Tails of Polynomials of Random Variables and Stable Limits for Nonconventional Sums

Abstract

First, we obtain decay rates of probabilities of tails of polynomials in several independent random variables with heavy tails. Then we derive stable limit theorems for sums of the form $\sum _{Nt\ge n\ge 1}F\big (X_{q_1(n)},\ldots ,X_{q_\ell (n)}\big )$ where F is a polynomial, $q_i(n)$ is either $n-1+i$ or ni and $X_n,n\ge 0$ is a sequence of independent identically distributed random variables with heavy tails. Our results can be viewed as an extension to the heavy tails case of the nonconventional functional central limit theorem from Kifer and Varadhan (Ann Probab 42:649–688, 2014).

Appendix

We will start here with a basic property of the $J_1$-convergence. Let X and Y be two Polish spaces and D([0, 1]; X) and D([0, 1]; Y) are spaces of X valued and Y valued functions that are right continuous, have left limits at every $t\in [0,1]$ and are, in addition, left continuous at 1. In general, if $x_n(\cdot )\in D([0,1];X)$ and $y_n(\cdot )\in D([0,1];Y)$ converge in $J_1$ topology respectively to x(t) and y(t) then it does not follow that $z_n(\cdot )=(x_n(\cdot ),y_n(\cdot ))$ converges to $z(\cdot )=(x(\cdot ),y(\cdot ))\in D([0,1];Z)$ where $Z=X\times Y$. It is possible that $x_n(\cdot )$ and $y_n(\cdot )$ have jumps at two distinct points $t_n, t^\prime _n$ that tend to a common point t. As functions with values in $Z=X\times Y$, $z_n(t)$ has two jumps that come together and this rules out convergence in D([0, 1]; Z). However we have the following

Lemma 7.1

Let $x_n(\cdot )$ and $y_n(\cdot )$ converge in $J_1$ topology to $x(\cdot )$ and $y(\cdot )$ respectively in D([0, 1]; X) and D([0, 1]; Y) . Let $x(\cdot )$ and $y(\cdot )$ do not have a jump at the same t, i.e the sets of discontinuity points of $x(\cdot )$ and $y(\cdot )$ are disjoint. Then $z_n(\cdot )= (x_n(\cdot ),y_n(\cdot ))$ converges in $J_1$ topology to $z(\cdot )=(x(\cdot ), y(\cdot ))$ in D([0, 1]; Z) where $Z=X\times Y$. In particular if $f:Z\rightarrow S$ is a continuous map then $f(x_n(\cdot ),y_n(\cdot ))$ converges in the $J_1$ topology to $f((x(\cdot ),y(\cdot ))\in D([0,1];S)$ .

Proof

Since $x_n(t)$ and $y_n(t)$ converge in the $J_1$ topologies on X and Y, respectively, there are compact sets $K_X$ and $K_Y$ in X and Y such that $x_n(t)\in K_X$, and $y_n(t)\in K_Y$ for all n and $t\in [0,1]$. Therefore $z_n(t)\in K=K_X\times K_Y$ for all n and $t\in [0,1]$. We need to control uniformly the D[0, 1] modulus of continuity $\omega _h(z_n(\cdot ))$ of $z_n(t)=(x_n(t),y_n(t))$ where

$$\begin{aligned} \omega _h(z(\cdot ))=\sup _{t_1,t_2: |t_1-t_2|\le h}\inf _{\tau \in (t_1,t_2)} \max [\Delta _{(t_1,\tau )}(z(\cdot )),\Delta _{(\tau ,t_2)}(z(\cdot ))] \end{aligned}$$

and

$$\begin{aligned} \Delta _{(a,b)}(z(\cdot ))=\sup _{t,s\in (a,b)} d(z(t),z(s)) \end{aligned}$$

is the oscillation of $z(\cdot )$ in the interval (a, b). We can take $d(z_1,z_2)=d_1(x_1,x_2)+d_2(y_1,y_2)$ where $d, d_1, d_2$ are the metrics in Z, X, Y respectively. Then with the obvious definitions of $\Delta _{(a,b)} (x(\cdot ))$ and $\Delta _{(a,b)}(y(\cdot ))$,

$$\begin{aligned} \omega _h(z(\cdot ))\le&\sup _{t_1,t_2: |t_1-t_2|\le h}\inf _{\tau \in (t_1,t_2)} \max [\Delta _{(t_1,\tau )}(x(\cdot )),\Delta _{(\tau ,t_2)}(x(\cdot ))]\\&\quad + \sup _{t_1,t_2: |t_1-t_2|\le h}\sup _{\tau \in (t_1,t_2)} \max [\Delta _{(t_1,\tau )}(y(\cdot )),\Delta _{(\tau ,t_2)}(y(\cdot ))]. \end{aligned}$$

