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do we at least have $Y_n=\Theta(\frac{\log n}{n})$ almost surely?

The answer to this is no. It is not even true that $Y_n=\Theta(\frac{h(n)}{n})$ almost surely (a.s.) for any (deterministic or random) function $h$ such that $h(n)\to\infty$ a.s. (as $n\to\infty$). Indeed, suppose the contrary: that $\liminf_n nY_n=\infty$ a.s. Then, by the Fatou lemma, $$ \infty=E\infty\le\liminf_n EnY_n =\liminf_n n\frac1{n+1}=1, $$ a contradiction. $\quad\Box$

Let us now show that $Y_n\not\sim f(n)$ a.s. for any deterministic positive function $f$.

To prove this, suppose the contrary: that $Y_n\sim f(n)$ a.s. for some deterministic positive function $f$. Take any positive sequence $(a_n)$ such that $\sum_n a_n=\infty$. Then $$\sum_n P(X_n<a_n/2)=\sum_n a_n/2=\infty.$$ So, by the Borel--Cantelli lemma, events $\{X_n<a_n/2\}$ a.s. occur infinitely often (i.o.), that is, for infinitely many values of $n$. Therefore and because $\{X_n<a_n/2\}\subseteq\{Y_n<a_n/2\}$, we see that events $\{Y_n<a_n/2\}$ a.s. occur i.o. Recalling now the assumption that $Y_n\sim f(n)$ a.s., we conclude that $f(n)\le a_n$ i.o. In particular, $f(n)\le \frac1{n\ln n}=o(1/n)$ i.o.

So, $$Z_n:=nY_n\sim nf(n)=o(1) \tag{1}\label{1}$$ a.s. for $n=n_k$ and $k\to\infty$, where $(n_k)$ is a strictly increasing deterministic sequence of positive integers. Also, $EZ_n^2<2$. So, for $n$ as above, $$\begin{aligned} 1\leftarrow EZ_n&=EZ_n\,1(Z_n<4)+EZ_n\,1(Z_n\ge4) \\ &\le EZ_n\,1(Z_n<4)+EZ_n^2/4 \\ &<EZ_n\,1(Z_n<4)+1/2\to1/2 \end{aligned}$$ by \eqref{1} and dominated convergence. So, we have a contradiction. $\quad\Box$

In fact, $Y_n\not\sim f(n)$ even in probability for any positive deterministic function $f$.

Indeed, suppose that $Y_n\sim f(n)$ in probability for some positive deterministic function $f$. Then, by the Fatou lemma, $$1=E\lim_n\frac{Y_n}{f(n)}\le \liminf_n\frac{EY_n}{f(n)} =\liminf_n\frac{1}{(n+1)f(n)},$$ so that $f(n)\lesssim1/n$. So, $nf(n)\to c$ for some real $c\ge0$, $n=n_k$, and $k\to\infty$, where $(n_k)$ is some strictly increasing sequence of positive integers. So, for such $n$ and $Z_n$ as above, $$Z_n\to c$$ in probability. Also, $EZ_n\to1$, $EZ_n^2\to2$, and $EZ_n^4\to24$. Take now any real $A>0$. Then $$E(Z_n-c)^2=E(Z_n^2-2cZ_n+c^2)\to C:=2-2c+c^2,$$ whereas, for $n$ as above, $$\begin{aligned} E(Z_n-c)^2 &=E(Z_n-c)^2\,1((Z_n-c)^2\le A)+E(Z_n-c)^2\,1((Z_n-c)^2>A) \\ &\le E(Z_n-c)^2\,1((Z_n-c)^2\le A)+E(Z_n-c)^4/A \\ &\le E(Z_n-c)^2\,1((Z_n-c)^2\le A)+2^3(EZ_n^4+c^4)/A \\ &\to 0+2^3(24+c^4)/A \end{aligned}$$ by dominated convergence. We conclude that $0<C\le 2^3(24+c^4)/A$ for all real $A>0$, a contradiction. $\quad\Box$

do we at least have $Y_n=\Theta(\frac{\log n}{n})$ almost surely?

The answer to this is no. It is not even true that $Y_n=\Theta(\frac{h(n)}{n})$ almost surely (a.s.) for any (deterministic or random) function $h$ such that $h(n)\to\infty$ a.s. (as $n\to\infty$). Indeed, suppose the contrary: that $\liminf_n nY_n=\infty$ a.s. Then, by the Fatou lemma, $$ \infty=E\infty\le\liminf_n EnY_n =\liminf_n n\frac1{n+1}=1, $$ a contradiction. $\quad\Box$

Let us now show that $Y_n\not\sim f(n)$ a.s. for any deterministic positive function $f$.

