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2 t /n 2 in .NET Draw code-128c in .NET 2 t /n 2




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1 2 t /n 2 using barcode creation for visual .net control to generate, create code-128c image in visual .net applications. Microsoft Office Word Website z(t/ n)t 2 /n (6.32). For n so large that t 2 < 2n we have n m=1 1 n m 1 2 t /n 2 z(t/ n)t 2 /n z(t/ n). t 2 /n = 1 + . z(t/ n). t 2 /n 1. (6.33).

The Mathematical and Statistical Foundations of Econometrics Now observe that, for any barcode standards 128 for .NET real-valued sequence an that converges to a,. lim ln ((1 + an /n )n ) = lim n ln(1 + an /n). = lim an lim ln(1 + an /n) ln(1) n n an /n ln(1 + ) ln(1) = a; = a lim 0 (6.34). hence, If we let an = z(t/ n). t 2 , which has limit a = barcode 128 for .NET 0, it follows from (6.34) that the right-hand expression in (6.

33) converges to zero, and if we let an = a = t 2 /2 it follows then from (6.32) that. n n lim an = a lim (1 + an /n)n = ea . lim n (t) = e t (6.35). The right-hand side of (6 .35) is the characteristic function of the standard normal distribution. The theorem follows now from Theorem 6.

22. Q.E.

D. There is also a multivariate version of the central limit theorem: Theorem 6.24: Let X 1 , .

. . , X n be i.

i.d. random vectors in Rk satisfying E(X j ) = , Var(X j ) = , where is nite, and let X = (1/n) n X j .

j=1 ) d Nk (0, ). Then n( X Proof: Let Rk be arbitrary but not a zero vector. Then it follows from Theorem 6.

23 that n T ( X ) d N (0, T ); hence, it follows from Theorem 6.22 that for all t R, limn E(exp[i t n T ( X )]) = 2 T exp( t /2). Choosing t = 1, we thus have that, for arbitrary Rk , limn E(exp[i T n( X )]) = exp( T /2).

Because the latter is the characteristic function of the Nk (0, ) distribution, Theorem 6.24 follows now from Theorem 6.22.

Q.E.D.

Next, let be a continuously differentiable mapping from Rk to Rm , and let the conditions of Theorem 6.24 hold. The question is, What is the limiting distribution of n( ( X ) ( )), if any To answer this question, assume for the time being that k = m = 1 and let var(X j ) = 2 ; thus, n( X ) d 2 N (0, ).

It follows from the mean value theorem (see Appendix II) that there exists a random variable [0, 1] such that n( ( X ) ( )) = n( X ) ( + ( X )). Because n( X ) d N (0, 2 ) implies ( X ) d 0, which by Theorem p , it follows that + ( X ) p . Moreover, 6.

16 implies that X is continuous in it follows now from Theorem 6.3 because the derivative. Modes of Convergence that ( + ( X )) p ( ). Therefore, it follows from Theorem 6.21 ) ( )) d N [0, 2 ( ( ))2 ].

Along similar lines, if we apply that n( ( X the mean value theorem to each of the components of separately, the following more general result can be proved. This approach is known as the -method. Theorem 6.

25: Let X n be a random vector in Rk satisfying n(X n ) d Nk [0, ], where Rk is nonrandom. Moreover, let (x) = ( 1 (x), . .

. , m (x))T with x = (x1 , . .

. , xk )T be a mapping from Rk to Rm such that the m k matrix of partial derivatives 1 (x)/ x1 . .

. 1 (x)/ xk . .

.. .

. (x) = (6.36) .

. . .

m (x)/ x 1 ...

. m (x)/ x k exists in an arbitrary, small, open neighborhood of and its elements are continuous in . Then n( (X n ) ( )) d Nm [0, ( ) ( )T ]. 6.

8. Stochastic Boundedness, Tightness, and the O p and o p Notations The stochastic boundedness and related tightness concepts are important for various reasons, but one of the most important is that they are necessary conditions for convergence in distribution. Definition 6.

7: A sequence of random variables or vectors X n is said to be stochastically bounded if, for every (0, 1), there exists a nite M > 0 such that inf n 1 P[ X n M] > 1 . Of course, if X n is bounded itself (i.e.

, P[. X n M] = 1 for all n), it is stochastically bounded as well, but the converse may not be true. For example, if the X n s are equally distributed (but not necessarily independent) random variables with common distribution function F, then for every (0, 1) we can choose continuity points M and M of F such that P[. X n M] = F(M) F( M) = 1 . Thus, the stochastic boundedness condition limits the heterogeneity of the X n s. Stochastic boundedness is usually denoted by O p (1) : X n = O p (1) means that the sequence X n is stochastically bounded.

More generally, Definition 6.8: Let an be a sequence of positive nonrandom variables. Then X n = O p (an ) means that X n /an is stochastically bounded and O p (an ) by itself represents a generic random variable or vector X n such that X n = O p (an ).

The necessity of stochastic boundedness for convergence in distribution follows from the fact that.
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