*605The defendant, Abimael Ramos, appeals from the judgment of conviction, rendered following a jury trial, of intentional manslaughter in the first degree with a firearm in violation of General Statutes §§ 53a-55 (a) (1) and 53a-55a. On appeal, he claims that (1) he was deprived of his rights to present a defense and to cross-examine witnesses, pursuant to the sixth amendment to the federal constitution, when the trial court prevented him from questioning police officers about alleged inadequacies in their investigation of the victim's death, and (2) the trial court abused its discretion in admitting into evidence, under the state of mind exception to the hearsay rule, testimony regarding the victim's relationship with the defendant prior to her death. We affirm the judgment of the trial court.
*606By way of a single count information, the state charged the defendant with murder with a firearm in violation of General Statutes §§ 53a-54a (a) and 53-202k. The charge stemmed from the death of Luz Morales, the victim, who died from a single gunshot wound to her abdomen. A jury found the defendant not guilty of murder, but guilty of the lesser included offense of manslaughter in the first degree with a firearm. The court accepted the verdict, rendered a judgment of conviction, and sentenced the defendant to a term of imprisonment of forty years, five of which are a mandatory minimum, to run concurrently with a sentence he then was already serving. This appeal followed. Additional facts and procedural history will be set forth as necessary.
I
We first address the defendant's claim that the trial court deprived him of *895his sixth amendment rights. According to the defendant, the court improperly prevented him from questioning the investigating police officers about their alleged failure to investigate a potential connection between the victim's death and a burglary at a former residence that he shared with the victim. He claims that the court deprived him of both his right to present a defense and to cross-examine witnesses.
The state argues that the defendant's proposed line of questioning addressed a ''purely speculative possibility [regarding third-party culpability that] was not relevant to the jury's determination ... and, furthermore, that ... carried with it a substantial risk of unfair prejudice to the state [by] diverting the jury's attention to collateral matters.'' (Citation omitted.) According to the state, the court properly exercised its discretion in limiting the inquiry into alleged deficiencies in the police investigation ''[absent] anything other than a bare suspicion *607that the victim's death ... was in any way related to the [previous] burglary ....''1 We agree with the state.
The following additional facts, which the jury reasonably could have found, and procedural history are relevant to our decision. For approximately five years, the defendant and the victim were in a romantic relationship. They lived together at 761 Wood Avenue in Bridgeport at the time of the victim's death, and previously had lived together at 222 Lenox Avenue in Bridgeport.
At approximately 11:30 p.m., on May 23, 2011, Christina Catlin heard the defendant, her neighbor, banging on her front door, stating ''help her, help her.'' After Catlin opened the door, the defendant ran back to his house, and she followed. When Catlin entered the defendant's residence, she saw the victim lying on her back, naked, and ''very, very pale,'' at the top of a staircase. The victim had a large cut near her left eyebrow, ''a tiny hole'' near her belly button, and blood underneath her. Catlin asked the defendant to call 911, but when he did not respond, she grabbed the cell phone from his hand and did so. Medical personnel subsequently took the victim to a hospital, where she later died from a gunshot wound to her abdomen.
In the course of their investigation into the victim's death, police officers questioned the defendant about the night of May 23, 2011. In various statements he made to the police, the defendant claimed that two unidentified Jamaican or Haitian men broke into his home, attacked him, and shot the victim before fleeing down his driveway. He provided partial descriptions of the men, noting that one had a missing tooth, the other *608had a scruffy beard, and they ''[were not] big dudes,'' standing at about five feet, eight inches or five feet, seven inches in height.
Within hours of the victim's death, William Simpson, a K-9 handler with the Bridgeport Police Department, and his K-9 dog, Balu, were dispatched to 761 Wood Avenue. According to Simpson, he responded to ''a claim of home invasion'' where ''two men had been involved.'' He also testified that Balu identified a trail of human scent that started in the rear of 761 Wood Avenue and continued ''[d]own Wood Avenue [for] two or three blocks'' until reaching another street, where Balu lost the scent.
