mean in articular. However, convergence of the subsamling distribution S itself is often much easier to directly establish under weak assumtions about the rocess {X t }; see Politis, Romano and Wolf [29] and Section 4 to follow. This raises a further question considered next: if one knows that subsamling estimator S is consistent (2.3) for a normal limit, then when will the subsamling variance ˆσ 2 be convergent as well, thereby guaranteeing (from Theorem 1) that the convolved estimator C n,kn is also consistent? As shown in Theorem 3, a general characterization is ossible as well as simle sufficient conditions based on moment roerties of subsamle statistics (e.g., Tb 2). For n 1, recall T n τ n (t n (X 1,..., X n ) t(p )) and additionally define T n,i τ n (t n (X i,..., X i+n 1 ) t(p )) for i 1 from the statistic alied to (X i,..., X i+n 1 ). Based on N n n b + 1 subsamle observations of length 1 b b n < n, define a distribution function (3.1) D n,b (x) 1 N n N n P (T b,i x), x R, i=1 as an average of subsamle-based robabilities. Theorem 3. Suose (2.3) and T n = o (τ n /τ b ) as n. (i) Then, ˆσ 2 σ 2 > 0 as n if and only if, for each ɛ > 0, ( ) (3.2) lim su 1 N n P T m b,i 2 n m N I( T b,i > m) > ɛ = 0. n i=1 (ii) Additionally, (3.2) holds whenever {Yb 2 : b 1} is uniformly integrable, where Y b denotes a random variable with distribution D n,b, n 1, from (3.1) (i.e., P (Y b x) = D n,b (x), x R). If (2.3) and T n = o (τ n /τ b ) hold, uniform integrability of {Yb 2 : b 1} is equivalent to x 2 dd n,b (x) = Nn 1 Nn i=1 ET b,i 2 σ2 as n. (iii) (3.2) also holds whenever {X t } is stationary and {Tb 2 : b 1} is uniformly integrable. Remark 3: As T n is tyically tight, the assumtion T n = o (τ n /τ b ) is often satisfied by a standard condition on block length: b with b/n + τ b /τ n 0. Block conditions are not, in fact, used or

9 CONVOLVED SUBSAMPLING 9 required in statements of Theorems 1-3 above. However, block assumtions are usually needed to show the original subsamling estimator S is convergent as in (2.3), and examles of Section 4 shall imose block length conditions for this urose. Theorem 3 connects convergence (2.3) of subsamling distributions S to the convergence of subsamling variances ˆσ 2 in a way involving no further conditions on the rocess or statistic beyond mild tyes of uniform integrability. For examle, with non-stationary rocesses {X t }, Theorem 3(ii) converts the roblem of robabilistic convergence ˆσ 2 σ 2 into a more aroachable one of subsamle-moment convergence Nn 1 Nn i=1 ET b,i 2 σ2. To frame another imlication of Theorem 3, note that many inference roblems with time series involve a stationary rocess {X t } and a statistic T n with a normal limit (2.2) such that {Tn 2 : n 1}, and consequently {Tb 2 : b 1}, is uniformly integrable; see Remark 2. In such roblems, it suffices to simly establish the consistency of the subsamling estimator S (2.3) and then the consistency of subsamling variance ˆσ 2 follows with no further effort (by Theorem 3(iii)) along with the consistency of the convolved subsamling estimator C n,kn (by Theorem 1). Again, with samle means, C n,kn is a block bootstra distribution and ˆσ 2 is a block bootstra variance estimator, so both will be consistent in this setting by showing that S is consistent. This strategy has two advantages with the block bootstra: showing the consistency of S is often an easier rosect than considering either C n,kn or ˆσ 2 directly, and the consistency of S (and thereby the bootstra) can tyically be established under weak rocess assumtions. To illustrate, Section 4 alies the basic results here for establishing the convolved subsamling estimator C n,kn, as well as the block bootstra for samle means, under different deendence structures. 