On Kesten s counterexample to the Cramér-Wold device for regular variation

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1 On Kesten s counterexample to the Cramér-Wold device for regular variation Henrik Hult School of ORIE Cornell University Ithaca NY 4853 USA hult@orie.cornell.edu Filip Lindskog Department of Mathematics Royal Institute of Technology 44 Stockholm Sweden lindskog@math.kth.se Abstract In 22 Basrak, Davis and Mikosch showed that an analogue of the Cramér-Wold device holds for regular variation of random vectors if the index of regular variation is not an integer. This characterization is of importance when studying stationary solutions to stochastic recurrence equations. In this paper we construct counterexamples showing that for integer-valued indices, regular variation of all linear combinations does not imply that the vector is regularly varying. The construction is based on unpublished notes by Harry Kesten. Key words: heavy-tailed distributions; linear combinations; multivariate regular variation

2 Introduction For R d -valued random vectors X n and X, the well-known Cramér-Wold Theorem says that a necessary and sufficient condition for X n X is that x T d d X n x T X for every x R d. We use the convention that x R d is a column vector and x T its transpose. In Basrak et al. (22a) it was shown that, for nonintegervalued indices of regular variation, there is a similar characterization of regular variation for a random vector in terms of regular variation of its linear combinations; meaning that for some α > and some function L which is slowly varying at infinity, for every x, t t α L(t) P(x T X > t) = w(x) exists, () w(x) > for some x. If () holds, then necessarily w(ux) = u α w(x) for all x and u >. The interest in this condition originates from a classical result by Kesten (973b) which (briefly) says that, under mild conditions, the stationary solution X of a multivariate stochastic recurrence equation X n = A n X n + B n satsifies (), where L(t) = and α is the unique solution to n n E log A n A α =. Here denotes the operator norm, A = sup x = Ax, and is the Euclidean norm on R d. A popular example is the stationary GARCH model which can be embedded in a stochastic recurrence equation (see Basrak et al., 22b). Other examples where the condition () appears are the stochastic recurrence equations with heavy-tailed innovations studied in Konstantinides and Mikosch (25) and the random coefficient AR(q) models of Klüppelberg and Pergamenchtchikov (24). Important and accessible papers on the extremal behavior of solutions to univariate stochastic recurrence equations are de Haan et al. (989), Goldie (99) and Borkovec and Klüppelberg (2). A random vector X is said to be regularly varying if there exist an α > and a probability measure σ on B(S d ), the Borel σ-field of S d = {x R d : x = }, such that, for every x >, as t, P( X > tx, X/ X ) P( X > t) v x α σ( ) on B(S d ). (2) Here v denotes vague convergence, and α and σ are respectively called the index of regular variation and spectral measure of X. For α (, 2) this formulation of multivariate regular variation is a necessary and sufficient condition for the convergence in distribution of normalized partial sums of independent and identically distributed random vectors to a stable random vector; see Rvačeva (962). It is also used for the characterization of the maximum domain of attraction of extreme value distributions and for weak convergence of point processes, see for example Resnick (987). In Basrak et al. (22a, Theorem.) it was proved that (2) implies () and the following statements hold: (A) If X satisfies (), where α is positive and noninteger, then (2) holds and the spectral measure σ is uniquely determined. 2

3 (B) If X assumes values in [, ) d and satisfies () for x [, ) d \{}, where α is positive and noninteger, then (2) holds and the spectral measure σ is uniquely determined. (C) If X assumes values in [, ) d and satisfies (), where α is an odd integer, then (2) holds and the spectral measure σ is uniquely determined. In Section 2 we construct a counterexample which shows that (A) cannot be extended to integer-valued indices of regular variation without additional assumptions on the distribution of X. In Section 3 we construct a counterexample which shows that (B) cannot be extended to integer-valued indices of regular variation without additional assumptions on the distribution of X. Whether (C) is true or not in the case of α belonging to the set of even integers is, to the best of the knowledge of the authors, still an open problem. Let us point out that there are several equivalent formulations of (2); many of them are documented in Basrak (2) and Resnick (24). See also Basrak et al. (22a), Hult (23), Lindskog (24) and Resnick (987) for more on multivariate regular variation. For a detailed treatment of the concept of regularly varying functions, see Bingham et al. (987). 2 Construction of the counterexamples The constructions of the counterexamples corresponding to (A) and (B) for integer-valued indices of regular variation are rather similar and consist of two steps. First note that due to the scaling property the it function w in () is determined by its values on S d. Therefore it is sufficient to consider linear combinations x T X with x S d. The first step consists of finding two bivariate regularly varying random vectors X and X with index of regular variation α > and spectral measures σ and σ, with σ σ, such that for every x S and t >, P(x T X > t) = P(x T X > t). (3) For the example corresponding to (B) we restrict x to S [, ) 2. In the second step we will construct the counterexamples by finding a random vector X such that, for every x S and x S [, ) 2, respectively, t tα P(x T X > t) = t α P(x T X > t) = t α P(x T X > t) =: w(x) t t and such that there are subsequences (u n ), (v n ), u n, v n, with the property that for every S B(S) with σ ( S) = σ ( S) =, P( X > u n, X/ X S) = σ (S), n P( X > u n ) (5) P( X > v n, X/ X S) = σ (S). n P( X > v n ) (6) The counterexamples are easily extended to R d -valued random vectors. Take X = (X (), X (2) ) T as above and Y = (Y (),..., Y (d 2) ) T independent of X with (4) 3