To prove this, suppose the contrary: that $Y_n\sim f(n)$ a.s. for some deterministic positive function $f$. Take any positive sequence $(a_n)$ such that $\sum_n a_n=\infty$. Then $$\sum_n P(X_n<a_n/2)=\sum_n a_n/2=\infty.$$ So, by the Borel--Cantelli lemma, events $\{X_n<a_n/2\}$ a.s. occur infinitely often (i.o.), that is, for infinitely many values of $n$. Therefore and because $\{X_n<a_n/2\}\subseteq\{Y_n<a_n/2\}$, we see that events $\{Y_n<a_n/2\}$ a.s. occur i.o. Recalling now the assumption that $Y_n\sim f(n)$ a.s., we conclude that $f(n)\le a_n$ i.o. In particular, $f(n)\le \frac1{n\ln n}=o(1/n)$ i.o.

So, $$Z_n:=nY_n\sim nf(n)=o(1) \tag{1}\label{1}$$ a.s. for $n=n_k$ and $k\to\infty$, where $(n_k)$ is a strictly increasing deterministic sequence of positive integers. Also, $EZ_n^2<2$. So, for $n$ as above, $$\begin{aligned} 1\leftarrow EZ_n&=EZ_n\,1(Z_n<4)+EZ_n\,1(Z_n\ge4) \\ &\le EZ_n\,1(Z_n<4)+EZ_n^2/4 \\ &<EZ_n\,1(Z_n<4)+1/2\to1/2 \end{aligned}$$ by \eqref{1} and dominated convergence. So, we have a contradiction. $\quad\Box$

In fact, $Y_n\not\sim f(n)$ even in probability for any positive deterministic function $f$.

Indeed, suppose that $Y_n\sim f(n)$ in probability for some positive deterministic function $f$. Then, by the Fatou lemma, $$1=E\lim_n\frac{Y_n}{f(n)}\le \liminf_n\frac{EY_n}{f(n)} =\liminf_n\frac{1}{(n+1)f(n)},$$ so that $f(n)\lesssim1/n$. So, $nf(n)\to c$ for some real $c\ge0$, $n=n_k$, and $k\to\infty$, where $(n_k)$ is some strictly increasing sequence of positive integers. So, for such $n$ and $Z_n$ as above, $$Z_n\to c$$ in probability. Also, $EZ_n\to1$, $EZ_n^2\to2$, and $EZ_n^4\to24$. Take now any real $A>0$. Then $$E(Z_n-c)^2=E(Z_n^2-2cZ_n+c^2)\to C:=2-2c+c^2,$$ whereas, for $n$ as above, $$\begin{aligned} E(Z_n-c)^2 &=E(Z_n-c)^2\,1((Z_n-c)^2\le A)+E(Z_n-c)^2\,1((Z_n-c)^2>A) \\ &\le E(Z_n-c)^2\,1((Z_n-c)^2\le A)+E(Z_n-c)^4/A \\ &\le E(Z_n-c)^2\,1((Z_n-c)^2\le A)+(EZ_n^4+c^4)/A \\ &\to 0+(24+c^4)/A \end{aligned}$$ by dominated convergence. We conclude that $0<C\le(24+c^4)/A$ for all real $A>0$, a contradiction. $\quad\Box$

do we at least have $Y_n=\Theta(\frac{\log n}{n})$ almost surely?

The answer to this is no. It is not even true that $Y_n=\Theta(\frac{h(n)}{n})$ almost surely (a.s.) for any (deterministic or random) function $h$ such that $h(n)\to\infty$ a.s. (as $n\to\infty$). Indeed, suppose the contrary: that $\liminf_n nY_n=\infty$ a.s. Then, by the Fatou lemma, $$ \infty=E\infty\le\liminf_n EnY_n =\liminf_n n\frac1{n+1}=1, $$ a contradiction. $\quad\Box$

Let us now show that $Y_n\not\sim f(n)$ a.s. for any deterministic positive function $f$.

To prove this, suppose the contrary: that $Y_n\sim f(n)$ a.s. for some deterministic positive function $f$. Take any positive sequence $(a_n)$ such that $\sum_n a_n=\infty$. Then $$\sum_n P(X_n<a_n/2)=\sum_n a_n/2=\infty.$$ So, by the Borel--Cantelli lemma, events $\{X_n<a_n/2\}$ a.s. occur infinitely often (i.o.), that is, for infinitely many values of $n$. Therefore and because $\{X_n<a_n/2\}\subseteq\{Y_n<a_n/2\}$, we see that events $\{Y_n<a_n/2\}$ a.s. occur i.o. Recalling now the assumption that $Y_n\sim f(n)$ a.s., we conclude that $f(n)\le a_n$ i.o. In particular, $f(n)\le \frac1{n\ln n}=o(1/n)$ i.o.