4. Alications of convolved subsamling estimation. Section 4.1 first develos consistency results for convolved subsamling estimators with strongly mixing rocesses and general statistics. The remaining subsections then consider convolved subsamling for the articular case of the samle mean with the goal of generalizing and extending results for the block bootstra across various tyes of deendent data, such as non-stationary mixing time rocesses (Section 4.2), linear time series (Section 4.3), long-range deendent rocesses (Section 4.4) and satial data (Section 4.5). Define the strong mixing coefficient of {X t } as α(k) = su i Z { P (A B) P (A)P (B) : A F i, B F k+i }, k 1, where F i and F k+i resectively denote σ-algebras generated by {X t : t i} and {X t : t k + i} (cf. [1], ch. 16.2). Recall {X t } is strongly mixing or α-mixing if lim k α(k) = Convolved subsamling for general statistics under mixing. For mixing stationary time series, Radulovic [30] roved consistency of block bootstra estimation for T n = τ n (t n (X 1,..., X n ) t(p )) based on the samle mean t n (X 1,..., X n ) = X n with t(p ) = EX 1 and τ n = n. The assumtions made were quite weak, requiring only (a1) a stationary, α-mixing rocess fulfilling (2.2) (i.e., T n d N(0, σ 2 )) and block lengths b 1 + b/n 0 as n ; (a2) uniformly integrable {T 2 n : n 1}. From results in Section 3 and the equivalence between the block bootstra and the convolved subsamling estimator C n,kn for the samle mean, a different ersective is ossible for the bootstra findings in Radulovic [30]. Under only assumtion (a1) above, the subsamling estimator S is consistent (i.e., (2.3) holds) for the asymtotically normal distribution of T n = n( X n EX 1 )

10 10 TEWES, POLITIS, NORDMAN (cf. Theorem 3.2.1, [29]), imlying, by Theorem 1 here, that the block bootstra estimator C n,kn would be consistent if the subsamling variance converges ˆσ 2 σ 2. But, if S is consistent for a normal limit by (a1), assumtion (a2) then guarantees that ˆσ 2 σ 2 holds by Theorem 3. Furthermore, under (a2) and with k n = n/b resamled blocks as in Radulovic [30, 31], convergence ˆσ 2 σ 2 becomes even necessary here by Theorem 2. Hence, α-mixing serves to show that the original subsamling estimator S is consistent; after which, uniform integrability and stationary assure both ˆσ 2 σ 2 and consistency of the block bootstra estimator C n,kn by Theorems 2-3. Under analogously weak assumtions as those of Radulovic [30], Theorem 4 next rovides the general consistency of convolved subsamling estimation for general statistics arising from mixing, and ossibly non-stationary, time rocesses. When alied to a samle mean t n (X 1,..., X n ) = X n, so that C n,kn is a block bootstra estimator, this result extends those of Radulovic [30] in two ways: by allowing otential non-stationarity series and by ermitting arbitrary levels k n of convolution/block resamling (rather than the single choice k n = n/b ). When the statistic t n (X 1,..., X n ) is not a samle mean, C n,kn may not again match the block bootstra but can have interest as an alternative block resamling estimator (cf. Section 5). Theorem 4. Let {X t } be a (ossibly non-stationary) strongly mixing sequence. Suose b 1 + d b/n + τ b /τ n 0 as n ; T n = o (τ n /τ b ); (3.2) holds; and that Y b N(0, σ 2 ) as n, for some σ 2 > 0, where each random variable Y b, b b n 1, has distribution function D n,b from (3.1). Then, as n, su S (x) Φ(x/σ) and, for any ositive integer sequence k n, 0 and ˆσ 2 su C n,kn (x) Φ(x/σ) 0. σ 2 Furthermore, if (2.2) additionally holds (i.e., T n d N(0, σ 2 )), then S and C n,kn (with any k n ) are consistent for the distribution F n of T n : su S (x) F n (x) 0 and su C n,kn (x) F n (x) 0. While roviding a broad result on the validity of convolved subsamling estimation for mixing rocesses, Theorem 4 also exands the general subsamling results of Politis, Romano and Wolf [29] (ch. 4.2), which focused on S for mixing series, to further include consistency of the subsamling variance ˆσ. 2 That is, when droing (3.2), the remaining Theorem 4 assumtions are minimal and match those of Theorem of Politis, Romano and Wolf [29] for the consistency of S to a normal limit; including (3.2) in Theorem 4 is then necessary for ˆσ 2 σ 2 by Theorem 3 and assures convergence of C n,kn by Theorem 1. If the rocess {X t } is actually stationary, we immediately obtain the following result. Corollary 2. Let {X t } be a stationary, strongly mixing sequence. Suose also b 1 + b/n + τ b /τ n 0 as n ; that (2.2) holds; and that (3.2) holds (e.g., uniform integrability of {T 2 n : n 1} suffices). Then, as n, the convergence results of Theorem 4 hold.

