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What should be really done is as follows. Suppose that $X_1,\dots,X_n$ are any real random variables for which individual distribution laws are known but no information is given on the joint distribut …

The argument below is a bit weird because it is a mixture of an explicit computation and a general handwaving (rigorous handwaving, of course), but, when figuring it out at 70 mph under medium strengt …

The simplest way is to couple your process with something obvious. It can be done in many ways. I would do the following. Consider $r$ bins represented as disjoint unit intervals on $\mathbb R$. Run …

If you condition on the (random) number of steps $n$ made by the time $t$, you'll see that the distribution is just a mixture of biased random walks with different number of steps. Each particular ran …

I hesitated a bit whether to use another answer window or to edit the old one but finally decided in favor of a new window. If moderators think it is a bad idea, they are welcome to merge. Also, it …

I just accidentally stumbled upon this nice question. I suspect that by now you know the answer yourself, but still, let me do one simple computation. If you like it, I'll think more of the question. …

It is always true. Split $x_i=y_i+z_i$ where $y_i$ are bounded and $Ez_i\le \frac \mu{10n}$. You have no problems with $y_i$ because if they were alone,$ES_j$ would be concentrated in a very strong se …

If I interpret your question correctly, the answer is that the exponential random variable $\xi$ (density $e^{-x}$) has the property that for two independent copies $\xi_1,\xi_2$ of $\xi$, one has $P( …

The answer is, indeed, "No" because there is an unbounded with probability $1$ stochastic process that satisfies the given inequality, namely, $X(t)=0.1\log|t-w|$ where $w$ is equidistributed on $[0,1 …

I'm not sure what you really want but here is a couple of simple minded inequalities that can serve as a baseline. Below $g=\sum_{k\in J}\gamma^k$, $M=\sum_{k\in J}k\gamma^k$, so $\mu=\frac Mg$. We'l …

Assume that the networks err independently and the corresponding probabilities are $(TP)_j, (TN)_j, (FP)_j, (FN)_j$ (false negative = the answer is yes and the network outputs no, etc.). Also assume t …

Here is a sketch. Choose a big constant $K$ and put in $Kn$ edges at random. That is essentially the same as to consider the random graph with the edge probability $p=2K/n$. The typical (in all senses …

The answer is "No, of course". Let us denote $(\omega,\omega',n)=(\omega_1,\dots,\omega_n,\omega'_{n+1},\dots)$. Take any large $m$ and consider the event $A_m$ that $\sum_{n=0}^{2m}\omega_n=m$. Then …

Suppose that $U_n,V$ are some random variables such that $V$ is uniformly distributed on $[0,1]$, $U_n$ converges to $V$ a.s. and for every $n$, $P(U_n\ne V)=1$ (To be honest, I got a bit confused try …

Let me know if I'm talking nonsense but it looks like if you just consider the usual entropy $H(P)=\sum_k P(k)\log P(k)$ for probability measures $P$ on $\mathbb N$, then $$ E[H(X_*Q_{n+1})]-E[H(X_* …