So, $$Z_n:=nY_n\sim nf(n)=o(1) \tag{1}\label{1}$$ a.s. for $n=n_k$ and $k\to\infty$, where $(n_k)$ is a strictly increasing deterministic sequence of positive integers. Also, $EZ_n^2<2$. So, for $n$ as above, $$\begin{aligned} 1\leftarrow EZ_n&=EZ_n\,1(Z_n<4)+EZ_n\,1(Z_n\ge4) \\ &\le EZ_n\,1(Z_n<4)+EZ_n^2/4 \\ &<EZ_n\,1(Z_n<4)+1/2\to1/2 \end{aligned}$$ by \eqref{1} and dominated convergence. So, we have a contradiction. $\quad\Box$

do we at least have $Y_n=\Theta(\frac{\log n}{n})$ almost surely?

The answer to this is no. It is not even true that $Y_n=\Theta(\frac{h(n)}{n})$ almost surely (a.s.) for any (deterministic or random) function $h$ such that $h(n)\to\infty$ a.s. (as $n\to\infty$). Indeed, suppose the contrary: that $\liminf_n nY_n=\infty$ a.s. Then, by the Fatou lemma, $$ \infty=E\infty\le\liminf_n EnY_n =\liminf_n n\frac1{n+1}=1, $$ a contradiction. $\quad\Box$

Let us now show that $Y_n\not\sim f(n)$ a.s. for any deterministic positive function $f$.

To prove this, suppose the contrary: that $Y_n\sim f(n)$ a.s. for some deterministic positive function $f$. Take any positive sequence $(a_n)$ such that $\sum_n a_n=\infty$. Then $$\sum_n P(X_n<a_n/2)=\sum_n a_n/2=\infty.$$ So, by the Borel--Cantelli lemma, events $\{X_n<a_n/2\}$ a.s. occur infinitely often (i.o.), that is, for infinitely many values of $n$. Therefore and because $\{X_n<a_n/2\}\subseteq\{Y_n<a_n/2\}$, we see that events $\{Y_n<a_n/2\}$ a.s. occur i.o. Recalling now the assumption that $Y_n\sim f(n)$ a.s., we conclude that $f(n)\le a_n$ i.o. In particular, $f(n)\le \frac1{n\ln n}=o(1/n)$ i.o.

So, $$Z_n:=nY_n\sim nf(n)=o(1) \tag{1}\label{1}$$ a.s. for $n=n_k$ and $k\to\infty$, where $(n_k)$ is a strictly increasing deterministic sequence of positive integers. Also, $EZ_n^2<2$. So, for $n$ as above, $$\begin{aligned} 1\leftarrow EZ_n&=EZ_n\,1(Z_n<4)+EZ_n\,1(Z_n\ge4) \\ &\le EZ_n\,1(Z_n<4)+EZ_n^2/4 \\ &<EZ_n\,1(Z_n<4)+1/2\to1/2 \end{aligned}$$ by \eqref{1} and dominated convergence. So, we have a contradiction. $\quad\Box$

In fact, $Y_n\not\sim f(n)$ even in probability for any positive deterministic function $f$.

Indeed, suppose that $Y_n\sim f(n)$ in probability for some positive deterministic function $f$. Then, by the Fatou lemma, $$1=E\lim_n\frac{Y_n}{f(n)}\le \liminf_n\frac{EY_n}{f(n)} =\liminf_n\frac{1}{(n+1)f(n)},$$ so that $f(n)\lesssim1/n$. So, $nf(n)\to c$ for some real $c\ge0$, $n=n_k$, and $k\to\infty$, where $(n_k)$ is some strictly increasing sequence of positive integers. So, for such $n$ and $Z_n$ as above, $$Z_n\to c$$ in probability. Also, $EZ_n\to1$, $EZ_n^2\to2$, and $EZ_n^4\to24$. Take now any real $A>0$. Then $$E(Z_n-c)^2=E(Z_n^2-2cZ_n+c^2)\to C:=2-2c+c^2,$$ whereas, for $n$ as above, $$\begin{aligned} E(Z_n-c)^2 &=E(Z_n-c)^2\,1((Z_n-c)^2\le A)+E(Z_n-c)^2\,1((Z_n-c)^2>A) \\ &\le E(Z_n-c)^2\,1((Z_n-c)^2\le A)+E(Z_n-c)^4/A \\ &\le E(Z_n-c)^2\,1((Z_n-c)^2\le A)+2^3(EZ_n^4+c^4)/A \\ &\to 0+2^3(24+c^4)/A \end{aligned}$$ by dominated convergence. We conclude that $0<C\le 2^3(24+c^4)/A$ for all real $A>0$, a contradiction. $\quad\Box$

do we at least have $Y_n=\Theta(\frac{\log n}{n})$ almost surely?

