Let $X_n \geq 0$ be independent for $n \geq 1$. Then the following are equivalent:   (i) $\sum_{n=1}^\infty X_n$ converges a.s.  (ii)$\sum_{n=1}^\infty  {[P(X_n &gt;1) + E[ X_n I_{X_n \leq 1} ] }]  &lt; \infty $  (iii)$\sum_{n=1}^\infty E [ X_n / (1+ X_n)] &lt; \infty $  How could we show the above? I tried using Kolmogorov's Three-Series Theorem but I was unable to show it.
Question  Let $X_n\geq 0$ be independent for $n\geq 1$. The following are equivalent.  $\sum_{n=1}^\infty X_n&lt;\infty$ a.s. $\sum_{n=1}^\infty[P(X_n&gt;1)+E(X_nI(X_n\leq 1)]&lt;\infty$ $\sum_{n=1}^\infty E(X_n/(1+X_n))&lt;\infty$   My attempt I was able to prove $(1)$ implies $(2)$ and $(2)$ implies $(3)$ but not $(3)$ implies $(1)$. For $(1)$ implies $(2)$, observe that by Borel-Cantelli $\sum_{n=1}^\infty P(X_n&gt;1)&lt;\infty$, else $X_n&gt;1$ infinitely often w.p.1 so that $(1)$ is violated. To see that $\sum_{n=1}^\infty E(X_nI(X_n\leq 1)]&lt;\infty$, apply Kolmogorov three-series theorem (I think). For $(2)$ implies $(3)$ we can write that $$ \begin{align} \frac{X_n}{1+X_n}&amp; =\frac{X_n}{1+X_n}I(X_n\leq 1)+\frac{X_n}{1+X_n}I(X_n&gt;1)\\ &amp;\leq X_nI(X_n\leq 1)+ I(X_n&gt;1). \end{align} $$ Now take the expectation of both sides and sum on $n$ using $(2)$ to conclude $(3)$. For the last implication $(3)$ implies $(1)$, I tried to use Kolmogorov three series theorem. To this end put $Y_n=X_nI(X_n\leq 1)$. First note that $\sum_{n=1}^\infty P(X_n&gt;1)&lt;\infty$ since $$ P(X_n&gt;1)\leq P\left(\frac{X_n}{1+X_n}&gt;\frac{1}{2}\right)\leq 2E\frac{X_n}{1+X_n} $$ by Markov's inequality. But I am unable to show that $\sum EY_n&lt;\infty$ and $\sum\text{Var}(Y_n)&lt;\infty$. Any help is appreciated.
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I'm trying to create a join/relate in QGIS where I have a shapefile of buildings and to that I would like to join a non-spatial table (.csv-file) containing people who work in each building. So I have multiple records in my table that I would like to be able to join to my single features in my shapefile. I can only run a join in QGIS where the first record is joined to the shapefile feature (ie building) but subsequent records get removed. Could someone let me know how to complete this join/relate in QGIS?