*896During a video-recorded interview on May 24, 2011, which was admitted into evidence, investigating officers questioned the defendant about a burglary that occurred at 222 Lenox Avenue in September, 2010, while the defendant and the victim lived there.2 In fact, Detective Todd Toth, one of the investigating officers who testified at trial, told the defendant, ''[T]he reason we asked about the break-in at Lenox Avenue is [that] we were wondering if it's the same people [whom you claimed were involved in shooting the victim].'' The defendant later informed police officers that the reason he and the victim moved to 761 Wood Avenue was because of the September, 2010 burglary at 222 Lenox Avenue and their fear that it might happen again.
Norman Pattis, counsel for the defendant, sought to cross-examine Toth about his knowledge of the Lenox Avenue burglary. Pattis asked Toth whether he had spoken to Carmen Rivera-Torres, the victim's aunt and the only eyewitness to the burglary, and specifically inquired whether he had ''asked her about the identifying characteristics of the persons who broke in *609....'' Toth testified, ''I believe we did.'' Pattis then asked, ''Did you ask her if they had Caribbean accents, Jamaican, Haitian, let's say?'' Toth did not answer that question, however, as the state's attorney immediately objected and stated that this particular matter was the subject of pretrial motions. The court held a sidebar discussion and stated that it would hear argument outside the presence of the jury.
In the absence of the jury, the court noted that the Lenox Avenue burglary was the subject of a motion in limine filed by the state3 and stated that it had sustained the state's objection ''because, based on what has transpired thus far, I don't believe that the defense has made the required showing for a third-party culpability [defense], and to get into that line of questioning would be to do so.'' Pattis nonetheless argued that he did not intend to argue third-party culpability; rather, ''[his] questions are going to the thoroughness of the investigation and whether [the investigating officers] prejudged things. And so, during the course of the interviews with [the defendant], the accents came up. [Toth] acknowledged going to see [Rivera-Torres] and acknowledged discussing this. If he didn't ask about the accents, and I don't know candidly what his answer will be, that will be probative ... in terms of the thoroughness of the investigation and leaving potentially exculpatory evidence on the table .... So I didn't-if he said yes, you know, I think I would have been stuck. But I don't think he did based on-I just don't think he did.'' Pattis also argued: ''Had he said no, I could have argued, *610perhaps, hey, you know, these [police officers] had made up their mind[s] and decided early to prejudge the case ....'' The court reiterated its prior ruling, and added that it also sustained the state's objection on the ground that there was no foundation establishing that Rivera-Torres ''either saw or even spoke to the individuals *897or individual that conducted the Lenox Avenue break-in.''4
Pattis also sought to cross-examine Detective Walberto Cotto, another investigating officer who testified at trial, about the police investigation into the possible connection between the Lenox Avenue burglary and the victim's death. During an offer of proof held outside the presence of the jury, Cotto testified that he was aware of the prior Lenox Avenue burglary, which possibly involved two black males, but that the defendant had informed him that the Lenox Avenue burglars did not have anything to do with the victim's death. According to Cotto, his understanding of the Lenox Avenue burglary was based solely on the defendant's statements. Pattis once again argued: ''The thoroughness of the investigation and the steps that officers took in investigating [the defendant] I think are fair game within the sixth amendment. We may well have had third-party culpability evidence had the officers done something with this information and investigated it. They didn't.''5 The court again ruled that it would not permit this line of questioning without a showing that the two incidents were somehow connected.
Pattis attempted to revisit the Lenox Avenue burglary for a third time during the defendant's case-in-chief. During an offer of proof held outside the presence of the jury, Pattis first questioned Rivera-Torres. She testified *611that she lived above the defendant and the victim at the 222 Lenox Avenue address when the burglary happened. According to her, on September 9, 2010, she noticed a familiar looking ''[six]-foot-[two], 220, 240 pound African-American male coming out of [the defendant's and the victim's] apartment.'' She later learned that their apartment had been burglarized, and spoke with police officers about her observations before and after the victim's death. She testified: ''I had spoken to [Cotto] back in 2010 when [the burglary] occurred. And then I did speak to him [after the victim's death] because I wanted to believe that that's what happened.'' Pattis indicated that he would not seek to elicit any further testimony from Rivera-Torres.
Later during the offer of proof, the following examination took place between Pattis and Toth regarding the police investigation into a possible connection between the Lenox Avenue burglary and the victim's death:
''[Pattis]: And did-was the topic of the identity or information about the burglary, did that come up in the interview with [Rivera-Torres]?