11 CONVOLVED SUBSAMPLING 11 Section 5 illustrates Theorem 4 for establishing convolved subsamling with mixing time series and several general classes of statistics. These reresent cases where C n,kn differs from the block bootstra estimator. However, Section 4.2 first rovides some further refinements with mixing rocesses in the samle mean case, where C n,kn matches the block bootstra Block bootstra for mixing non-stationary time rocesses. Consider a strongly mixing, otentially non-stationary sequence {X t } having a common mean arameter EX t = µ R, which is estimated by the samle mean X n. In this setting and under conditions where T n n( X n µ) has a normal limit (2.2), Fitzenberger [12] established the consistency of the block bootstra for estimating the distribution of T n. The result, however, required the existence of a (4 + δ)-moment (i.e., su t E X t 4+δ < for some δ > 0) along with stringent mixing conditions and restrictions on the block length b = o(n 1/2 ). Politis, Romano and Wolf [29] (examle 4.4.1) showed that the subsamling estimator S is consistent under weaker conditions, including only a (2 + δ)-moment. For the block bootstra with k n = n/b resamled blocks, Politis, Romano and Wolf [29] also roved bootstra consistency by alying convolved subsamling in this roblem, using a weaker block assumtion b = o(n) than Fitzenberger [12] but otherwise with same remaining strong assumtions about the rocess. However, [29] (remark 4.4.4) conjectured that the block bootstra might be established under non-stationarity using the same weak moment/mixing conditions as the subsamling estimator S, just as in the case of stationary mixing rocesses (cf. [30]). We confirm this by the following Theorem 5. Theorem 5. Let {X t } be a sequence of (not necessarily stationary) strongly mixing random variables with common mean µ. For some δ > 0, suose that su t E X t 2+δ < and k=1 α(k)δ/(2+δ) <. Assume also that, for some σ 2 > 0, ( ) i+n 1 lim su n Var n 1/2 X t σ 2 = 0. i 1 Then, as n, T n = n( X n µ) d N(0, σ 2 ) (i.e., (2.2) holds). Additionally, if b 1 + b/n 0 as n, then su S (x) Φ(x/σ) 0 and ˆσ 2 σ 2 and, for any ositive integer sequence k n, t=i su C n,kn (x) Φ(x/σ) 0. Hence, with any number k n of concatenated blocks, the block bootstra estimator C n,kn is valid for the distribution of the samle mean under mild assumtions for mixing, and ossibly nonstationary, rocesses. Note that the assumtions of Theorem 5 resemble those essentially needed for a central limit theorem (CLT) itself (cf. Theorem , [1]). In articular, the assumtions also match those commonly used in the stationary case for establishing the block bootstra; see Section 3.2 of Lahiri [19]. With the same moment condition as Politis, Romano and Wolf [29] (Theorem 4.4.1), Theorem 5 additionally shows that the original subsamling estimator S is consistent under non-stationarity with even weaker mixing assumtions than considered reviously k=1 (k + 1)2 α(k) δ/(8+δ) <. The central message of Theorem 5, however, is that the convolved for the samle mean to be es- subsamling aroach allows the block bootstra estimator C n,kn tablished under weak conditions similarly to S.