The answer to this is no. It is not even true that $Y_n=\Theta(\frac{h(n)}{n})$ almost surely (a.s.) for any (deterministic or random) function $h$ such that $h(n)\to\infty$ a.s. (as $n\to\infty$). Indeed, suppose the contrary: that $\liminf_n nY_n=\infty$ a.s. Then, by the Fatou lemma, $$ \infty=E\infty\le\liminf_n EnY_n =\liminf_n n\frac1{n+1}=1, $$ a contradiction. $\quad\Box$

Let us now show that $Y_n\not\sim f(n)$ a.s. for any deterministic positive function $f$.

To prove this, suppose the contrary: that $Y_n\sim f(n)$ a.s. for some deterministic positive function $f$. Take any positive sequence $(a_n)$ such that $\sum_n a_n=\infty$. Then $$\sum_n P(X_n<a_n/2)=\sum_n a_n/2=\infty.$$ So, by the Borel--Cantelli lemma, events $\{X_n<a_n/2\}$ a.s. occur infinitely often (i.o.), that is, for infinitely many values of $n$. Therefore and because $\{X_n<a_n/2\}\subseteq\{Y_n<a_n/2\}$, we see that events $\{Y_n<a_n/2\}$ a.s. occur i.o. Recalling now the assumption that $Y_n\sim f(n)$ a.s., we conclude that $f(n)\le a_n$ i.o. In particular, $f(n)\le \frac1{n\ln n}=o(1/n)$ i.o.

So, $$Z_n:=nY_n\sim nf(n)=o(1) \tag{1}\label{1}$$ a.s. for $n=n_k$ and $k\to\infty$, where $(n_k)$ is a strictly increasing deterministic sequence of positive integers. Also, $EZ_n^2<1$. So, for $n$ as above, $$\begin{aligned} 1\leftarrow EZ_n&=EZ_n\,1(Z_n<2)+EZ_n\,1(Z_n\ge2) \\ &\le EZ_n\,1(Z_n<2)+EZ_n^2/2 \\ &\le EZ_n\,1(Z_n<2)+1/2\to1/2 \end{aligned}$$ by \eqref{1} and dominated convergence. So, we have a contradiction. $\quad\Box$

do we at least have $Y_n=\Theta(\frac{\log n}{n})$ almost surely?

The answer to this is no. It is not even true that $Y_n=\Theta(\frac{h(n)}{n})$ almost surely (a.s.) for any (deterministic or random) function $h$ such that $h(n)\to\infty$ a.s. (as $n\to\infty$). Indeed, suppose the contrary: that $\liminf_n nY_n=\infty$ a.s. Then, by the Fatou lemma, $$ \infty=E\infty\le\liminf_n EnY_n =\liminf_n n\frac1{n+1}=1, $$ a contradiction. $\quad\Box$

Let us now show that $Y_n\not\sim f(n)$ a.s. for any deterministic positive function $f$.

To prove this, suppose the contrary: that $Y_n\sim f(n)$ a.s. for some deterministic positive function $f$. Take any positive sequence $(a_n)$ such that $\sum_n a_n=\infty$. Then $$\sum_n P(X_n<a_n/2)=\sum_n a_n/2=\infty.$$ So, by the Borel--Cantelli lemma, events $\{X_n<a_n/2\}$ a.s. occur infinitely often (i.o.), that is, for infinitely many values of $n$. Therefore and because $\{X_n<a_n/2\}\subseteq\{Y_n<a_n/2\}$, we see that events $\{Y_n<a_n/2\}$ a.s. occur i.o. Recalling now the assumption that $Y_n\sim f(n)$ a.s., we conclude that $f(n)\le a_n$ i.o. In particular, $f(n)\le \frac1{n\ln n}=o(1/n)$ i.o.

So, $$Z_n:=nY_n\sim nf(n)=o(1) \tag{1}\label{1}$$ a.s. for $n=n_k$ and $k\to\infty$, where $(n_k)$ is a strictly increasing deterministic sequence of positive integers. Also, $EZ_n^2<2$. So, for $n$ as above, $$\begin{aligned} 1\leftarrow EZ_n&=EZ_n\,1(Z_n<4)+EZ_n\,1(Z_n\ge4) \\ &\le EZ_n\,1(Z_n<4)+EZ_n^2/4 \\ &<EZ_n\,1(Z_n<4)+1/2\to1/2 \end{aligned}$$ by \eqref{1} and dominated convergence. So, we have a contradiction. $\quad\Box$