Questions in probability theory

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8
votes
1answer
69 views

$k$-wise independent probability spaces

I have been having a great deal of difficulty finding a reference that gives simple and straightforward explanation of the following: Suppose we have $n$ random variables $Y_1, \dots, Y_n$, each of ...
2
votes
0answers
56 views

Convergence of iterated stochastic matrices

(This question has been unsuccessfully asked on mathoverflow: http://mathoverflow.net/questions/161728/convergence-of-iterated-stochastic-matrices) It is well-known that for a stochastic aperiodic ...
5
votes
2answers
136 views

Subgraph of G whose maximum degree and minimum degree are of the same order

Consider a graph $G$ with max degree $\Delta_G$, min degree $\delta_G$ and average degree $d_G$. Is it possible to obtain a subgraph of $G$, say $G'$, such that $\Delta_{G'} = c_1d_{G}$, ...
5
votes
0answers
55 views

Fast convergence of a contagion process in special graphs

The process: Given is a clique $C_n$ of size $n$. Consider the following synchronous process, also known as the (synchronous) voter model (e.g., Even-Dar and Shapira): Define an indicator variable ...
5
votes
1answer
100 views

Independent set size in triangle-free graphs

Consider a triangle-free graph $G$. The notations used are: $\alpha(G) = $ the size of a largest independent set of $G$. $n(G) = $ the number of vertices in $G$. Theorem (Ajtai et al.): For a ...
1
vote
0answers
26 views

What is entropy of a variable described by Knightian uncertainty?

Given a discrete variable whose value is characterized by Knightian uncertainty, that is, belief and plausibility, as in Dempster-Shafer theory, what is its entropy?
3
votes
3answers
171 views

The use of crossovers in Genetic Algorithm

My questions concern the use of crossovers in genetic algorithms. The three basic ingredients of genetic algorithms are: selection mutation crossover If we think of genetic algorithm acting on ...
0
votes
1answer
58 views

Multiple independent random number streams

Having multiple streams of pseudo-random numbers known to be independent and with a uniform distribution I want to do Monte Carlo simulations in parallel. In other words, one thread will have a ...
5
votes
2answers
118 views

Concentration inequality for function of independent Bernoulli r.v.'s (related to random graph)

Consider a random undirected graph on a set of $n$ nodes, say $\{1,2,\ldots,n\}$, such that the probability of edge between nodes $i$ and $j$ is $p_{ij}$ (we may assume $p_{ij}=o(1)$ for all $i,j$, ...
-3
votes
1answer
58 views

Why is the zero value ignored while deriving a ranking function for query terms in Probabilistic IR?

Take for instance equation 67 and 68 from this chapter: the value of $P(q|R=1,q)$ can become zero if the term is not present in the document, and as all probabilities are multiplied, the ...
1
vote
1answer
41 views

A theorem regarding statistically-hiding commitment schemes

Let $C_n$ be a non-interactive statistically-hiding commitment scheme, able to commit to an $n$-bit string. To commit to $m \in \{0,1\}^n$, the sender picks a random $r$ (of proper length), and sends ...
4
votes
1answer
145 views

Probability for an element to appear in at least one set

Say that we have $k$ sets, each with cardinality $N$, where the elements in each set are taken at random from $M \ge N$ possible ones. The elements in each set are known to be distinct. What is the ...
6
votes
1answer
155 views

correlation in an almost independent set of random variables

Suppose I have a set of $n$ binary random variables $X_1, \ldots, X_n$ that sit on a line, and assume that $\Pr(X_i=0)=\delta$ for all $i$. In addition, assume that any two subsets of variables that ...
0
votes
0answers
52 views

On introductive material of set systems

I am reading Jukna's textbook "Extremal Combinatorics", and I want to some introductive material various theory of set systems including optimizational area like matroid theory. Are there a lecture ...
4
votes
0answers
120 views

Sums of products of bernoulli random variables

Let $x_1 \ldots x_a,y_1 \ldots y_b$ be independent random variables taking values +1 or -1. Consider the sum $$S = \sum_{i,j} x_iy_j.$$ I wish to upper bound the probability $P(|S| > t)$. The ...
7
votes
1answer
500 views

Sum of Independent Exponential Random Variables

Can we prove a sharp concentration result on the sum of independent exponential random variables, i.e. Let $X_1, \ldots X_r$ be independent random variables such that $Pr(X_i < x) = 1 - ...
5
votes
0answers
118 views

Squares of random binary matrices

Are there results/techniques pertaining to the analysis of squares of random matrices ? More specifically, let $A$ be an $n\times n$ matrix such that each entry is $1$ or $-1$ independently and with ...
1
vote
1answer
60 views

Terminology for games with incomplete information and no prior beliefs

Can anyone please tell me what is the term used for games with incomplete information and there are no prior beliefs about other players' private information. For example, let $ v_i(a_i,\theta_i) $ ...
0
votes
1answer
109 views

Capacity planning algorithm resources

Let's say I have a machine with many boxes of different sizes. I want to put packages inside those boxes. Packages arrive at different time and then stay in the box for specific period of time. I need ...
26
votes
4answers
2k views

Does a noisy version of Conway's game of life support universal computation?