12 12 TEWES, POLITIS, NORDMAN Next consider the block bootstra in another imortant examle of non-stationarity, involving certain eriodically correlated time series. Here the mean function µ(t) EX t is not constant, as in Theorem 5, but rather an almost eriodic function. A real-valued function f is almost eriodic if, for every ɛ > 0, there is an n(ɛ) N such that in every interval I n(ɛ) of length n(ɛ) or greater, there is an integer I n(ɛ) such that su f(t + ) f(t) < ɛ; t Z see [7]. For such functions the limit M(f) lim n n 1 s+n 1 i=s f(i) exists and does not deend on s. Moreover, if the set Λ = {λ [0, 2π) : M(g λ ) 0} is finite for g λ (t) f(t)e ıλt, t Z (ı = 1), then s+n 1 1 (4.1) (f(i) M(f)) n C n i=s holds for some C > 0 not deending on n or s by Cambanis et al. [6]. Hence, M(f) reresents the mean value of an almost eriodic function f. A time series is called almost eriodically correlated (APC) if its mean and autocovariance functions are almost eriodic, that is, for every fixed τ Z, µ(t) = EX t and ρ τ (t) = EX t X t+τ are almost eriodic as functions of t; see [14]. For an ACP series {X t }, a arameter of interest is then t(p ) M(µ) = lim n n 1 s+n 1 i=s µ(i) as a summary of the rocess mean structure, which is estimated by X n. Synowiecki [34] showed that the block bootstra consistently estimates the samling distribution of T n = n 1/2 ( X n M(µ)) under aroriate conditions. By alying the convolved subsamling technique, we may extend the bootstra results of Synowiecki [34] (Corollary 3.2) by substantially weakening the assumtions made there about (4 + δ)-moments and k=1 kα(k)δ/(4+δ) <. Corollary 3. Let {X t } be an APC sequence of strongly mixing random variables such that su t E X t 2+δ < and k=1 α(k)δ/(δ+2) < for some δ > 0, and suose the set Λ = {λ [0, 2π) : M(g λ ) 0} is finite for g λ (t) µ(t)e ıλt, t Z, with µ(t) = EX t. Then, all conclusions of Theorem 5 hold for T n = n 1/2 ( X n M(µ)) as n Block bootstra for linear time rocesses. Based on a samle X 1,..., X n, next consider inference about the mean EX t = µ R of a stationary time rocess {X t } rescribed as (4.2) X t = µ + j Z a j ε t j, t Z, in terms of iid variables {ε t } with mean zero and finite variance Eε 2 t (0, ) and a real-valued sequence {a j } of constants where j Z a2 j <. The linear series {X t} need not be mixing and, deending on constants {a j }, can otentially exhibit either weak or strong forms of time deendence. Using the samle mean X n to estimate the rocess mean µ, suose that (4.3) lim n nα Var( X n ) = σ 2 for some σ 2 > 0 and exonent α (0, 1] deending on the rocess {X t }. When α = 1, the samle mean s variance decays at a rate O(n 1 ) with the samle size, as tyical for weakly, or

13 CONVOLVED SUBSAMPLING 13 short-range, deendent rocesses. However, when α (0, 1), the samle mean has a variance with comaratively slower decay O(n α ), which may be associated with rocesses exhibiting strong, or long-range, forms of deendence. Long-range deendent rocesses are commonly characterized by slowly decaying covariances involving a long-memory exonent α (0, 1), which results in less recision (4.3) for a samle mean comared to the weak deendence case [3]. Classes of strongly deendent rocesses that satisfy (4.2)-(4.3) include fractional Gaussian models [26] and fractional autoregressive integrated moving averages [13]. Based on (4.3), define T n n α/2 ( X n µ) in terms of scaling τ n n α/2. In this setting, the convolved subsamling C n,kn once again corresonds to the block bootstra estimator based on k n resamled blocks, but there is a wrinkle to note. Recalling from (2.5) that the bootstra samle mean X n1 is created from a bootstra samle of length n 1 = k n b, the bootstra rendition of T n here is given by (4.4) T n b (1 α)/2 (n 1 ) α/2 ( X n 1 E X n1 ) rather than the analog Tn = (n 1 ) α/2 ( X n 1 E X n1 ) of (2.5). While intuitive, the latter is incorrect under long-memory and roduces a degenerate bootstra [18]. Instead, the bootstra form (4.4) requires an adjustment b (1 α)/2, which disaears under weak deendence α = 1 whereby bootstra versions of T n then match in (2.5) and (4.4). Interestingly, convolved subsamling estimator C n,kn automatically corresonds to the correct bootstra