4 Y satisfying () with the same α as X, L(t) =, and it function w Y. Put Z = (X (), X (2), Y (),..., Y (d 2) ) T. Then, by independence (c.f. Davis and Resnick, 996, Lemma 2.), t tα P(z T Z > t) = t t α P((z (), z (2) )X > t) + t t α P((z (3),..., z (d) )Y > t) = w(z (), z (2) ) + w Y (z (3),..., z (d) ). Hence, Z satisfies (). However, Z does not satisfy (2). Indeed, assume on the contrary that Z satisfies (2) with spectral measure σ Z. Then since X = T (Z) with T : R d R 2 is given by T (z) = (z (), z (2) ) T it follows that (e.g. Basrak et al., 22b, Proposition A.) X satisfies (2) for some spectral measure σ. This is a contradiction. 2. Construction of X and X We will now focus on the construction of X and X in the counterexample corresponding to (A) when α is a positive integer. We will construct two regularly varying random vectors X and X with index of regular variation α and different spectral measures such that (3) is satisfied. A different construction in the case α = can be found in Meerschaert and Scheffler (2, Example 6..35). Let Θ be a [, 2π)-valued random variable with density f satisfying, for some ε >, f (θ) > ε for all θ [, 2π). Take v (, ε) and let Θ have density f given by f (θ) = f (θ) + v sin((α + 2)θ), θ [, 2π). Let R Pareto(α), that is P(R > x) = x α for x, be independent of Θ i, i =,, and put X i d = (R cos Θ i, R sin Θ i ) T. Obviously X i is regularly varying with σ i ( ) = P((cos Θ i, sin Θ i ) ). x S and let β [, 2π) be given by x = (cos β, sin β) T. Then, for t >, Take P(x T X > t) P(x T X > t) = v β+arccos(t/r) t = vα α + 2 β arccos(t/r) = 2vα sin((α + 2)β) α + 2 t αr α sin((α + 2)θ)dθdr r α ( cos{(α + 2)(β + arccos(t/r)} ) cos{(α + 2)(β arccos(t/r))} dr t r α sin{(α + 2) arccos(t/r)}dr. Using standard variable substitutions and trigonometric formulae the integral 4

5 can be rewritten as follows: t r α sin{(α + 2) arccos(t/r)}dr = t α r α sin{(α + 2) arccos(r)}dr π/2 = t α cos α (r) sin((α + 2)r) sin(r)dr π/2 π/2 = t α cos α (r) cos((α + )r)dr t α cos α (r) cos((α + 2)r)dr. The two last integrals equal zero for every α {, 2,... }, see Gradshteyn and Ryzhik (2) p Hence, for t >, P(x T X > t) = P(x T X > t), which proves (3). The following construction of X satisfying (4)-(6) is based on unpublished notes by Harry Kesten (973a) relating to Remark 4, p. 245, in Kesten (973b). 2.2 The counterexample Consider the random vectors X and X above. Let g and g denote their densities. We construct a random vector X satisfying () which is not regularly varying; we will show that it satisfies (4) - (6). Take y R 2 with y >. There exist unique integers j, n such that { n n y (j!, (j + )!] and j 2 k +,..., 2 k}. Let X be an R 2 -valued random vector with density g given by ( ( n j 2 k) 2 n) ( n g b(n) (y) + j 2 k) 2 n g b(n+) (y), k= for y (j!, (j + )!], where b(n) = k= k= {, if n is odd,, if n is even. That is, the density g is given by g (y) =, y (, ], 2 g (y) + 2 g (y), y (, 2], k= g(y) = g (y), y (2, 3!], 4 g (y) g (y), y (3!, 4!], 2 g (y) + 2 g (y), y (4!, 5!], 3 4 g (y) + 4 g (y), y (5!, 6!], etc. 5

