Let us say that a RV $X$ with mean $\mu$ and variance $\sigma^2$ satisfies Bernstein condition with a parameter $\beta>0$, if for all $k \ge 2$, it holds that $$ |\mathbb{E}[(X - \mu)^k]| \le \frac 12 k! \sigma^2 \beta^{k-2}. $$ Trivially, if $X$ satisfies the Bernstein condition with a parameter $\beta$, then it also satisfies it with any $\beta' > \beta$.

If there are $n$ independent RVs $X_1,X_2,\dotsc,X_n$ with means $\mu_1,\dotsc,\mu_n$, variances $\sigma_1^2,\dotsc,\sigma_n^k$, and satisfying Bernstein condition with parameters $\beta_1,\dotsc,\beta_n$, respectively, what can we say about their sum $X=X_1+\dotsb+X_n$?

I think I can show that $X$ satisfies Bernstein condition with the parameter $$ \beta = \sqrt[3]{n} \max(\sigma_1,\dotsc,\sigma_n,\beta_1,\dotsc,\beta_n). $$

But I also believe this parameter can be decreased. What could be a smaller value of the Bernstein parameter of $X$?

UPD: I have a suspicion that it can be shown that $\beta = \max(\sigma_1,\dotsc,\sigma_n,\beta_1,\dotsc,\beta_n)$ but is it true?

Let us say that a RV $X$ with mean $\mu$ and variance $\sigma^2$ satisfies Bernstein condition with a parameter $\beta>0$, if for all $k \ge 2$, it holds that $$ |\mathbb{E}[(X - \mu)^k]| \le \frac 12 k! \sigma^2 \beta^{k-2}. $$ Trivially, if $X$ satisfies the Bernstein condition with a parameter $\beta$, then it also satisfies it with any $\beta' > \beta$.

If there are $n$ independent RVs $X_1,X_2,\dotsc,X_n$ with means $\mu_1,\dotsc,\mu_n$, variances $\sigma_1^2,\dotsc,\sigma_n^2$, and satisfying Bernstein condition with parameters $\beta_1,\dotsc,\beta_n$, respectively, what can we say about their sum $X=X_1+\dotsb+X_n$?

I think I can show that $X$ satisfies Bernstein condition with the parameter $$ \beta = \sqrt[3]{n} \max(\sigma_1,\dotsc,\sigma_n,\beta_1,\dotsc,\beta_n). $$

But I also believe this parameter can be decreased. What could be a smaller value of the Bernstein parameter of $X$?

UPD: I have a suspicion that it can be shown that $\beta = \max(\sigma_1,\dotsc,\sigma_n,\beta_1,\dotsc,\beta_n)$ but is it true?

Let us say that a RV $X$ with mean $\mu$ and variance $\sigma^2$ satisfies Bernstein condition with a parameter $\beta>0$, if for all $k \ge 2$, it holds that $$ |\mathbb{E}[(X - \mu)^k]| \le \frac 12 k! \sigma^2 \beta^{k-2}. $$ Trivially, if $X$ satisfies the Bernstein condition with a parameter $\beta$, then it also satisfies it with any $\beta' > \beta$.

If there are $n$ independent RVs $X_1,X_2,\dotsc,X_n$ with means $\mu_1,\dotsc,\mu_n$, variances $\sigma_1^2,\dotsc,\sigma_n^k$, and satisfying Bernstein condition with parameters $\beta_1,\dotsc,\beta_n$, respectively, what can we say about their sum $X=X_1+\dotsb+X_n$?

I think I can show that $X$ satisfies Bernstein condition with the parameter $$ \beta = \sqrt[3]{n} \max(\sigma_1,\dotsc,\sigma_n,\beta_1,\dotsc,\beta_n). $$

But I also believe this parameter can be decreased. What could be a smaller value of the Bernstein parameter of $X$?

Let us say that a RV $X$ with mean $\mu$ and variance $\sigma^2$ satisfies Bernstein condition with a parameter $\beta>0$, if for all $k \ge 2$, it holds that $$ |\mathbb{E}[(X - \mu)^k]| \le \frac 12 k! \sigma^2 \beta^{k-2}. $$ Trivially, if $X$ satisfies the Bernstein condition with a parameter $\beta$, then it also satisfies it with any $\beta' > \beta$.

If there are $n$ independent RVs $X_1,X_2,\dotsc,X_n$ with means $\mu_1,\dotsc,\mu_n$, variances $\sigma_1^2,\dotsc,\sigma_n^k$, and satisfying Bernstein condition with parameters $\beta_1,\dotsc,\beta_n$, respectively, what can we say about their sum $X=X_1+\dotsb+X_n$?

I think I can show that $X$ satisfies Bernstein condition with the parameter $$ \beta = \sqrt[3]{n} \max(\sigma_1,\dotsc,\sigma_n,\beta_1,\dotsc,\beta_n). $$

But I also believe this parameter can be decreased. What could be a smaller value of the Bernstein parameter of $X$?

UPD: I have a suspicion that it can be shown that $\beta = \max(\sigma_1,\dotsc,\sigma_n,\beta_1,\dotsc,\beta_n)$ but is it true?