The convergence of $x_n(\cdot )$ and $y_n(\cdot )$ in the corresponding $J_1$ topologies guarantees that

$$\begin{aligned} \lim _{h\rightarrow 0}&\limsup _{n\rightarrow \infty }\bigg [\sup _{t_1,t_2: |t_1-t_2|\le h} \inf _{\tau \in (t_1,t_2)} \max [\Delta _{(t_1,\tau )}(x_n(\cdot )), \Delta _{(\tau ,t_2)}(x_n(\cdot ))]\\&\quad + \sup _{t_1,t_2: |t_1-t_2|\le h}\inf _{\tau \in (t_1,t_2)} \max [\Delta _{(t_1,\tau )}(y_n(\cdot )),\Delta _{(\tau ,t_2)}(y_n(\cdot ))]\bigg ]=0. \end{aligned}$$

Since the jumps of $x_n(\cdot )$ and $y_n(\cdot )$ converge individually to the jumps of $x(\cdot )$ and $y(\cdot )$ while $x(\cdot )$ and $y(\cdot )$ do not have any common jumps, for any $\epsilon >0$ there is a $\delta >0$ such that all the jumps of $x_n(\cdot )$ and $y_n(\cdot )$ of size at least $\epsilon >0$ are uniformly separated from one another by some $\delta =\delta (\epsilon )>0$ . We can now estimate the D[0, 1] modulus continuity of $z_n(\cdot )$. If $h<\delta$ any interval of length h will have at most one jump of size larger than $\epsilon$. Therefore of the two components $x_n(\cdot )$ and $y_n(\cdot )$ only one of them can have a jump larger than $\epsilon$. If $y_n(t)$ does not have a jump of size larger than $\epsilon$ in $(t_1,t_2)$ and $|t_2-t_1|<h$ then

$$\begin{aligned}&\sup _{\tau \in (t_1,t_2)} \max \left[ \Delta _{(t_1,\tau )}(y_n(\cdot )),\Delta _{(\tau , t_2)}(y_n(\cdot ))\right] \le \Delta _{(t_1,t_2)}(y_n(\cdot ))\\&\le 2\inf _{\tau \in (t_1,t_2)} \max \left[ \Delta _{(t_1,\tau )}(y_n(\cdot )), \Delta _{(\tau ,t_2)}(y_n(\cdot ))\right] +\epsilon \end{aligned}$$

Therefore

$$\begin{aligned} \lim _{h\rightarrow 0}\limsup _{n\rightarrow \infty } \omega _h(z_n(\cdot ))=0 \end{aligned}$$

and we are done. $\square$

Corollary 7.2

• (i) If $x^i_n$ converge to $x^i$, $i=1,...,d$ in the $J_1$ topology on $D([0,1]; X_i)$, where $X_i$ are Polish spaces, and for any pair $i\not =j$ the limits $x^i$ and $x^j$ have no common jumps, i.e the discontinuity points $U_l=\{t: x^l(t-0)\not = x^l(t\})$ for $l=i$ and $l=j$ are disjoint, then as $n\rightarrow \infty$ the d-vector functions $\{ x_n^i,\, i=1,...,d\}$ converge to $\{ x^i,\, i=1,...,d\}$ in the $J_1$ topology of the product space $D([0,1];\prod ^d_{i=1}X_i)$.

• (ii) Let $x_n^i$ and $x^i$ satisfy conditions of (i) with $X_i=R^q,\, i=1,...,d$ for some integer $q\ge 1$. Then $\sum _{i=1}^dx_n^i$ converges in the $J_1$ topology of $D([0,1];R^q)$ to $\sum _{i=1}^dx^i$.

• (iii) Let $\{P_n,\, n\ge 1\}$ be probability measures on $D([0,1]; \prod _{i=1}^dX_i)$, where $X_i, i=1,...,d$ are the same as in (i), and suppose that the marginals on $D([0,1];X_i),\, i=1,...,d$ of $P_n$’s converge weakly with respect to the $J_1$ topology while the joint finite dimensional distributions of $P_n$ converge on $D([0,1];\prod _{i=1}^dX_i)$ to a limit P. If P almost surely the components of d-vector functions $(x_1(t),...,x_d(t))$ have no common jumps pairwise then $P_n$ weakly converges to P as $n\rightarrow \infty$ in the $J_1$ topology.

While considering real valued random variables in the domain of attraction of a stable law it is natural to consider tail behavior of the form

$$\begin{aligned} \lim _{T\rightarrow \infty } T^{\alpha }(\ln T)^{-k} P\{\pm X\ge T\}=c_\pm . \end{aligned}$$

(7.1)

If X is $R^d$ valued then a tail behavior similar to the one dimensional case above will be to require that for every continuous function f on the unit sphere $S^{d-1}$ the limit

$$\begin{aligned} \lim _{\rho \rightarrow \infty }\rho ^{\alpha } (\ln \rho )^{-k}E\left[ \mathbf{1}_{|X|\ge \rho } f\left( \frac{X}{|X|}\right) \right] =\int _{S^{d-1}} f(s)\nu (ds) \end{aligned}$$

(7.2)

exists where $\nu$ is a finite nonnegative measure on $S^{d-1}$. To make the connection we need only to think of $S^0$ as $\pm 1$ and $\nu (\{\pm 1\})=c^\pm$. The following result essentially coincides with Theorem 3.6 of [13] but for readers’ convenience we provide its proof here.