6 Note that in each disc y (j!, (j + )!] the density g is a convex combination of the densities g and g. Therefore, g(y)dy = g(y)dy R 2 j= y (j!,(j+)!] (j+)! = αr α dr = αr α dr =, j= j! so g is indeed a probability density. Take x S and t ((j )!, j!]. Then there are two possibilities: (i) j n } 2 k +,..., 2 k { n k= n (ii) j = 2 k. k= Suppose (i) holds. Then, with γ = Hence, k= or ( j n k= 2k )2 n [, ], we have P(x T X > t) = ( γ + 2 n) P(x T X b(n) > t, X b(n) (t, j!]) + ( γ 2 n) P(x T X b(n+) > t, X b(n+) (t, j!]) + ( γ) P(x T X b(n) > t, X b(n) (j!, (j + )!]) + γ P(x T X b(n+) > t, X b(n+) (j!, (j + )!]) + P(x T X > t, X > (j + )!). P(x T X > t) = 2 n{ P(x T X b(n) > t, X b(n) (t, j!]) } P(x T X b(n+) > t, X b(n+) (t, j!]) + ( γ) P(x T X b(n) > t) + γ P(x T X b(n+) > t) }{{} B n ( γ) P(x T X b(n) > t, X b(n) > (j + )!) γ P(x T X b(n+) > t, X b(n+) > (j + )!) + P(x T X > t, X > (j + )!). We have 2 n{ } P(x T X b(n) > t, X b(n) (t, j!]) P(x T X b(n+) > t, X b(n+) (t, j!]) 2 n{ } P( X b(n) > t) + P( X b(n+) > t) = 2 n+ t α. Moreover, since P(x T X > t) = P(x T X > t) we have B n = P(x T X > t). The absolute value of each of the remaining terms is less than or equal to ((j + )!) α (j j!) α (j t) α, so we conclude that t α P(x T X > t) P(x T X > t) 2 n+ + 3j α. 6

7 Since j = j(t) and n = n(t) as t, we have That is, tα P(x T X > t) P(x T X > t) =. t t tα P(x T X > t) = t α P(x T X > t) = t α P(x T X > t). t t Suppose now that (ii) holds. Then P(x T X > t) = P(x T X b(n) > t, X b(n) (t, j!]) + ( 2 n ) P(x T X b(n) > t, X b(n) (j!, (j + )!]) + 2 n P(x T X b(n+) > t, X b(n+) (j!, (j + )!]) + P(x T X > t, X > (j + )!) = P(x T X b(n) > t) P(x T X b(n) > t, X b(n) > (j + )!) + 2 n{ P(x T X b(n+) > t, X b(n+) (j!, (j + )!]) } P(x T X b(n) > t, X b(n) (j!, (j + )!]) + P(x T X > t, X > (j + )!). By arguments similar to those for case (i) we obtain It follows that t α P(x T X > t) P(x T X > t) 2(2 n + j α ). t tα P(x T X > t) = t α P(x T X > t) = t α P(x T X > t), t t which proves (4). Finally, we find subsequences (u n ) and (v n ) satisfying (5) and (6). S B(S) with σ ( S) = σ ( S) =. Put Take c n = 2n k= 2 k and d n = 2n+ Note that for c n! < y (c n + )! we have g(y) = g b(2n+) (y) = g (y), whereas for d n! < y (d n + )! we have g(y) = g b(2n+2) (y) = g (y). It follows that, with u n = c n!, u α n P( X > u n, X/ X S) = u α n P (c n! < X (c n + )!, X/ X S) Since the second term is less than or equal to as n, it follows that k= 2 k. + u α n P ( X > (c n + )!, X/ X S). u α n P ( X > (c n + )!) = (c n!) α [(c n + )!] α, n uα n P( X > u n, X/ X S) = n uα n = σ (S). ( u α n [(c n + )!] α) σ (S) 7

