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1.
Let Q be the lexicographic sum of finite ordered sets Q
x over a finite ordered set P. For some P we can give a formula for the jump number of Q in terms of the jump numbers of Q
x and P, that is,
, where s(X) denotes the jump number of an ordered set X. We first show that
where w(X) denotes the width of an ordered set X. Consequently, if P is a Dilworth ordered set, that is, s(P) = w(P)–1, then the formula holds. We also show that it holds again if P is bipartite. Finally, we prove that the lexicographic sum of certain jump-critical ordered sets is also jump-critical. 相似文献
2.
Jean-Marie De Koninck 《Monatshefte für Mathematik》1993,116(1):13-37
Let
and, for each integern such that (n)k, denote byP
k
(n) itsk
th largest prime factor. Further, given a set of primesQ of positive density <1 satisfying a certain regularity condition, defineP(n, Q), as the largest prime divisor ofn belonging toQ, assuming thatP(n,Q)=+ if no such prime factor exists. We provide estimates of
, fork2, and of
. We also study the median value of the functionP(n,Q) and that of the functionP
k
(n) for eachk1. 相似文献
3.
Let
be i.i.d. random variables and let, for each
and
. It is shown that
a.s. whenever the sequence of self-normalized sums S
n
/V
n is stochastically bounded, and that this limsup is a.s. positive if, in addition, X is in the Feller class. It is also shown that, for X in the Feller class, the sequence of self-normalized sums is stochastically bounded if and only if
相似文献
4.
Qingfeng Sun 《Central European Journal of Mathematics》2011,9(2):328-337
Let λ(n) be the Liouville function. We find a nontrivial upper bound for the sum
$
\sum\limits_{X \leqslant n \leqslant 2X} {\lambda (n)e^{2\pi i\alpha \sqrt n } } ,0 \ne \alpha \in \mathbb{R}
$
\sum\limits_{X \leqslant n \leqslant 2X} {\lambda (n)e^{2\pi i\alpha \sqrt n } } ,0 \ne \alpha \in \mathbb{R}
相似文献
5.
Define
, where
is a symmetric U-type statistic, H
k() is the Hermite polynomial of degree k, and {X, X
n, n1} are independent identically distributed binary random variables with Pr(X{–1, 1}})=1. We show that
according as EX=0 or EX0, respectively. 相似文献
6.
Xia Chen 《Journal of Theoretical Probability》1999,12(2):421-445
Let {X
n
}
n0 be a Harris recurrent Markov chain with state space E, transition probability P(x, A) and invariant measure , and let f be a real measurable function on E. We prove that with probability one,
7.
Suppose that
(j) is the lag-j autocorrelation of the squared residuals computed from a realization of length n under the assumption that the observations follow a GARCH(1,1) model. We study the asymptotic distribution of the statistics of the form
, where the j are nonnegative summable weights and the matrix
, can be estimated from the data. We show that, under weak assumptions on model errors, the statistic Q
n converges in distribution to
, where the N
i are iid standard normal. We discuss choices of the weights j for which the distribution of Q is tabulated. Our results lead to and provide a rigorous justification for Portmanteau goodness-of-fit tests for GARCH(1,1) specification. 相似文献
8.
Vidmantas Bentkus 《Journal of Theoretical Probability》1994,7(2):211-224
LetF be the distribution function of a sumS
n ofn independent centered random variables, denote the standard normal distribution function and its density. It follows from our results that
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