Lemma 7.3

The relation (7.2) holds true for every bounded continuous f on $S^{d-1}$ if and only if

$$\begin{aligned} \lim _{\rho \rightarrow \infty } \rho ^{\alpha }(\ln \rho )^{-k} E\left[ W\left( \frac{X}{\rho }\right) \right] = \int _{S^{d-1}}\int _0^\infty W(sr)\frac{\alpha \nu (ds)\,dr}{r^{1+\alpha }} \end{aligned}$$

(7.3)

for every W from the space ${\mathcal W}$ of bounded continuous functions satisfying (2.2).

Proof

In (7.2) we can replace $\rho$ by $\rho z$ with $z>0$, to get

$$\begin{aligned} \lim _{\rho \rightarrow \infty }\rho ^{\alpha } (\ln \rho )^{-k}E\left[ \mathbf{1}_{\frac{|X|}{\rho }\ge z}f\left( \frac{X}{|X|}\right) \right]&=\frac{1}{z^{\alpha }}\int _{S^{d-1}} f(s)\nu (ds)\nonumber \\&=\int _{S^{d-1}} f(s)\nu (ds)\int _z^\infty \frac{\alpha }{u^{1+\alpha }}du \end{aligned}$$

(7.4)

It is now easy to conclude that if V(r, s) is a continuous function of $r>0$ and $s\in S^{d-1}$ and for some $\delta >0$ it is identically 0 if $r\le \delta$ then

$$\begin{aligned} \lim _{\rho \rightarrow \infty }\rho ^{\alpha } (\ln \rho )^{-k}E\left[ V\left( \frac{|X|}{\rho }, \frac{X}{|X|}\right) \right] =\int _0^\infty \int _{S^{d-1}} V(u,s)\nu (ds)\frac{\alpha }{u^{1+\alpha }}du. \end{aligned}$$

(7.5)

We now take $V(r,s)=W(rs)$ and obtain (7.3). To control the contribution near 0 for $W\in \mathcal W$ we denote by R(x) the tail probability $P[|X|\ge x]$ and obtain

$$\begin{aligned} \rho ^\alpha (\ln \rho )^{-k} E\left[ W\left( \frac{X}{\rho }\right) \mathbf{1}_{\frac{|X|}{\rho } \le \delta }\right]\le & {} C\rho ^\alpha (\ln \rho )^{-k} E\left[ \left( \frac{X^2}{\rho ^2}\right) \mathbf{1}_{\frac{|X|}{\rho }|\le \delta }\right] \\= & {} C \rho ^{\alpha -2} (\ln \rho )^{-k} E\left[ X^2\mathbf{1}_{|X|\le \delta T}\right] \\= & {} -C T^{\alpha -2} (\ln T)^{-k} \int _0^{\delta \rho } x^2 dR(x)\\\le & {} C \rho ^{\alpha -2} (\ln \rho )^{-k} \int _0^{\delta \rho } 2\,x\,R(x)\,dx\\\le & {} C\rho ^{{\alpha }-2} (\ln \rho )^{-k}\int _0^{\delta \rho } x (1+x)^{-\alpha } (\ln (2+x))^kdx\\\le & {} C\delta ^{2-\alpha } \end{aligned}$$

is uniformly controlled because $\alpha <2$. Finally to go from (7.3) to (7.2), we take $W(x)=\mathbf{1}_{[1,\infty ]}(x) f(\frac{x}{|x|})$ which can be justified by approximating $\mathbf{1}_{[1,\infty ]}$ by continuous functions. $\square$

Remark 7.4

Let $\{X_n\}$ be a sequence of independent and identically distributed random vectors in $R^d$ that satisfy (7.3). Let

$$\begin{aligned} S_N(t)=\sum _{1\le n\le Nt} X_n \end{aligned}$$

and

$$\begin{aligned} \** _N(t)=\frac{1}{b_N}[S_N(t)-Nta_N] \end{aligned}$$

(7.6)

where the normalizer $b_N$ is given by

$$\begin{aligned} b_N=N^{\frac{1}{{\alpha }}}\left( \frac{\ln N}{{\alpha }}\right) ^{\frac{k}{{\alpha }}} \end{aligned}$$

and the centering $a_N$ is given by

$$\begin{aligned} a_N=E\left[ \frac{b_N^2 X_1}{b_N^2+|X_1|^2}\right] . \end{aligned}$$

Then, according to standard limit theorems for sums of independent random vectors (see, for instance, [14] and Sect. 7.2 in [13]), the processes $\** _N(t)$ converge in the Skorokhod $J_1$ topology on $D[[0,T]; R^d]$ to a limiting stable process $\**$ with the characteristic function of the increments $\** (t)-\** (s)$ given by

$$\begin{aligned} E[e^{i\langle \xi ,\** (t)-\** (s)\rangle }]=\exp [(t-s)\psi (\xi )] \end{aligned}$$

(7.7)

where

$$\begin{aligned} \psi (\xi )&=\int _{R^d\setminus \{0\}} \left[ e^{i\langle \xi , y\rangle }-1- \frac{i\langle \xi ,y\rangle }{1+|y|^2}\right] \mu (dy)\\&=\int _{S^{d-1}}\int _0^\infty \left[ e^{i\langle \xi , sr\rangle }-1- \frac{i\langle \xi ,sr\rangle }{1+|r|^2}\right] \frac{\nu (ds) dr}{r^{1+\alpha }} \end{aligned}$$

It follows that if $\nu$ is concentrated on axes then components of the process X(t) are independent. The proof relies on the calculation

$$\begin{aligned} \lim _{N\rightarrow \infty } N\ln [E[e^{i\langle \xi , X^\prime _N\rangle }]]=\psi (\xi ) \end{aligned}$$

where $X^\prime _N=\frac{1}{b_N}[X_1-a_N]$.