rendition Tn in (4.4) under both weak α = 1 and strong α (0, 1) deendence. Considering the samle mean from stationary linear rocesses (4.2) ranging over short- or longrange deendence, Kim and Nordman [16] showed the consistency of the block bootstra distribution C n,kn (when k n = n/b ) and bootstra variance ˆσ. 2 Via convolved subsamling, we may generalize their results. For linear rocesses {X t } satisfying (4.2)-(4.3), the samle mean T n n α/2 ( X n µ) has a normal limit (2.2) (cf. [8]) and the subsamling estimator S is also consistent (i.e., (2.3) holds) under mild assumtions (cf. [27]). Hence, by rimitively assuming (2.2)- (2.3) to hold, Corollary 4 next extends the block bootstra to general stationary rocesses with samle means satisfying a variance condition (4.3), which includes results of Kim and Nordman [16] for linear rocesses as a secial case. Corollary 4. Let {X t } be a stationary rocess with mean µ R satisfying (4.3) for some α (0, 1], and suose that (2.2)-(2.3) hold for T n n α/2 ( X n µ). Then, as n, ˆσ 2 for any ositive integer sequence k n. σ 2 and su C n,kn (x) Φ(x/σ) Corollary 4 is an alication of Theorems 1 and 3 for stationary rocesses which may not be strongly mixing. Our exosition has assumed the exonent α (0, 1] to be known. Uon relacing α with an estimator ˆα ˆα(X 1,..., X n ) where ˆα α log n 0, the conclusions of Corollary 4 still hold; see Remark 3 of [16] for further details Block bootstra under long-range deendence. This section briefly mentions the block bootstra with additional tyes of long-memory sequences. Beyond linear rocesses, the samle mean of a long-range deendent sequence may converge to a non-normal limit, such as the case for certain subordinated Gaussian rocesses considered by Taqqu [35] and Dobrushin and Major [10] (e.g., X t = G(Z t ) as a function G of a long-range deendent Gaussian series {Z t }). For such time series, 0

14 14 TEWES, POLITIS, NORDMAN Lahiri [18] roved that the block bootstra samle mean always has a normal limit, so that the block bootstra fails if the original samle mean is asymtotically non-normal. This result is in concordance with our Theorem 2(ii). Zhang et al. [36] considered subsamling for a wider class of long-memory series that includes both subordinated Gaussian rocesses as well tyes of linear rocesses (4.2). Namely, sequences X t = K(Z t ), t Z, formed by a measurable transformation K of a long-range deendent linear rocess Z t = ε t + j β L(j)ε t j, t Z, j=1 defined with iid mean zero, finite variance innovations {ε t }, an index arameter 1/2 < β < 1, and slowly varying function L( ). They distinguish two cases, deending on β and the so-called ower rank 1 of K. In the first case (i.e., (2β 1) > 1), the transformation K diminishes long-range deendence, and the samle mean converges to a normal limit. In the second case (i.e., (2β 1) < 1), the transformed rocess X t = K(Z t ) remains strongly deendent and the samle mean has a normal limit only when = 1. Assuming a constant function L( ) = C in the above formulation, the variance of a samle mean satisfies (4.3) (i.e., lim n n α Var( X n ) = σ 2 > 0) with a long-memory exonent α min{1, (2β 1)} (0, 1] that changes between cases of weak α = 1 or strong α = (2β 1) (0, 1) deendence (cf. Lemma 1, [36]). For the samle mean, Zhang et al. [36] established consistency of several subsamling estimators as well as convergence of ˆσ 2. Thus, by slightly re-casting results of [36] and alying our Corollary 4, we may show the validity of the block bootstra C n,kn for estimating the distribution of T n n α/2 ( X n µ), µ = EX t, for transformed linear rocesses exhibiting either short- or long-range deendence. To the best of our knowledge, the bootstra has not yet been investigated for such rocesses. Corollary 5. For X t = K(Z t ), t Z, as above, suose (4.3) holds for α = min{1, (2β 1)} (0, 1] along with conditions of Theorem 1 in [36] (involving a block b n a for some a (0, 1)) with either (2β 1) > 1, or = 1 and (2β 1) < 1. Then, for T n = n α/2 ( X n µ) as n, both (2.2)-(2.3) hold and ˆσ 2 for any ositive integer sequence k n. σ 2 and su C n,kn (x) Φ(x/σ) As with subordinated Gaussian rocesses [18], consistency of the block bootstra or convolved estimator