Affine stochastic equation with triangular matrices

References

• G. Alsmeyer and S. Mentemeier, Tail behavior of stationary solutions of random difference equations: the case of regular matrices, J. Differ. Equ. Appl. 18 (2012), pp. 1305–1332.
   Web of Science ®Google Scholar
• P. Bougerol and N. Picard, Strict stationarity of generalized autoregressive processes, Ann. Probab. 20 (1992), pp. 1714–1730.
   Web of Science ®Google Scholar
• A. Brandt, The stochastic equation Yn+1=AnYn+Bn with stationary coefficients, Adv. Appl. Probab. 18 (1986), pp. 211–220.
   Web of Science ®Google Scholar
• D. Buraczewski and E. Damek, Regular behavior at infinity of stationary measures of stochastic recursion on NA groups, Colloq. Math. 118 (2010), pp. 499–523.
   Web of Science ®Google Scholar
• D. Buraczewski and E. Damek, A simple proof of heavy tail estimates for affine type Lipschitz recursions, Stoch. Process. Appl. 127 (2017), pp. 657–668.
   Web of Science ®Google Scholar
• D. Buraczewski, E. Damek, Y. Guivarch, A. Hulanicki, and R. Urban, Tail-homogeneity of stationary measures for some multidimensional stochastic recursions, Probab. Theory Related Fields 145 (2009), pp. 385–420.
   Web of Science ®Google Scholar
• D. Buraczewski, E. Damek, and T. Mikosch, Stochastic Models with Power-law Tails: The Equation X=_dAX+B, Springer International, Switzerland, 2016.
   Google Scholar
• E. Damek and B. Kolodziejek, Stochastic recursions: between Kesten’s and Grey’s assumptions, submitted for publication. 2016. Available at https://arxiv.org/pdf/1701.02625.pdf.
   Google Scholar
• E. Damek, M. Matsui, and W. \’{S}wi\c{a}tkowski, Componentwise different tail solutions for bivariate stochastic recurrence equations - with application to GARCH(1,1) processes, submitted for publication. 2016. Available at https://arxiv.org/abs/1706.05800.pdf.
   Google Scholar
• L. Gerencser, G. Michaletzky, and Z. Orlovits, Stability of blocktriangular stationary random matrices, Syst. Cintr. Lett. 57(8) (2008), pp. 620–625.
   Web of Science ®Google Scholar
• C.M. Goldie, Implicit renewal theory and tails of solutions of random equations, Ann. Appl. Probab. 1(1) (1991), pp. 126–166.
   Google Scholar
• D.R. Grey, Regular variation in the tail behavior of solutions of random difference equations, Ann. Appl. Probab. 4(1) (1994), pp. 169–183.
   Web of Science ®Google Scholar
• Y. Guivarc’h and E. Le Page, Spectral gap properties for linear random walks and Pareto’s asymptotics for affine stochastic recursions, Ann. Inst. H. Poincaré Probab. Statist. 52 (2016), pp. 503–574.
   Web of Science ®Google Scholar
• H. Kesten, Random difference equations and renewal theory for products of random matrices, Acta. Math. 131 (1973), pp. 207–248.
   Web of Science ®Google Scholar
• P. Kevei, A note on the Kesten--Grincevicius--Goldie theorem, Electron. Commun. Prob. 21(51) (2016), pp. 1–12.
   Google Scholar
• C. Klüppelberg and S. Pergamenchtchikov, The tail of the stationary distribution of a random coefficient AR(q) model, Ann. Appl. Probab. 14 (2004), pp. 971–1005.
   Web of Science ®Google Scholar
• M. Matsui and T. Mikosch, The extremogram and the cross-extremogram for a bivariate GARCH process, Adv. Appl. Probab. 48A (2016), pp. 217–233.
   Google Scholar
• M. Matsui and W. \’{S}wi\c{a}tkowski, Tail indices for upper triangular matrices, preprint.
   Google Scholar
• T. Mikosch and C. Stărică, Limit theory for the sample autocorrelations and extremes of a GARCH(1,1) process, Ann. Statist. 28 (2000), pp. 1427–1451.
   Web of Science ®Google Scholar
• M. Mirek, Heavy tail phenomenon and convergence to stable laws for iterated Lipschitz maps, Probab. Theory Related Fields 151(3--4) (2011), pp. 705–734.
   Web of Science ®Google Scholar
• V. Petrov, On the probabilities of large deviations for sums of independent random variables, Theory Probab. Appl. 10 (1965), pp. 287–298.
   Google Scholar
• V. Petrov, Limit Theorems of Probability Theory, Vol. 4, Oxford Studies in Probability, Clarendon Press, Oxford, 1995.
   Google Scholar
• D. Straumann, Estimation in Conditionally Heteroscedasic Time Series Models, Vol. 181, Lecture Notes in Statistics, Springer-Verlag, Berlin, 2005.
   Google Scholar
• W. Vervaat, On a stochastic difference equation and a representation of non-negative infinitely divisible random variables, Adv. Appl. Prob. 11 (1979), pp. 750–783.
   Web of Science ®Google Scholar