Remark 7.5

The basic assumptions involved in proving the convergence to the stable process with independent increments is independence of the random variables, a common distribution with the correct tail behavior that puts them in the domain of attraction of the stable distribution with Lévy measure $\frac{\nu (ds)dr}{r^{1+\alpha }}$ on $R^d$. It is possible that we have a random vector in which two components are not independent but in the limiting distribution they become independent. If the Lévy measure is $\frac{\nu (ds)dr}{r^{1+\alpha }}$, then for $R^d$ to split as a sum $\oplus V_j$ with components of the random vector corresponding to different $V_j$ being mutually independent it is necessary and sufficient that $\nu$ is supported on the union of the subspaces $V_j$. In other words $\nu [ |x_j|>0, |x_{j'}|>0] =0$ where $x=\sum x_j$ is the natural decomposition of x into components from $\{V_j\}$. This requires that

$$\begin{aligned} \lim _{\rho \rightarrow \infty } \rho ^{{\alpha }}(\ln \rho )^{-k} P\{|X_{1j}|>\rho , |X_{1j'}|> \rho \}=0 \end{aligned}$$

for any $j\ne j'$ where $X_1=(X_{11},X_{12},...,X_{1d})$ and $X_1$ is the same as in Remark 7.4.

Remark 7.6

Let $\{ X_n\}$ be as in Remark 7.4 and T be a linear map $R^d\rightarrow R^m$. Then the process

$$\begin{aligned} Y_N(t)=T\** _N(t)=\sum _{1\le n\le Nt} \frac{1}{b_N}(TX_n-Ta_N) \end{aligned}$$

will converge to $Y(t)=T\** (t)$, a process with independent increments given by

$$\begin{aligned} E\big [e^{i<\xi ',Y(t)-Y(s)}\big ]=\exp \big [(t-s)\big [i \langle \gamma , \xi ' \rangle +\psi '(\xi ') \big ]\big ] \end{aligned}$$

where

$$\begin{aligned} \psi ^\prime (\xi ^\prime )=\int _{R^m\setminus \{0\}} \left[ e^{i\langle \xi ', s'r \rangle }-1- \frac{i\langle \xi ',sr \rangle }{1+|r|^2}\right] \frac{\nu '(ds') dr}{r^{1+\alpha }} \end{aligned}$$

and

$$\begin{aligned} \gamma '=\int _{R^d} \bigg [\frac{T(sr)}{1+|T(sr)|^2} - \frac{T(sr)}{1+|r|^2} \bigg ] \frac{\nu (ds) dr}{r^{1+\alpha }}. \end{aligned}$$

The amount by which the process needs centering is only unique up to a constant and this requires us to make the adjustment with the term $\gamma ^ \prime t$ which is the difference between two possibly infinite terms. It is not hard to see that as $N\rightarrow \infty$, $b_N\rightarrow \infty$ and by the bounded convergence theorem, $\frac{a_N}{b_N}\rightarrow 0$.

In Sect. 5 we proved convergence of the finite dimensional distributions to a stable Lévy process, by appealing to [16]. This required a rearrangement of the monomial terms, that make up the polynomial. The new process converged in $J_1$ topology. If condition (7.13) below were satisfied then the rearrangement was not needed and the original process converged in the $J_1$ topology. We provide here an alternate proof that does not require this modification, but yields directly finite dimensional convergence for stationary $\ell$ dependent sums as well as convergence in $J_1$ topology under the additional assumption (7.13).

Let H be a finite dimensional Euclidean space and $\{X_i,\, i\ge 1\}$ be a stationary sequence of H valued random variables that are $\ell$-dependent and have regularly varying heavy tails with an index ${\alpha }\in (0,2)$ and a Lévy measure M, i.e. (7.2) holds true with $X=X_1$ and $\nu =M$. We will assume that for $d=2\ell -1$ and all $f\in {\mathcal W}(H^d)$,

$$\begin{aligned} \lim _{N\rightarrow \infty } N E\left[ f\left( \frac{X_1}{b_N},\ldots , \frac{X_d}{b_N}\right) \right] =\int _{H^d} f(x_1,\ldots ,x_d) M^d (dx). \end{aligned}$$

(7.8)

Observe that when the stationary $\ell$-dependent sequence of random vectors above is obtained in the framework of Theorem 2.3 then relying on Lemma 4.1 and Corollary 4.4 we see that the condition (7.8) is automatically satisfied.