C n,kn for the samle mean only follows in cases where a CLT holds. For subordinated Gaussian rocesses and statistics other than the samle mean, Betken and Wendler [4] roved the general consistency of the subsamling estimator, and Bai and Taqqu [2] established weak conditions for the subsamle size. When the original statistic has a normal limit, consistency of convolved subsamling will follow by our Theorem 2 by showing convergence of ˆσ 2 (which, as [2] and [4] consider stationary rocesses, can hold by Theorem 3 and uniform integrability) Satial data. While convolved subsamling results have been resented for rocesses {X t } indexed by time t to ease the exosition, Theorems 1-3 also aly to more general rocesses, including satial random fields. In the sulement [15], we illustrate this with satial data on a grid, for which various authors have considered block bootstra and subsamling; see Lahiri 0

15 CONVOLVED SUBSAMPLING 15 [19] (ch. 12) and Politis, Romano and Wolf [29] (ch. 5) and references therein. Under aroriate assumtions for the stationary random field, the satial samle mean has a normal limit and we establish convolved subsamling under mixing conditions from Lahiri [20] (sec. 4.2) which are almost otimal, or minimal, for a satial CLT. The result given also demonstrates the satial block bootstra for the samle mean under weaker mixing/moment conditions than considered reviously (cf. Theorem 12.1, [19]). 5. Convolved subsamling in other contexts. We briefly outline relationshis between convolved subsamling and some recent literature about block resamling for statistics outside of the samle mean cases in Sections As alternatives to bootstra, such works have considered generalized aroaches to resamling that are convolved subsamling. When viewed as such, these revious develoments may be unified and simlified by general results here for mixing time series (or rocesses) (cf. Section 4.1), as illustrated in Section 5.1 for U-statistics and Section 5.2 for sectral estimators. Section 5.3 mentions extensions to further statistics, such as L-estimators. For the classes of statistics next considered, our results with convolved subsamling cannot be used to directly justify the block bootstra. However, our findings may still contribute to this end, as exlained in Section U-statistics. U-statistics are a class of nonlinear functionals for rescribing statistics, such as the samle variance. Suose that X 1,..., X n arise from a stationary rocess and, based on a symmetric kernel h : R 2 R, define a (bivariate) U-statistic as t n t n (X 1,..., X n ) = 2 n(n 1) 1 i<j n h(x i, X j ), which estimates a target arameter t(p ) h(x, y)dg(x)dg(y), where G denotes the marginal distribution of X t. Consider the roblem of estimating the distribution of T n n(t n t(p )), with scaling τ n = n, under weak time deendence. The subsamling distribution S is defined by comuting the U-statistic t n,b,i = [b(b 1)] 1 2 i j 1 <j 2 i+b 1 h(x j 1, X j2 ) on each length b subsamle {(X i,..., X i+b 1 )} Nn n b+1 i=1 in (2.1). In contrast, block bootstra versions of U-statistics have a formulation similar to (2.5); see Dehling and Wendler [9], Shariov and Wendler [33] and Leucht [23]. That is, a bootstra samle X1,..., X n 1, n 1 = k n b, is generated by resamling k n blocks of length b (tyically k n = n/b ) and then the U-statistic t n 1 t n1 (X1,..., X n 1 ) is calculated from the comlete bootstra samle to create a bootstra rendition Tn = n 1 (t n 1 E t n 1 ) of T n. In this setting, the bootstra distribution Tn would not generally corresond to that of a k n -fold convolution C n,kn of the subsamling distribution S, as occurred in the samle mean case (Section 2.2). However, Shariov, Tewes and Wendler [32] recently considered an alternative block resamling estimator for U-statistics, which matches the convolved subsamling estimator C n,kn here based on the subsamling estimator S for T n described above. Note that, for stationary mixing data, Dehling and Wendler [9] (Theorem 1.8-Lemma 3.6) rovide a CLT for the relevant U-statistic: d T n N(0, σ 2 ) and ETn 2 σ 2 as n where σ 2 4 k= Cov(h 1(X 0 ), h 1 (X k )) for h 1 (x) = h(x, y)dg(y). Under mixing conditions and with kn = n/b, Shariov, Tewes and Wendler [32] established that C n,kn catures this limiting normal distribution of T n and also showed the consistency of the variance ˆσ 2 of C n,kn. The argument there involved decomosing the bootstra U-statistic Tn into a linear art, coinciding with a samle mean from the usual block bootstra, and degenerate art shown to be negligible. However, the general convolution result in Theorem 4 d for mixing rocesses rovides an alternative, and much simler, aroach. From T n N(0, σ 2 ) and

