Csdrhrt

I'm working on the following exercise from Achim Klenke's "Probability Theory: A Comprehensive Course" (exercise 6.1.4):

Let $X_1, X_2, ldots$ be independent, square integrable, centered random variables with $sum_{i=1}^infty mathbf{Var}[X_i] < infty$. Show that there exists a square integrable $X$ with $X = lim_{n to infty} sum_{i=1}^n X_i$ almost surely.

Chebyshev's inequality gives us
$$
mathbf Pleft[|S_m - S_n| > epsilonright] leq epsilon^{-2} mathbf{Var}left[ sum_{i=m+1}^n X_iright] = epsilon^{-2} sum_{i=m+1}^n mathbf{Var}left[X_iright] xrightarrow{m,n to infty} 0.
$$
whence $(S_n)_{n in mathbb N}$ is a Cauchy sequence in probability. Thus $S_n xrightarrow{mathbf P} X$. Using a similar strategy, we can in fact show that $S_n to X$ in $L^2$.

Now, to prove almost sure convergence, I'd like to use the following result (Corollary 6.13 in Klenke):

Let $(E,d)$ be a separable metric space. Let $f, f_1, f_2, ldots$ be measurable maps $Omega to E$. Then the following statements are equivalent.

(i)$quad f_n to f$ in measure as $n to infty$.

(ii)$quad$For any subsequence of $(f_n)_{n in mathbb N}$, there exists a sub-subsequence that converges to $f$ almost everywhere.

and somehow use the fact that we're working with a sum of centered random variables to show that in fact every subsequence converges a.s. But I'm not sure how to do this since our $X_i$ are not nonnegative. I tried reconstructing the proof of this theorem, but I've only been able to show once again that there are a.e. convergent subsequences.

My other thought was to apply the Borel-Cantelli lemma to the events $B_n(epsilon) := left{ |X - S_n| > epsilonright}$ and prove that $limsup_{n to infty} B_n(epsilon) =: B(epsilon)$ has probability $0$, but in the latter case I don't know how to approximate the probability of $B_n(epsilon)$. Chebyshev doesn't seem available to us since strictly speaking we don't know what $X$ looks like, only that $S_n$ converges in $L^2$ to it. Even if we could say $X - S_n = sum_{i=n+1}^infty X_i$, the above approximation using Chebyshev with $|X - S_n|$ instead of $|S_m - S_n|$ would work out to
$$
mathbf Pleft[|X - S_n| > epsilonright] leq epsilon^{-2} sum_{i=n+1}^infty mathbf{Var}[X_i]
$$
which would sum to $epsilon^{-2} sum_{n=1}^infty nmathbf{Var}[X_n]$, but I don't see why this series converges.

Any thoughts on how to prove $S_n to X$ almost surely?

Since
$$
lim_{ntoinfty}mathsf{P}left(sup_{kge n}|S_n-S_k|> epsilonright)=0,
$$
the set on which the sequence ${S_n}$ is not Cauchy,
$$
N=bigcup_{epsilon>0}bigcap_{nge 1}left{sup_{j,kge n}|S_j-S_k|>epsilonright}
$$
is a null set ($because sup_{j,kge n}|S_j-S_k|le 2sup_{kge n}|S_n-S_k|$). So you define $X:=lim_{ntoinfty} S_n1_{N^c}$.

$var (sum _{i=n}^{m} X_i) =sum _{i=n}^{m} var(X_i) to 0$ as $n,m to infty$so the partial sums of $sum X_i$ form a Cauchy sequence in $L^{2}$. Hence there is a square integrable random variable $X$ such that $sum _1^{n} X_i to X$ in $L^{2}$. Now convergence in mean square implies convergence in probability and for series of independent random variables convergence in probability implies almost sure convergence.

Your try is not bad, but I doubt that those methods will give you the result. We should show that the sum of independent random variables
$$S_n = sum_{i=1}^n X_i to S_infty
$$ almost surely. But we know that in general $$
P(sum_{i=1}^infty |X_i|<infty) =1
$$ fails. This means that the series is conditionally convergent in most cases, and it is known that some kind of maximal inequality such as $$
P(max_{nin mathbb{N}} |S_n|>lambda) leq frac{C}{lambda^2}sum_{n=1}^infty operatorname{Var}(X_n)
$$ provides sufficient and necessary condition for the almost sure convergence. It is necessary since we need to control the oscillation of the sequence
$$
nmapsto S_n(omega)
$$ for almost all $omegain Omega$. Fortunately there are several known maximal inequalities such as Kolmogorov's maximal inequality, Etemadi's inequality or martingale maximal inequalities. In particular, Kolmogorov's inequality can establish that $$
S_n text{ converges a.s.} iff S_n text{ converges in probability.}
$$ (Or you can see this: https://en.wikipedia.org/wiki/Kolmogorov%27s_two-series_theorem.) If you are allowed to use more powerful tools such as martingale convergence theorem, then $$S_n = sum_{i=1}^n X_i to S_infty
$$ almost surely follows immediately.