Lemma 7.7

Suppose that (7.8) holds true. Then for any integer $k\ge 1$ the limit

$$\begin{aligned} \lim _{N\rightarrow \infty } N E\left[ f\left( \frac{X_1}{b_N},\ldots , \frac{X_k}{b_N}\right) \right] =\int _{H^d} f(x_1,\ldots ,x_k) M^k (dx) \end{aligned}$$

(7.9)

exists and $M^k$ can be computed from $M^d$.

Proof

If for $k>d$ the limits do not exist then because of stationarity we can always select a subsequence such that the limits exist for all values of k and $f\in {\mathcal W}(H^k)$ and we will continue to denote corresponding limiting measures by $M^k$. If we can recover $M^k$ from $M^d$, then all subsequences will have the same limit and hence the limit as $N\rightarrow \infty$ will exist. For $-\infty<a\le b<\infty$ with $b-a\le 2\ell -2$, $J=\{i: a\le i\le b\}$ will be of size at most $2\ell -1$ and the limits $M^J$ on $H^J$ will exist and be translation invariant. For any partition of J into disjoint subsets A and B we can write $H^J\backslash \{0\}$ as the disjoint union

$$\begin{aligned} H^J\backslash \{0\}=\cup _{ A \in {\mathcal A}(J)}H^J_A \end{aligned}$$

where ${\mathcal A}(J)=\{ A:\, (A,B)\in {\mathcal P}(J)\}$ and ${\mathcal P}(J)$ is the set of partitions with nonempty A and

$$\begin{aligned} H^J_A=\{x\in H^J: x_i\not =0\ \forall \ i\in A;\ x_i=0\ \forall i\in B\}. \end{aligned}$$

We can write $M^J=\sum _{A\in {\mathcal A}(J)}M^J_A$, where $M^J_A$ is the restriction of $M^J$ to $H^J_A$. By the $\ell$ dependence we obtain that if $|i-j|\ge \ell$ then for any $\delta >0$

$$\begin{aligned} \lim _{N\rightarrow \infty } N P\{|X_i|\ge \delta b_N, |X_j|\ge \delta b_{N}\}=0 \end{aligned}$$

which in turn implies that $M^J[ x_i\not =0, x_j\not =0]=0$ whenever $i,j\in J$ and $|i-j|\ge \ell$. Any set $A\in \mathcal A(J)$ can be ordered $a_1<a_2<\ldots a_r$ and integers r, and $\sigma _i= a_{i+1}-a_i$ for $i=1,\ldots ,r-1$ determine A up to a translation. Set $sp(A)=1+\sigma _1+\cdots +\sigma _r=a_r-a_1+1$. Then $M^J_A=0$ unless $sp(A)\le \ell$. For any A consider the extended interval ${\widehat{A}}=\{a_r-\ell +1\le i\le a_1+\ell -1\}$. It is the set of integers j such that $|j-i|\le \ell -1$ for all $i\in A$. In particular $M^J_A$ is determined by $M^{\widehat{A}}_A$ if $J\supset \widehat{A}$. We write

$$\begin{aligned} M^J=\sum _{A\in {\mathcal A}(J)}M^J_A=\sum _{\begin{array}{c} A\in {\mathcal A}(J)\\ {\widehat{A}} \subset J \end{array}}M^J_A+\sum _{\begin{array}{c} A\in {\mathcal A}(J)\\ {\widehat{A}}\not \subset J \end{array}}M^J_A. \end{aligned}$$

Since $M^J$ determines $M^J_A$ it determines also $M^{\widehat{A}}_A$ if ${\widehat{A}}\subset J$. If ${\widehat{A}}\not \subset J$ we can replace J by $I=J\cup {\widehat{A}}$ and

$$\begin{aligned} M^J_A=\sum _{B\supset A} M^I_B=\sum _{B\supset A} M^{\widehat{B}}_B \end{aligned}$$

Clearly, $M^{\widehat{B}}_B=0$ unless $sp(B)\le \ell$ and so $sp(\widehat{B}) \le 2\ell -1$. By translation invariance all these measures are determined by $M^{2\ell -1}$. $\square$

Sets with $sp(A)\le \ell$ can be characterized by r and $a_1<a_2 \ldots <a_r$ with $a_r-a_1\le \ell -1$ and up to a translation by r and $\sigma _i\ge 1$ for $i=1,\ldots ,r-1$ with $\sum _{i=1}^{r-1}\sigma _i\le 1$. We map $H^{\widehat{A}}$ into H by $x=\sum _{i\in {\widehat{A}}} x_i$ and the push forward of the measure $M^{\widehat{A}}_A$ is denoted by $M^{r, \{\sigma _i\}}$ on H. Let

$$\begin{aligned} M^*=\sum _{r,\sigma _1,\ldots ,\sigma _r} M^{r,\{\sigma _i\}}. \end{aligned}$$

(7.10)

For each $r,\{\sigma _i\}$ we set

$$\begin{aligned} \gamma _{r,\{\sigma _i\}}=\int _{H^{\widehat{A}} }\left[ \frac{\sum _{i\in A} x_i}{1+|\sum _{i\in A} x_i|^2}-\sum _{i\in A} \frac{x_i}{1+|x_i|^2}\right] M^{\widehat{A}}_A( dx) \end{aligned}$$

and let

$$\begin{aligned} \gamma =\sum _{r,\{\sigma _i\}} \gamma _{r,\{\sigma _i\}}. \end{aligned}$$

We note that $\gamma _{1,\{\sigma _i\}}=0$ because A contains only one integer.

