Survival probabilities of weighted random walks
... 1.3. Related work. Let us briefly summarize some important known results on survival probabilities. For Brownian motion, the survival exponent is easily seen to be θ = 1/2 by the reflection principle. The probability that a Brownian motion does not hit a moving boundary has also been studied in vari ...
... 1.3. Related work. Let us briefly summarize some important known results on survival probabilities. For Brownian motion, the survival exponent is easily seen to be θ = 1/2 by the reflection principle. The probability that a Brownian motion does not hit a moving boundary has also been studied in vari ...
A CELL COMPLEX IN NUMBER THEORY 1. Introduction Let M(n
... Remark 2.4. (i) Gegenbauer’s estimate of the error term was O( n). The sharper exponent cited here is due to Jia. (ii) Landau’s asymptotic formula for σk (x) was conjectured by Gauss. Note that the k = 1 case is the Prime Number Theorem. Estimates of the error term exist but will not be used in this ...
... Remark 2.4. (i) Gegenbauer’s estimate of the error term was O( n). The sharper exponent cited here is due to Jia. (ii) Landau’s asymptotic formula for σk (x) was conjectured by Gauss. Note that the k = 1 case is the Prime Number Theorem. Estimates of the error term exist but will not be used in this ...
Very simply explicitly invertible approximations of normal cumulative
... Remark 6. Usually the approximations of Φ(x) are not designed to be explicitly invertible by means of elementary functions, but sometimes they are, solving cubic or quartic equations (after obvious substitutions) or rarely in simpler manners. Remark 7. As well known, it is possible to explicitly sol ...
... Remark 6. Usually the approximations of Φ(x) are not designed to be explicitly invertible by means of elementary functions, but sometimes they are, solving cubic or quartic equations (after obvious substitutions) or rarely in simpler manners. Remark 7. As well known, it is possible to explicitly sol ...
Conditional Inapproximability and Limited Independence
... To answer this, we must first elaborate on what we mean by “efficient algorithm”. We say that an algorithm A is efficient, if there is some number x such that, if we increase the size of the input to A by 1%, the running time of A increases by at most x%. Another way of characterizing this kind of perfor ...
... To answer this, we must first elaborate on what we mean by “efficient algorithm”. We say that an algorithm A is efficient, if there is some number x such that, if we increase the size of the input to A by 1%, the running time of A increases by at most x%. Another way of characterizing this kind of perfor ...
Central limit theorem
In probability theory, the central limit theorem (CLT) states that, given certain conditions, the arithmetic mean of a sufficiently large number of iterates of independent random variables, each with a well-defined expected value and well-defined variance, will be approximately normally distributed, regardless of the underlying distribution. That is, suppose that a sample is obtained containing a large number of observations, each observation being randomly generated in a way that does not depend on the values of the other observations, and that the arithmetic average of the observed values is computed. If this procedure is performed many times, the central limit theorem says that the computed values of the average will be distributed according to the normal distribution (commonly known as a ""bell curve"").The central limit theorem has a number of variants. In its common form, the random variables must be identically distributed. In variants, convergence of the mean to the normal distribution also occurs for non-identical distributions or for non-independent observations, given that they comply with certain conditions.In more general probability theory, a central limit theorem is any of a set of weak-convergence theorems. They all express the fact that a sum of many independent and identically distributed (i.i.d.) random variables, or alternatively, random variables with specific types of dependence, will tend to be distributed according to one of a small set of attractor distributions. When the variance of the i.i.d. variables is finite, the attractor distribution is the normal distribution. In contrast, the sum of a number of i.i.d. random variables with power law tail distributions decreasing as |x|−α−1 where 0 < α < 2 (and therefore having infinite variance) will tend to an alpha-stable distribution with stability parameter (or index of stability) of α as the number of variables grows.