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626
HIGH MOMENT PARTIAL SUM PROCESSES OF RESIDUALS IN GARCH MODELS AND THEIR APPLICATIONS 1
, 2006
"... In this paper we construct high moment partial sum processes based on residuals of a GARCH model when the mean is known to be 0. We consider partial sums of kth powers of residuals, CUSUM processes and selfnormalized partial sum processes. The kth power partial sum process converges to a Brownian p ..."
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Cited by 14 (0 self)
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In this paper we construct high moment partial sum processes based on residuals of a GARCH model when the mean is known to be 0. We consider partial sums of kth powers of residuals, CUSUM processes and selfnormalized partial sum processes. The kth power partial sum process converges to a Brownian
Sums and Differences of Three kth Powers
"... If k ≥ 2 is a positive integer the number of representations of a positive integer N as either x k 1 + x k 2 = N or x k 1 − x k 2 = N, with integers x1 and x2, is finite. Moreover it is easily shown to be Oε(N ε), for any ε> 0. It is known that if k = 2 or 3 then the number of representations is ..."
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Cited by 7 (1 self)
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If k ≥ 2 is a positive integer the number of representations of a positive integer N as either x k 1 + x k 2 = N or x k 1 − x k 2 = N, with integers x1 and x2, is finite. Moreover it is easily shown to be Oε(N ε), for any ε> 0. It is known that if k = 2 or 3 then the number of representations is unbounded as N varies, but it
SUMS AND DIFFERENCES OF FOUR kTH POWERS
, 2009
"... We prove an upper bound for the number of representations of a positive integer N as the sum of four kth powers of integers of size at most B, using a new version of the Determinant method developed by HeathBrown, along with recent results by Salberger on the density of integral points on affine ..."
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We prove an upper bound for the number of representations of a positive integer N as the sum of four kth powers of integers of size at most B, using a new version of the Determinant method developed by HeathBrown, along with recent results by Salberger on the density of integral points
New bounds for Gauss sums derived from kth powers, and for Heilbronn’s exponential sum’, Quart
 J. Math
"... This paper is concerned with the Gauss sums and with Heilbronn’s sum G(a) = Gp(a, k) = H(a) = Hp(a) = ..."
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Cited by 58 (3 self)
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This paper is concerned with the Gauss sums and with Heilbronn’s sum G(a) = Gp(a, k) = H(a) = Hp(a) =
Matrices over orders in algebraic number fields as sums of kth powers
 Proc. Amer. Math. Soc
"... Abstract. David R. Richman proved that for n ≥ k ≥ 2 every integral n × n matrix is a sum of seven kth powers. In this paper, in light of a question proposed earlier by M. Newman for the ring of integers of an algebraic number field, we obtain a discriminant criterion for every n × n matrix (n ≥ k ..."
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Cited by 2 (1 self)
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Abstract. David R. Richman proved that for n ≥ k ≥ 2 every integral n × n matrix is a sum of seven kth powers. In this paper, in light of a question proposed earlier by M. Newman for the ring of integers of an algebraic number field, we obtain a discriminant criterion for every n × n matrix (n ≥ k
On the 2kth power mean value of the generalized quadratic Gauss sum
 Bull Korean Math. Soc
"... Abstract. The main purpose of this paper is using the elementary and analytic methods to study the properties of the 2kth power mean value of the generalized quadratic Gauss sums, and give two exact mean value formulae for k = 3 and 4. 1. ..."
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Cited by 1 (0 self)
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Abstract. The main purpose of this paper is using the elementary and analytic methods to study the properties of the 2kth power mean value of the generalized quadratic Gauss sums, and give two exact mean value formulae for k = 3 and 4. 1.
A NOTE ON SUM OF kTH POWER OF HORADAM’S SEQUENCE
, 2003
"... Let wn+2 = pwn+1 + qwn for n ≥ 0 with w0 = a and w1 = b. In this paper we find an explicit expression, in terms of determinants, for ∑ n≥0 wk nxn for any k ≥ 1. As a consequence, we derive all the previously known results for this kind of problems, as well as many new results. ..."
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Let wn+2 = pwn+1 + qwn for n ≥ 0 with w0 = a and w1 = b. In this paper we find an explicit expression, in terms of determinants, for ∑ n≥0 wk nxn for any k ≥ 1. As a consequence, we derive all the previously known results for this kind of problems, as well as many new results.
THE PARTIAL SUMS OF POWER SERIES* By J.
, 1966
"... A function f (z) is said to belong to the class J if it is regular in I z I < 1 but not in any larger disc. If fe. and f (z) 1:a 7,z'n ( ~ z I < 1) we investigate an aspect of the behaviour of the zeros of the partial sums S,(z) n = 2, a„z vo Let p„(f) be the laruest number r such that ..."
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A function f (z) is said to belong to the class J if it is regular in I z I < 1 but not in any larger disc. If fe. and f (z) 1:a 7,z'n ( ~ z I < 1) we investigate an aspect of the behaviour of the zeros of the partial sums S,(z) n = 2, a„z vo Let p„(f) be the laruest number r
Results 1  10
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626