Theorem 7.8

Under the condition (7.8) the finite dimensional distributions of the process

$$\begin{aligned} \xi _N(t)=\sum _{1\le i\le Nt}\frac{1}{b_N} [X_i-a_N] \end{aligned}$$

(7.11)

converge weakly to those of a stable process with the logarithm of its characteristic function at time t given by

$$\begin{aligned} \log \psi _t(u)=it\langle {\gamma },u\rangle +t\int _H \left[ e^{i\langle u,x \rangle }-1- \frac{i\langle u, x\rangle }{1+\Vert x\Vert ^2} \right] M^*(dx) \end{aligned}$$

(7.12)

where $M^*$ is computed from $M^d$ as in (7.10). In addition, if for all $i\not = j$

$$\begin{aligned} \lim _{N\rightarrow \infty } N P\{|X_i|\ge \delta b_N, |X_j|\ge \delta b_{N}\}=0 \end{aligned}$$

(7.13)

then there is no need in the assumption (7.8) as (7.9) automatically holds true for any integer $k\ge 1$ and the weak convergence as $N\rightarrow \infty$ of the processes $\xi _N$ takes place in the $J_1$ topology. Furthermore, in the latter case $M^*=M$.

Note that since $\frac{a_N}{b_N}\rightarrow 0$ as $N\rightarrow \infty$ (7.13), is equivalent to

$$\begin{aligned} \lim _{N\rightarrow \infty } N P\{|X_i-a_N|\ge \delta b_N, |X_j-a_N|\ge \delta b_{N}\}=0 \end{aligned}$$

Proof

Let k be an integer that will eventually get large. To exploit $\ell$ dependence we want to sum over blocks of size k and leave gaps of size $\ell -1$. We divide the set of positive integers into blocks of size $k+\ell -1$. $B(r)=\{i: (r-1)(k+\ell -1)+1\le i\le (r-1)(k+\ell -1)\}$. Each B(r) consists of the initial segment $B^+(r)$ of length k and the gap $B^-(r)$ of size $\ell -1$. $Z^+=\cup _r B^+(r)$ and $Z^-(r)=\cup _r B^-(r)$. We define

$$\begin{aligned} \xi ^k_N(t)=\frac{1}{b_N}\sum _{\begin{array}{c} 1\le i\le Nt\\ i\in Z^+ \end{array}}[X_i-a_N] \end{aligned}$$

and

$$\begin{aligned} \eta ^k_N(t)=\frac{1}{b_N}\sum _{\begin{array}{c} 1\le i\le Nt\\ i\in Z^- \end{array}}[X_i-a_N] \end{aligned}$$

so that $\xi _N(t)=\xi ^k_N(t)+\eta _N^k(t)$. It follows from Lemma 6.2 considered with $X_i$’s in place of $Y_i(\theta )$’s (as the proof in this lemma does not rely on a specific monomial form of summands there) that for any $\epsilon >0$,

$$\begin{aligned} \lim _{k\rightarrow \infty } \limsup _{N\rightarrow \infty }P\left\{ \sup _{0\le t\le T} \big |\eta _N^k(t)\big |\ge \epsilon \right\} =0 \end{aligned}$$

(7.14)

It is enough to show that for fixed k the limit theorem is valid for $\xi _N^k(t)$ as $N\rightarrow \infty$. We can then let $k\rightarrow \infty$ similarly to the end of Sect. 6. We denote the block sums by $Y_i=\sum _{j\in B^+(i)} X_j$ and observe that $Y_1, Y_2,...$ are i.i.d. by the $\ell$-dependence and stationarity. Then

$$\begin{aligned} \xi _N^k(t)=\frac{1}{b_N}\sum _{j: j(k+\ell -1)\le Nt} [Y_j-ka_N]+\frac{1}{b_N} \sum _{i\in I(t)}[X_i-a_N]=\zeta ^k_N(t)+R^k_N(t)\quad \quad \quad \end{aligned}$$

(7.15)

where I(t) is an incomplete block at the end. If we want to show convergence of finite dimensional distributions to the Lévy process given by (7.10) we need to show that

$$\begin{aligned} \lim _{k\rightarrow \infty }\lim _{N\rightarrow \infty } \frac{N}{k}E\left[ f\left( \frac{X_1+\cdots +X_k }{b_N}\right) \right] =\int _H f(x) M^*(dx). \end{aligned}$$

In order to complete the proof of finite dimensional convergence we need to check only that for each fixed t the second term in (7.15) is negligible in probability. Since k is fixed and $N\rightarrow \infty$ this is obvious. The tails behavior of $\frac{X_1+\cdots +X_k}{b_N}$ is given by the image of $M^k$ by the map $x_1+x_2+\cdots +x_k\rightarrow x$ of $H^k\rightarrow H$. Since $M^k=\sum _A M^k_A$ and they are 0 unless $sp(A)\le \ell$, averaging over translations of A, ignoring a few terms at the ends, produces as $k\rightarrow \infty$, $M^*$ for the Lévy measure. The effect of the gap is a factor of $\frac{k}{k+\ell -1}$ that multiplies $M^*$ which tends to 1 as $k\rightarrow \infty$. Any partial block consists of a sum of at most k terms of $\frac{X_i}{b_N}$ and is negligible for large N. So is $\frac{a_N}{b_N}$.