Quoting Wikipedia, "[Conway's Game of Life] has the power of a universal Turing machine: that is, anything that can be computed algorithmically can be computed within Conway's Game of Life." Do such ...
4
votes
0answers
110 views

The balls and bins model: bounding the marginal contributions in the m>>n regime

Consider the standard balls and bins process, where $m$ balls are thrown into $n$ bins, and consider the case where $m >> n$. Denote the load on bin $i$ by the RV $L_i$. Given a set $S ...
7
votes
0answers
109 views

Convergence speed in Lévy’s continuity theorem

I suppose the answer to my question follows from the proof of the Lévy’s continuity theorem, but probably one could suggest me a direct reference to the corresponding answer. The question is as ...
7
votes
1answer
133 views

Looking for a reference: equivalence of (semi)computable (semi)measures and PTMs

I'm working with computable probability distributions over all finite strings. These are usually formalized as the space of all semicomputable semimeasures. Informally: a probability distribution is ...
2
votes
0answers
80 views

Concentration Bounds for Dependent Rounding

Consider the following random process which is defined on $n$ numbers $0\leq x_1,\ldots,x_n\leq 1$: At each step, pick an arbitrary number, say $x_i$. Then randomly (and independently) change its ...
2
votes
0answers
46 views

Set-systems with some version of independence

Let $S \subset [N]$ be a fixed set of size $n$. Suppose $p$ is the probability that a random set $T$ of size $m$ intersects $S$ in $k$ or more points. That is, $$ \Pr_{\substack{T\subset [N]\\|T| = ...
1
vote
2answers
156 views

Chernoff Bounds for settings with limited dependence

Could someone point me to a way of bounding the tail probability of sums of bernoulli variables each generated by the same distribution but the condition of independence is only partially satisfied. ...
3
votes
0answers
64 views

A sampling and learning question

Suppose there is an oracle that returns a number $b \in \mathbb{Z}_{n}$ whenever I press the button. We have $b = a + e$, where $a \in \mathbb{Z}_n$ is a fixed number and $e$ is sampled according to ...
16
votes
1answer
659 views

The complexity of sampling (approximately) the Fourier transform of a Boolean function

One thing that quantum computers can do (possibly even with just BPP + log-depth quantum circuits) is to approximate-sample the Fourier transform of a Boolean $\pm 1$-valued function in P. Here and ...
5
votes
2answers
149 views

Behaviour of Labelled Markov Processes

Labelled Markov Processes (LMP) seem to be a generalization of Probabilistic Automata (PA) studied by Segala to the case of the general state space. Namely, any LMP is given by a be a finite set of ...
3
votes
1answer
178 views

Boundedness of expected reward Markov chain

This is a repost of a question I asked on math.SE. The problem: I have an infinite Markov chain $M$ over the natural numbers, with transition probabilities $$P(n,m)=\sum_{i=0}^{min(m,n)} {n\choose ...
24
votes
2answers
311 views

Current tightest bounds for critical 3-SAT density

I'm interested in the critical 3-satisfiability (3-SAT) density $\alpha$. It's conjectured that such $\alpha$ exists: if the number of randomly generated 3-SAT clauses is $(\alpha + \epsilon) n$ or ...
20
votes
2answers
818 views

Balls and Bins analysis in the $m \gg n$ regime: gaps

Suppose we are throwing $m$ balls into $n$ bins, where $m \gg n$. Let $X_i$ be the number of balls ending up in bin $i$, $X_\max$ be the heaviest bin, $X_\min$ be the lightest bin, and ...
12
votes
4answers
657 views

Reverse Chernoff bound

Is there an reverse Chernoff bound which bounds that the tail probability is at least so much. i.e if $X_1,X_2,\ldots,X_n$ are independent binomial random variables and $\mu=\mathbb{E}[\sum_{i=1}^n ...
3
votes
1answer
188 views

