3 votes
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Estimation of the expected number of sites visited by i.i.d

Denote $p_k=P(X=k)$. Then $E(R_n)=\sum_k P(k\in \{X_1,\ldots,X_n\})\leqslant \sum_k \min(1,np_k)$. We are given that $\sum kp_k<\infty$. Fix $\varepsilon>0$. The sum of $\min(1,np_k)$ over $k<...
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1 vote

Carne-Varopoulos bound and stationary measure

I strongly recommand Woess' book Random walks on infinite graphs and groups for all this. In the first chapters, the authors takes time to explain all basic concepts, such as (positive) recurrence (...
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1 vote

Random probability following a log concave distribution of order p

$\newcommand{\tla}{\tilde\lambda}\newcommand{\Ga}{\Gamma}$By Definition 4.1 in the paper by Bobkov and Madiman (BM), a positive random variable (r.v.) $\xi$ has a log-concave distribution of order $p\...
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1 vote

Cumulants of a sequence of variables with zero mean and variance

Take a bernoulli variable with weight $\log^{-4} n$ at $-\log n$ and weight $1-\log^{-4} n$ at $\log n/(\log^4 n-1)$. Then the mean is 0, the variance is $O(\log^{-2} n)$, the 4th moment converges to ...
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1 vote

Computing moments of discrete probability distribution

Here's a bit more direct way than with Vandermonde transpose. Consider the generating function $$ f(x) = \sum\limits_{k=0}^\infty m_k x^k $$ It rewrites as $$ f(x) = \sum\limits_{k=0}^\infty \sum\...
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