Next, we need to verify the centering. We use the truncated mean $\frac{x}{1+|x|^2}$ which then appears in the representation as a counter term in the integrand

$$\begin{aligned} \int \Big [e^{i\langle \xi ,x \rangle }-1-\sum _j \frac{ i\,\xi _j\,x_j}{1+|x_j|^2}\Big ] M^{\widehat{A}}_A(dx) \end{aligned}$$

(7.16)

But when we push forward the measure $M^{\widehat{A}}$ from $H^{\widehat{A}} \rightarrow H$ by taking the sum $y=\sum _{i\in A} x_i$ we end up with

$$\begin{aligned}&\int \Big [e^{i\,\xi \,y}-1- \frac{ i\,\xi \,y}{1+|y|^2}\Big ] M^{r, \{\sigma _i\}}(dy)\nonumber \\&\quad =\int \Big [e^{i\,\xi \,y}-1-\frac{ i\,\xi \,\sum _i x_i}{1+ |\sum _i x_i|^2}\Big ] M^{\widehat{A}}_A(dx) \end{aligned}$$

(7.17)

The difference between the two counter terms in (7.16) and (7.17) is $\gamma _{r,\{\sigma _i\}}$ and it adds up to

$$\begin{aligned} \gamma =\sum _{r,\{\sigma _i\}}\gamma _{r,\{\sigma _i\}} \end{aligned}$$

which appears outside of the integral in (7.12) and does not influence the limiting measure $M^*$.

Now, if the condition (7.13) is satisfied then convergence in the $J_1$ topology follows from Proposition 5.1 obtained in [16] as a corollary of a more general result but we will still give an alternative direct proof below. Since convergence of finite dimensional distributions was already obtained above it remains to establish tightness of the processes $\xi _N^k,\, N\ge 1$ and the conclusion of the proof is similar to Sect. 6 by letting $k\rightarrow \infty$. Observe though that the convergence of finite dimensional distributions was established above under the condition (7.8) while we claim that (7.13) already implies (7.8), and so we discuss this issue first.

Set $m=$dimH and let $Z_i=(X_j,\, j\in B^+(i))$ be mk-dimensional random vectors whose m-dimensional components are $X_j$’s and their sum amounts to $Y_i$. The random vectors $Z_i,\, i=1,2,...$ are i.i.d. and in view of (7.13) we conclude by Lemma 4.1 that they have regularly varying tails and, in particular, that (7.8) holds true. Thus by [14] (or by Sect. 7.2 in [13]) the vector process

$$\begin{aligned} \Psi ^k_N(t)=\frac{1}{b}_N\sum _{i:\, i(k+\ell -1)\le Nt}\big (X_j-a_N,\, j\in B^+(i)\big ) \end{aligned}$$

converges weakly in the $J_1$ topology as $N\rightarrow \infty$ to a vector Lévy process. Considering the linear map $(x_1,...,x_k)\longrightarrow x_1+\cdots +x_k$ of $H^k$ to H we see by Remark 7.6 that

$$\begin{aligned} \zeta ^k_N(t)=\frac{1}{b_N}\sum _{i:\, i(k+\ell -)\le Nt}(Y_i-ka_N) \end{aligned}$$

also converges weakly in the $J_1$ topology as $N\rightarrow \infty$ to a corresponding Lévy process which implies also tightness of the sequence of processes $\{\zeta ^k_N,\, N\ge 1\}$.

Now, we just need to make sure that the summation that has been carried out over blocks $Y=X_1+\cdots +X_k$ will still allow us to derive tightness in the $J_1$ topology of the processes $\xi _N^k$ which amounts to boundedness and modulus of continuity estimates (see, for instance, [10], Theorem 3.21 in Ch.VI). The $J_1$ tightness of processes $\{\zeta ^k_N,\, N\ge 1\}$ explained above yields estimates of D[0, T] modulus of continuity of these processes in the following form.