Coupon collector - the effect of randomization

Given a set $S = \{1, ..., n\}$, an element is randomly drawn with replacement from $S$. Let's say we got the following sequence $X = \{x _1, ..., x _k\}$ until all elements of $S$ are seen, where $x ...
3
votes
1answer
248 views

Estimator for sum of independent and identically distributed (iid) variables

This is a repost of a question at math.stackexchange, but I was told by a reliable source that people around here might be able to help me, so I thought I'd give it a shot. Consider the Chernoff ...
1
vote
0answers
107 views

Distribution of number of unique items in a sample

Suppose we're sampling a discrete random variable from a distribution f, n times. Is there a simple analytical formulation for the expected number of unique items we obtain, or for the distribution of ...
7
votes
2answers
151 views

Guessing a low entropy value in multiple attempts

Suppose Alice has a distribution $\mu$ over a finite (but possibly very large) domain, such that the (Shannon) entropy of $\mu$ is upper bounded by an arbitrarily small constant $\varepsilon$. Alice ...
5
votes
1answer
227 views

Random walk returning probability

Consider a two-dimensional random walk, but this time the probabilities are not $1/4$, but some values $p_1, p_2, p_3, p_4$ with $\sum_{i=1}^4 p_i=1$. For example, from $(0,0)$, it goes to $(1,0)$ ...
21
votes
3answers
956 views

Number of distinct nodes in a random walk

Commute time in a connected graph $G=(V,E)$ is defined as the expected number of steps in a random walk starting at $i$, before node $j$ is visited and then node $i$ is reached again. It is basically ...
1
vote
2answers
202 views

question about probability ranking principle

I am studying IR and I am not clearly understand "probability ranking principle" ( I tried to google the definition, but i couldn't find clear answer.) I am assuming that it's system which ...
7
votes
1answer
271 views

expected number of sets generated by greedy set cover ?

I see most of the analysis for the greedy set cover analyses the approximation ratio. However, assume that each element in $T$ belong with a constant probability to one of the sets of $S$ (where $S = ...
8
votes
0answers
155 views

Testing the degree of a vertex

Let's say we have a graph $G$ with $n$ vertices. Given $\epsilon>0$ and a specific vertex $v$, consider the problem of deciding whether $\mathrm{deg}(v) < \frac{\epsilon}{3}n$ or ...
8
votes
2answers
459 views

High probability events without low probability coordinates

Let $X$ be a random variable taking values in $\Sigma^n$ (for some large alphabet $\Sigma$), which has very high entropy - say, $H(X) \ge (n- \delta)\cdot\log|\Sigma|$ for an arbitrarily small ...
5
votes
0answers
131 views

Tree search guided by a probabilistic oracle

I'm trying to find a solution for the following problem. I have a tree $T$ of branching factor $b$ and depth $d$. For the moment, I only care about the case where I restrict $b=2$, but I would be ...
9
votes
0answers
357 views

Sampling from Multivariate Gaussian with Graph Laplacian (inverse) Covariance

We know from e.g. Koutis-Miller-Peng (based on work of Spielman & Teng), that we can very quickly solve linear systems $A x = b$ for matrices $A$ that are the graph Laplacian matrix for some ...
10
votes
0answers
191 views

Cheeger's inequality for directed graphs?

Cheeger's inequality can be used to relate the size of the worst cut in the graph to the eigenvalue gap of a simple random walk on that graph. I am wondering if it possible to extend this result to ...
1
vote
0answers
208 views

application for the Kchinchine inequality in Computer Science

The classical Kchinchine inequality states that for vector $a=(a_1, \ldots, a_{2m})\in R^{2m}$, for $p\geq 2$, and for independent Rademacher random variables $r_1, \ldots, r_{2m}$, one has $$ ...
12
votes
2answers
254 views

What's the bias of random polynomials with low degree over GF(2)?

I have a question concerning low-degree polynomials and probability: What is the (assyptotic behavior of the) probability that a random* polynomial, $p$, over GF(2), with degree $\le d$ and n ...
9
votes
1answer
217 views

Cover time and spectral gap for reversible random walks

I am looking for a theorem which say something like this: if the cover time of a reversible Markov chain is small, then the spectral gap is large. Here the spectral gap means $1-|\lambda_2|$, that is, ...
4
votes
0answers
93 views

Problem about machine scheduling [closed]

I am re-posting the problem http://mathoverflow.net/questions/92783/random-task-scheduling-problem here because I think people here are more familiar with this topic. Assume there are $m$ tasks, each ...