8 By a similar argument, with v n = d n!, n vα n P( X > v n, X/ X S) = σ (S). Thus, we have found sequences (u n ) and (v n ) satisfying (5) and (6) and the counterexample is complete. 3 Vectors with nonnegative components In this section we construct a counterexample corresponding to (B) in the case of integer-valued indices of regular variation. The following construction of X and X was given in Basrak et al. (22a) for α = 2 but can, as we will see, be extended to any positive integer α. Take α {, 2,... } and let Θ, Θ be two [, π/2]-valued random variables with unequal distributions satisfying E(cos k Θ sin α k Θ ) = E(cos k Θ sin α k Θ ), k =,,..., α. (7) Let R Pareto(α), that is P(R > x) = x α for x. Suppose R is independent of Θ i, i =, and put X i d = (R cos Θ i, R sin Θ i ) T. For x [, ) 2 \{}, we have t α P(x T X i > t) = t α P(x R cos Θ i + x 2 R sin Θ i > t) = t α P(x cos Θ i + x 2 sin Θ i > t/r)αr α dr = = t α α k= P((x cos Θ i + x 2 sin Θ i ) α > v)dv ( ) α x k k x α k 2 E(cos k Θ i sin α k Θ i ) for t sufficiently large. We can now construct the vector X as in Section 2.2, with x S + = S [, ) 2 and the new densities g and g of X and X. It remains to show that we can find unequal distributions of the [, π/2]-valued random variables Θ and Θ satisfying (7). Let Θ have density f satisfying, for some ε >, f (θ) > ε for all θ [, π/2]. We will show that the density f of Θ can be chosen as f (θ) = f (θ) + vf(θ), where v (, ε) and f is chosen such that sup θ [,π/2] f(θ) =, π/2 f(θ)dθ =, and (7) holds. Let A := span{, sin α (θ), cos(θ) sin α (θ),..., cos α (θ)} C 2 ([, π/2]), where C 2 ([, π/2]) is the space of real-valued continuous functions on [, π/2] equipped with the inner product (h, h 2 ) = π/2 h (s)h 2 (s)ds, which makes it an inner product space. For any nonzero f / A with f C 2 ([, π/2]) we can choose f Proj f := A ( f) sup θ [,π/2] { f Proj A ( f)}(θ). Then f A, f is a density function and (7) holds. Since A is a finitedimensional subspace of the infinite-dimensional space C 2 ([, π/2]) it is clear that its orthogonal complement is nonempty. Hence, the density f above exists. 8

9 4 Acknowledgements An essential part of the construction of the counterexamples in this paper is based on unpublished notes by Harry Kesten. The authors wish to thank Laurens de Haan for providing them with these notes and Harry Kesten for supporting this publication. The authors thank the anonymous referee for comments that led to an improvement of the presentation of the material. The research of the first author was supported by the Swedish Research Council. References Basrak, B., 2. The sample autocorrelation function of non-linear time series. Ph.D. thesis, Department of Mathematics, University of Groningen. Basrak, B., Davis, R., Mikosch, T., 22a. A characterization of multivariate regular variation. Ann. Appl. Probab. 2 (3), Basrak, B., Davis, R. A., Mikosch, T., 22b. Regular variation of GARCH processes. Stochastic Process. Appl. 99, Bingham, N., Goldie, C., Teugels, J., 987. Regular Variation. No. 27 in Encyclopedia of mathematics and its applications. Cambridge University Press, Cambridge. Borkovec, M., Klüppelberg, C., 2. The tail of the stationary distribution of an autoregressive process with ARCH() errors. Ann. Appl. Probab., Davis, R., Resnick, S., 996. Limit theory for bilinear processes with heavytailed noise. Ann. Appl. Probab. 6 (4), 9 2. de Haan, L., Resnick, S., Rootzén, H., de Vries, C., 989. Extremal behaviour of solutions to a stochastic difference equation with applications to ARCHprocesses. Stochastic Process. Appl. 32, Goldie, C. M., 99. Implicit renewal theory and tails of solutions of random equations. Ann. Appl. Probab., Gradshteyn, I., Ryzhik, I., 2. Table of Integrals, Series, and Products, 6th Edition. Academic Press, Orlando. Hult, H., 23. Topics on fractional Brownian motion and regular variation for stochastic processes. Ph.D. thesis, Department of Mathematics, Royal Institute of Technology, Stockholm. Kesten, H., 973a. Unpublished notes. Kesten, H., 973b. Random difference equations and renewal theory for products of random matrices. Acta Math. 3, Klüppelberg, C., Pergamenchtchikov, S., 24. The tail of the stationary distribution of a random coefficient AR(q) model. Ann. Appl. Probab. 4 (2),

10 Konstantinides, D., Mikosch, T., 25. Large deviations and ruin probabilities for solutions to stochastic recurrence equations with heavy-tailed innovations. Ann. Probab. To appear. Lindskog, F., 24. Multivariate extremes and regular variation for stochastic processes. Ph.D. thesis, Department of Mathematics, Swiss Federal Institute of Technology, Zürich, available at: Meerschaert, M. M., Scheffler, P., 2. Limit Distributions for Sums of Independent Random variables: Heavy Tails in Theory and Practice. Wiley, New York. Resnick, S., 24. On the foundations of multivariate heavy-tail analysis. J. Appl. Probab. 4A, Resnick, S. I., 987. Extreme Values, Regular Variation, and Point Processes. Springer, New York. Rvačeva, E., 962. On domains of attraction of multi-dimensional distributions. Select. Transl. Math. Statist. Probab., American Mathematical Society, Providence, R.I. 2,