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Theory and applications of probability and stochastic processes: e.g. central limit theorems, large deviations, stochastic differential equations, models from statistical mechanics, queuing theory.

2 votes
0 answers
137 views

Expected value of number of collisions for a matrix valued random function

Consider the function $$f(x_1, x_2, \ldots, x_k) = S_1^{x_1} S_2^{x_2} \cdots S_k^{x_k}.$$ Each $x_1, x_2, \ldots, x_k \in \{0, 1\}$ and each $S_i \in \mathbb{F}_q^{n \times n}$ is a randomly chosen …
RandomMatrices's user avatar
1 vote
0 answers
80 views

Calculating the mean squared error for an estimate of a large sum

Consider the set of all Boolean function $f: \{0, 1\}^{n} \rightarrow \{-1, 1\}$. Now, let's pick a function uniformly at random from this set. Let $F$ be the random variable corresponding to the rand …
RandomMatrices's user avatar
0 votes
1 answer
289 views

Joint distribution of random Fourier coefficients

Consider choosing a Boolean function $f : \{0, 1\}^{n} \rightarrow \{-1, 1\}$ uniformly at random from the set of all Boolean functions and consider the random variable $\left(\hat f(z_{1}), \hat f(z_ …
RandomMatrices's user avatar
1 vote
1 answer
332 views

Posterior expected value for squared Fourier coefficients of random Boolean function

Let $f : \{0, 1\}^{n} \rightarrow \{-1, 1\}$ be a Boolean function. Let the Fourier coefficients of this function be given by $$ \hat f(z) = \frac{1}{2^{n}} \sum_{x \in \{0, 1\}^{n}} f(x)(-1)^{x \cdot …
RandomMatrices's user avatar