''[Toth]: I don't-I don't recall that. I don't think it did.
''[Pattis]: Did you take any steps to-you questioned [the defendant] about [the Lenox Avenue burglary], correct?
''[Toth]: Correct.
''[Pattis]: In part to see whether the same people might have returned to Wood Avenue, correct?
''[Toth]: Yeah, to see if, you know, they were related in any way.
''[Pattis]: Did-beyond talking to [the defendant], did you do anything to see whether there was any relationship between the two?
*612''[Toth]: Beyond speaking with him?
''[Pattis]: Yes.
*898''[Toth]: No.
''[Pattis]: Nothing further, Judge.''6
On redirect, Pattis also asked Toth: ''So, did you make any effort to determine whether the people who came to Wood Avenue were related to the people who went into Lenox Avenue in any way?'' Toth testified: ''I don't believe so.''
Through his questioning during the latter offer of proof, Pattis sought to establish that the alleged home invasion on May 23, 2011, might have been drug related because the defendant was a known drug dealer with a substantial amount of cash, firearms, and drugs at the Wood Avenue residence. Pattis further sought to establish that the defendant may have denied knowing who was involved in either the Lenox Avenue burglary or the alleged May 23, 2011 home invasion because individuals involved in illegal narcotics activities generally try to resolve their disputes without the help of police.
Following his final offer of proof, Pattis argued that the evidence pertaining to the alleged inadequacy of the police investigation into a possible connection between the Lenox Avenue burglary and the victim's death was admissible, principally relying on this court's decision in State v. Wright , 152 Conn. App. 260, 96 A.3d 638 (2014), rev'd, 322 Conn. 270, 140 A.3d 939 (2016). He argued: ''[T]he failure to investigate this potential linkage may well have deprived us of third-party culpability [evidence] if the police had established the link. But *613their failure to even consider it is a ... deficiency in the investigation that we think ... we should be permitted to explore ....''7 Throughout the trial, Pattis argued that preventing him from pursuing this line of questioning abridged both the defendant's right to present a defense and his right to confrontation under the sixth amendment to the federal constitution.
The court again sustained the state's objections to this line of questioning. According to the court, ''there's no other purpose to elicit this testimony other than to back door in a third-party culpability [defense]. The defendant's ability to present a defense that somebody else did this, that the police failed to pursue leads, that they had made up their mind[s] that he was the murderer, all of that he is permitted to argue, [and] has argued by way of cross-examination.'' The defendant took his exception and maintains on appeal that the court's rulings deprived him of his constitutional rights.
We now set forth the relevant legal principles governing our review of the defendant's claims. ''It is fundamental that the defendant's rights to confront the witnesses against him and to present a defense are guaranteed by the sixth amendment to the United States constitution....
''In plain terms, the defendant's right to present a defense is the right to present the defendant's version of the facts as well as the prosecution's to the jury so that it may decide where the truth lies....
*899It guarantees the right to offer the testimony of witnesses, and *614to compel their attendance, if necessary .... Therefore, exclusion of evidence offered by the defense may result in the denial of the defendant's right to present a defense....
''Although it is within the trial court's discretion to determine the extent of cross-examination and the admissibility of evidence, the preclusion of sufficient inquiry into a particular matter tending to show motive, bias and interest may result in a violation of the constitutional requirements [of the confrontation clause] of the sixth amendment....
''These sixth amendment rights, although substantial, do not suspend the rules of evidence .... A court is not required to admit all evidence presented by a defendant; nor is a court required to allow a defendant to engage in unrestricted cross-examination. ... Instead, [a] defendant is ... bound by the rules of evidence in presenting a defense .... Nevertheless, exclusionary rules of evidence cannot be applied mechanistically to deprive a defendant of his rights .... Thus, [i]f the proffered evidence is not relevant ... the defendant's right[s] to confrontation [and to present a defense are] not affected, and the evidence was properly excluded.'' (Citation omitted; internal quotation marks omitted.) State v. Holley , 327 Conn. 576, 593-94, 175 A.3d 514 (2018).
