the strong law of large numbers when the mean is undefined
... since Xx++ ■• ■+Xn+> Xx+ • • ■+Xn = S„.lt follows from (4.6H4.8) that EXX+ > E\XX\ which, since we are assuming P(XX< 0) > 0, is impossible unless EXX+= E\XX\= oo. From (4.6) and (4.7) it now follows that lim sup(S„/n) = oo with probability 1. ...
... since Xx++ ■• ■+Xn+> Xx+ • • ■+Xn = S„.lt follows from (4.6H4.8) that EXX+ > E\XX\ which, since we are assuming P(XX< 0) > 0, is impossible unless EXX+= E\XX\= oo. From (4.6) and (4.7) it now follows that lim sup(S„/n) = oo with probability 1. ...
The binomial theorem
... of this triangle of numbers. The triangular numbers, the square numbers and the numbers of the Fibonacci sequence can be found from the triangle, and many interesting identities can be established. For example, the triangular numbers occur in Pascal’s triangle along the diagonal shown in the followi ...
... of this triangle of numbers. The triangular numbers, the square numbers and the numbers of the Fibonacci sequence can be found from the triangle, and many interesting identities can be established. For example, the triangular numbers occur in Pascal’s triangle along the diagonal shown in the followi ...
ON THE ERD¨OS-STRAUS CONJECTURE
... corresponding to classes C1 , C2 , C3 , C4 , C8 , C15 , C27 , . . .. which shows a steep increase in the size of classes relative to the number of jumps. In [11], Yamamoto has a different approach from ours and obtains a lesser number of exceptions at least for the primes involved in Theorem 4. For e ...
... corresponding to classes C1 , C2 , C3 , C4 , C8 , C15 , C27 , . . .. which shows a steep increase in the size of classes relative to the number of jumps. In [11], Yamamoto has a different approach from ours and obtains a lesser number of exceptions at least for the primes involved in Theorem 4. For e ...
Nyquist–Shannon sampling theorem
In the field of digital signal processing, the sampling theorem is a fundamental bridge between continuous-time signals (often called ""analog signals"") and discrete-time signals (often called ""digital signals""). It establishes a sufficient condition for a sample rate that permits a discrete sequence of samples to capture all the information from a continuous-time signal of finite bandwidth.Strictly speaking, the theorem only applies to a class of mathematical functions having a Fourier transform that is zero outside of a finite region of frequencies (see Fig 1). Intuitively we expect that when one reduces a continuous function to a discrete sequence and interpolates back to a continuous function, the fidelity of the result depends on the density (or sample rate) of the original samples. The sampling theorem introduces the concept of a sample rate that is sufficient for perfect fidelity for the class of functions that are bandlimited to a given bandwidth, such that no actual information is lost in the sampling process. It expresses the sufficient sample rate in terms of the bandwidth for the class of functions. The theorem also leads to a formula for perfectly reconstructing the original continuous-time function from the samples.Perfect reconstruction may still be possible when the sample-rate criterion is not satisfied, provided other constraints on the signal are known. (See § Sampling of non-baseband signals below, and Compressed sensing.)In some cases (when the sample-rate criterion is not satisfied), utilizing additional constraints allows for approximate reconstructions. The fidelity of these reconstructions can be verified and quantified utilizing Bochner's theorem.The name Nyquist–Shannon sampling theorem honors Harry Nyquist and Claude Shannon. The theorem was also discovered independently by E. T. Whittaker, by Vladimir Kotelnikov, and by others. It is thus also known by the names Nyquist–Shannon–Kotelnikov, Whittaker–Shannon–Kotelnikov, Whittaker–Nyquist–Kotelnikov–Shannon, and cardinal theorem of interpolation.