Lecture 10
... Theorem 3.1 (Stationary measures) If X is a recurrent Harris chain, then there exists a unique (modulo constant multiple) stationary measure. If X is furthermore aperiodic with stationary distribution π, then for any x ∈ S with Px (τα < ∞) = 1, we have kΠn (x, ·)−π(·)k → 0, where k · k denotes the ...
... Theorem 3.1 (Stationary measures) If X is a recurrent Harris chain, then there exists a unique (modulo constant multiple) stationary measure. If X is furthermore aperiodic with stationary distribution π, then for any x ∈ S with Px (τα < ∞) = 1, we have kΠn (x, ·)−π(·)k → 0, where k · k denotes the ...
Ramsey`s Theorem and Compactness
... any set of size at least 6, then for every 2-coloring of X in 2 colors, there is a monochromatic subset Y of size 3. Theorem 2.2 (Finitary Ramsey’s Theorem (Ramsey, 1930) [3]). For all positive natural numbers n, k, and a, there is a natural number b such that if X is any set of size at least b, the ...
... any set of size at least 6, then for every 2-coloring of X in 2 colors, there is a monochromatic subset Y of size 3. Theorem 2.2 (Finitary Ramsey’s Theorem (Ramsey, 1930) [3]). For all positive natural numbers n, k, and a, there is a natural number b such that if X is any set of size at least b, the ...
Numerical integration for complicated functions and random
... In general, to each smoothness class of integrands, there corresponds an appropriate numerical integration method, and the error is estimated in terms of the norm of the integrand and the number of sampling points. The most widely applicable numerical integration method by means of deterministic sam ...
... In general, to each smoothness class of integrands, there corresponds an appropriate numerical integration method, and the error is estimated in terms of the norm of the integrand and the number of sampling points. The most widely applicable numerical integration method by means of deterministic sam ...
Corpus-based estimates of word association predict - clic
... independence. For two words to have high PMI, it is not sufficient nor necessary to co-occur frequently in absolute terms (this is also true for most other association measures). Rather, the observed cooccurrence count of the two words must be higher than what is expected by chance given their indepe ...
... independence. For two words to have high PMI, it is not sufficient nor necessary to co-occur frequently in absolute terms (this is also true for most other association measures). Rather, the observed cooccurrence count of the two words must be higher than what is expected by chance given their indepe ...
The Time Scale of Evolutionary Innovation
... searches fail upto b steps is at least expð{(Mb)=d Þ (i.e., the success probability within b steps of any of the searches is at most 1{ expð{(Mb)=d Þ), when the starting sequence is far away from the target center. In such a case, one could quickly exhaust the physical resources of an entire planet. ...
... searches fail upto b steps is at least expð{(Mb)=d Þ (i.e., the success probability within b steps of any of the searches is at most 1{ expð{(Mb)=d Þ), when the starting sequence is far away from the target center. In such a case, one could quickly exhaust the physical resources of an entire planet. ...
N - The University of Texas at Dallas
... brackets. If the comparison of colors depends on the sequence of colors, the sequence matters and customers perceive [W, B, L, R, Y ] different from [ B, L, R, Y, W ]. If the sequence does not matter both [W, B, L, R, Y ] and [ B, L, R, Y, W ] have the same colors and can be mapped to the set {W, B, ...
... brackets. If the comparison of colors depends on the sequence of colors, the sequence matters and customers perceive [W, B, L, R, Y ] different from [ B, L, R, Y, W ]. If the sequence does not matter both [W, B, L, R, Y ] and [ B, L, R, Y, W ] have the same colors and can be mapped to the set {W, B, ...
Randomly Supported Independence and Resistance
... a balanced pairwise independent distribution supported on the inputs accepted by P is, assuming the UGC, hereditarily approximation resistant. Using this they proved that without assumptions on the form of t there are predicates that accept t + o(t) inputs which satisfy this property. Furthermore if ...
... a balanced pairwise independent distribution supported on the inputs accepted by P is, assuming the UGC, hereditarily approximation resistant. Using this they proved that without assumptions on the form of t there are predicates that accept t + o(t) inputs which satisfy this property. Furthermore if ...
Lecture 6: Borel-Cantelli Lemmas 1.) First Borel
... We will show that the Ak are pairwise independent with P(Ak ) = 1/k. We first observe that because F is continuous, P(Xj = Xk ) = 0 for any j 6= k, so we can let Y1n > Y2n > · · · > Ynn be the (decreasing) order statistics of X1 , · · · , Xn . This induces a random permutation of {1, · · · , n} defi ...
... We will show that the Ak are pairwise independent with P(Ak ) = 1/k. We first observe that because F is continuous, P(Xj = Xk ) = 0 for any j 6= k, so we can let Y1n > Y2n > · · · > Ynn be the (decreasing) order statistics of X1 , · · · , Xn . This induces a random permutation of {1, · · · , n} defi ...
(pdf)
... In this case, given any x ∈ Zd , there is at least one possible path from the origin to x in a finite number of steps; say this number of steps is N . There is a (2d)−N chance of our simple random walk following this precise path from the first step to the N th step. Thus, certainly P (SN = x) ≥ (2d ...
... In this case, given any x ∈ Zd , there is at least one possible path from the origin to x in a finite number of steps; say this number of steps is N . There is a (2d)−N chance of our simple random walk following this precise path from the first step to the N th step. Thus, certainly P (SN = x) ≥ (2d ...
SEEDSM12_4final
... • Careful experimental design to eliminate differences not caused by the techniques being compared. • Must take a large number of users in each group & randomize the way the users are assigned to groups. • Once other differences have been eliminated as far as possible, remaining difference will hope ...
... • Careful experimental design to eliminate differences not caused by the techniques being compared. • Must take a large number of users in each group & randomize the way the users are assigned to groups. • Once other differences have been eliminated as far as possible, remaining difference will hope ...
Infinite monkey theorem
The infinite monkey theorem states that a monkey hitting keys at random on a typewriter keyboard for an infinite amount of time will almost surely type a given text, such as the complete works of William Shakespeare.In this context, ""almost surely"" is a mathematical term with a precise meaning, and the ""monkey"" is not an actual monkey, but a metaphor for an abstract device that produces an endless random sequence of letters and symbols. One of the earliest instances of the use of the ""monkey metaphor"" is that of French mathematician Émile Borel in 1913, but the first instance may be even earlier. The relevance of the theorem is questionable—the probability of a universe full of monkeys typing a complete work such as Shakespeare's Hamlet is so tiny that the chance of it occurring during a period of time hundreds of thousands of orders of magnitude longer than the age of the universe is extremely low (but technically not zero). It should also be noted that real monkeys don't produce uniformly random output, which means that an actual monkey hitting keys for an infinite amount of time has no statistical certainty of ever producing any given text.Variants of the theorem include multiple and even infinitely many typists, and the target text varies between an entire library and a single sentence. The history of these statements can be traced back to Aristotle's On Generation and Corruption and Cicero's De natura deorum (On the Nature of the Gods), through Blaise Pascal and Jonathan Swift, and finally to modern statements with their iconic simians and typewriters. In the early 20th century, Émile Borel and Arthur Eddington used the theorem to illustrate the timescales implicit in the foundations of statistical mechanics.