16 16 TEWES, POLITIS, NORDMAN ETn 2 σ 2, all of the conditions of Theorem 4 automatically hold, roving that C n,kn is consistent for the distribution of T n for any convolution level k n and also that ˆσ 2 σ 2. This aroach also weakens the block assumtions used by [32] (i.e., b = O(n ɛ ) for some ɛ (0, 1)) to b 1 + b/n 0 under Theorem Sectral estimators for non-stationary time series. As described in Section 4.2, almost eriodically correlated (APC) time series {X t } are an imortant examle of non-stationary sequences. Beyond the mean function, inference about the correlation structure is also of interest. Based on a samle X 1,..., X n, a symmetric kernel w( ) and a bandwidth choice L n, Lenart [21, 22] considered kernel estimators n n t n (X 1,..., X n ) 1 2πn t=1 s=1 1 L n w ( t s L n ) X t X s e ıυt e ıωs for an extended sectral density t(p ) t(p )(υ, ω), (υ, ω) (0, 2π] 2, used to reresent the almost eriodic covariance function c τ (t) Cov(X t, X t+τ ), t Z, for a given τ Z; see [21, 22] for details. For T n τ n (t n t(p )) with scaling τ n = n/l n, Lenart [21] (Theorems ) roved a d CLT T n N(0, σ 2 ) and moment convergence ETn 2 σ 2 with mixing APC series, which was extended in Lenart [22] to multivariate data. Due to the comlicated form of σ 2, a subsamling estimator S for the distribution of T n may comuted as in (2.1) with analog statistics t n,b,i and scaling τ b = b/l b defined from subsamles {(X i,..., X i+b 1 )} Nn n b+1 i=1. Lenart [21] roved the consistency of the estimator S, while Lenart [22] roosed a generalized resamling method which essentially corresonds to a convolved subsamling estimator C n,kn induced from S. In articular, Lenart [22] established the consistency of C n,kn through bootstra arguments requiring d much stronger mixing and moment assumtions than needed for the convergence T n N(0, σ 2 ) and ETn 2 σ 2. However, the general convolved subsamling result in Theorem 4 may alternatively be used here with mixing non-stationary ACP series. To aly Theorem 4 with blocks where b 1 + b/n + τ b /τ n 0 as n, one requires that d N(0, σ 2 ) and that (3.2) holds, where Y b, b b n 1, denotes a sequence of variables with Y b distribution D n,b ( ) from (3.1). But, the same conditions needed for T n d N(0, σ 2 ) and ET 2 n σ 2 d also yield Y b N(0, σ 2 ) and EYb 2 σ2 (cf. Theorems and 4.1, [21]). Furthermore, mixing d and Y b N(0, σ 2 ), along with T n = O (1) and τ n /τ b, guarantee that (2.3) holds (i.e., su S (x) Φ(x/σ) 0) and that consequently (3.2) follows from Theorem 3(ii) by EYb 2 σ 2. That is, the same minimal conditions for a CLT with APC series suffice for the consistency of convolved subsamling C n,kn by the general result of Theorem L-estimators and other statistics beyond the samle mean. Other classes of statistics with normal limits, where resamling may be helful, include M-estimators, L-statistics and generalizations such as GL-statistics. Just as for the U-statistics and sectral estimators in the revious sections, convolved subsamling may be alied as a resamling method which is neither classical subsamling nor the block bootstra for general statistics. As an advantage in such cases, convolved subsamling is often verifiable under mild assumtions (cf. Sections 3, 4.1), which we mention for L-estimators. If X 1,..., X n denotes a stationary stretch with marginal quantile function G 1, an L-estimator t n = 1 0 Ĝ 1 n (u)j(u)du of a arameter t(p ) = 1 0 G 1 (u)j(u)du reresents a linear combinations of order statistics, defined by the quantiles of the emirical distribution Ĝ n (x) = n 1 n t=1 I(X t x), x R, and a weighting function J : [0, 1] R. Under the mild mixing/moment conditions used for samle means in Theorem 5, the same conclusions for subsamling