For any $\epsilon>0,\eta >0$, there is a $\theta >0$ and a set $\Delta (N,\theta , \epsilon )$ such that

$$\begin{aligned} \limsup _{N\rightarrow \infty } P\{\Delta (N, \theta ,\epsilon )\}<\eta \end{aligned}$$

and on the complement $[\Delta (N,\theta ,\epsilon )]^c$, given any u, v with $|u-v|<\theta$, either

$$\begin{aligned} \sup _{\frac{Nu}{k_\ell }\le j\le \frac{Nv}{k_\ell }}\left| \frac{1}{b_N}\sum _{\frac{Nu}{k_\ell }\le r \le j} (Y_r-ka_N)\right| <\epsilon \end{aligned}$$

(7.18)

or there is an integer q such that $Nu\le q\le Nv$ and both

$$\begin{aligned} \sup _{\frac{Nu}{k_\ell }\le j\le q-1}\left| \frac{1}{b_N}\sum _{\frac{Nu}{k_\ell } \le r \le j} (Y_r-ka_N)\right| <\epsilon \end{aligned}$$

(7.19)

and

$$\begin{aligned} \sup _{q+1\le j\le \frac{Nv}{k_\ell }}\left| \frac{1}{b_N}\sum _{q+1\le r \le j} (Y_r-ka_N)\right| <\epsilon \end{aligned}$$

(7.20)

where for brevity we set $k_\ell =k+\ell -1$. We need a similar estimate for $\xi ^k_N(t)$ the process of partial sums of $\{\frac{1}{b_N}[X_i-a_N]\}$. From (7.13), we can assume that for any $\eta>0,\delta >0$, there is a $\theta >0$ such that

$$\begin{aligned} \limsup _{N\rightarrow \infty } P\{F(N,\theta ,\delta )\}<\eta \end{aligned}$$

where

$$ \begin{aligned} F(N,\theta ,\delta )=\cup _{i,j:|i-j|\le N\theta }\{||X_i-a_N]\ge \delta b_N\ \& \ |X_j-a_N|\ge \delta b_N\}. \end{aligned}$$

Suppose $Y=X_1+\cdots X_k$ is the sum over a block. On $[F_{N,\theta , \delta }]^c$, if some $|X_i-a_N|\ge \delta b_N$, then $|X_j-a_N|\le \delta b_N$ for $j\not =i$ and therefore

$$\begin{aligned} |X_i-a_N|\le |Y-ka_N|+\sum _{j: j\not =i} |X_j-a_N|\le |Y-ka_N|+(k-1)\delta b_N \end{aligned}$$

In particular, if $|Y-ka_N|\le \epsilon b_N$ then $\sup _{1\le i\le k}|X_i-a_N| \le ((k-1) \delta +\epsilon ) b_N$. On the other hand, if $|Y-ka_N|\ge \epsilon b_N$ then $|X_i-a_N|\ge \frac{\epsilon }{k} b_N$ for some i. If $\delta \le \frac{\epsilon }{k}$ then $|X_i-a_N|\ge \delta b_N$ and on $[F(N,\theta , \delta )]^c$, for $j\not =i$, $|X_j-a_N|\le \delta b_N$.

Now we can estimate the D[0, T] modulus of continuity of the $\xi ^k_N$ process. Let $\theta$ be small enough and $|i-j|<N\theta$. Then there are blocks $B^+(r_i)$ and $B^+(r_j)$ to which i and j belong. We consider the rescaled partial sums of the process $Y_r$ in the interval $r_i\le r \le r_j$. We restrict ourselves to the set $[\Delta (N,\theta ,\epsilon )]^c$. Suppose the alternative (7.18) holds. In particular $|Y_r-ka_N|\le \epsilon b_N$ for every r in the range. Any $X_q$ belonging to any of the blocks will satisfy $\frac{1}{b_N}|X_q-a_N|\le (\epsilon +(k-1)\delta )$. Therefore the analog of (7.18) holds and

$$\begin{aligned} \sup _{i \le q\le j}\left| \frac{1}{b_N}\sum _{i\le r \le q} (X_q-a_N)\right| <\epsilon +k (\epsilon +(k-1)\delta )=f_k(\epsilon ,\delta ) \end{aligned}$$

(7.21)

which goes to 0 with $\epsilon$ and $\delta$.

Suppose the alternatives (7.19) and (7.20) hold. This provides us a block $B^+(q)$. Suppose $\frac{1}{b_N}|Y_q-ka_N|<\epsilon$ we are in the previous situation of (7.18) with $3\epsilon$ replacing $\epsilon$. If $\frac{1}{b_N}|Y_q-ka_N|>\epsilon$ and $\delta <\frac{\epsilon }{k}$, there is a $q\prime$ in the block with $\frac{1}{b_N}|X^\prime _q-a_N|\ge \delta b_N$. On $[F(N,\theta ,\delta )]^c$, for any $q^{\prime \prime }\not = q^\prime$ in $B^+(q)$, $\frac{1}{b_N}|X^{\prime \prime }_q-a_N|\le \delta b_N$, and combined with (7.19) and (7.20) this proved the modulus of continuity estimate for the process $\xi ^k_N(\cdot )$ in the $J_1$ topology. Now the tightness follows (see Theorem 3.21, Ch.VI in [10]) since we obtain also uniform boundedness in probability of processes $\xi ^k_N$ from the corresponding result for normalized sums $\zeta ^k_N$ of independent blocks together with the above estimates.

As mentioned above, letting $k\rightarrow \infty$ similarly to the end of Sect. 6 we obtain weak convergence in the $J_1$ topology of processes $\xi _N$ to a Lévy process with the measure $M^*$. Finally, we observe that under the condition (7.13) the right hand side of (7.10) contains only $r=1$, and so $M^*=M$, completing the proof. $\square$