''It is axiomatic that [t]he trial court's ruling on the admissibility of evidence is entitled to great deference.... In this regard, the trial court is vested with wide discretion in determining the admissibility of evidence, including issues of relevance and the scope of cross-examination.... Accordingly, [t]he trial court's ruling on evidentiary matters will be overturned only upon a showing of a clear abuse of the court's discretion.... In determining whether there has been an abuse of discretion, every reasonable presumption should be *615made in favor of the correctness of the trial court's ruling, and we will upset that ruling only for a manifest abuse of discretion.'' (Internal quotation marks omitted.) State v. Calabrese , 279 Conn. 393, 406-407, 902 A.2d 1044 (2006).
Additionally, ''[our Supreme Court] has recognized that defendants may use evidence regarding the inadequacy of the investigation into the crime with which they are charged as a legitimate defense strategy.'' State v. Wright , 322 Conn. 270, 282, 140 A.3d 939 (2016), citing State v. Collins , 299 Conn. 567, 599-600, 10 A.3d 1005, cert. denied, 565 U.S. 908, 132 S.Ct. 314, 181 L. Ed. 2d 193 (2011).8 ''Conducting a thorough, professional investigation is not an element of the government's case.... A defendant may, however, rely upon relevant deficiencies or lapses in the police investigation to raise the specter of reasonable doubt, and the trial court violates his right to a fair trial by precluding the jury from considering evidence to that effect.'' (Internal quotation marks omitted.)
*900State v. Wright , supra, at 282, 140 A.3d 939.
''A defendant, however, does not have an unfettered right to elicit evidence regarding the adequacy of the police investigation. The reference in Collins to relevant deficiencies or lapses in the police investigation suggests that the defendant must do more than simply seek to establish that the police could have done more.... Even when such evidence has some probative value, the court must consider whether the probative *616weight of the ... evidence exceed[s] the risk of unfair prejudice to the [state] from diverting the jury's attention to collateral matters....
''All of these factors must be evaluated by the trial court in determining whether the particular inadequate investigation evidence should be admitted. That evaluation necessarily is framed by the theory of the proffering party. It is well settled that [t]he proffering party bears the burden of establishing the relevance of the offered testimony. Unless a proper foundation is established, the evidence is irrelevant.... Relevance may be established in one of three ways. First, the proffering party can make an offer of proof.... Second, the record can itself be adequate to establish the relevance of the proffered testimony.... Third, the proffering party can establish a proper foundation for the testimony by stating a good faith belief that there is an adequate factual basis for his or her inquiry.'' (Citations omitted; emphasis in original; internal quotation marks omitted.) Id., at 284-85, 140 A.3d 939 ; see also State v. Johnson , 171 Conn. App. 328, 349-50, 157 A.3d 120, cert. denied, 325 Conn. 911, 158 A.3d 322 (2017).
The defendant claims that he was precluded from presenting relevant evidence of an inadequate police investigation into the possible connection between the 222 Lenox Avenue burglary and the victim's death. According to him: ''It is clear the investigating police officers were aware that [he] believed that the same two individuals who robbed him earlier [had] committed the instant offense. This is not only a claim that the police could have done more, but rather ... [he] was precluded from making the claim that the police had actionable, definable information of other individuals involved and simply did not bother to follow up on those leads or develop the information further.'' (Emphasis added.)
*617We conclude that the trial court properly excluded the proffered testimony as irrelevant and, therefore, the defendant's constitutional claims fail.9 Under *901Wright , a defendant must demonstrate that further investigation reasonably may have led to additional evidence bearing on the defendant's guilt or innocence. See State v. Wright , supra, 322 Conn. at 284, 140 A.3d 939 (citing cases); see also id., at 287-88, 140 A.3d 939 (offer of proof deemed inadequate based on, inter alia, failure to elicit evidence demonstrating ''the possibility that adherence to such practices or procedures could have led to material evidence of the defendant's guilt or innocence''); Commonwealth v. Alcantara , 471 Mass. 550, 562, 31 N.E.3d 561 (2015) (proposed inadequate police investigation into sperm and drug evidence lacked probative value because there was no indication that murder victim engaged in sexual intercourse around time of attack, ''nor was there any evidence ... suggesting that the killing arose from a sexual relationship,'' and no evidence existed that ''the drugs or supplier of the drugs played any role in causing *618[the victim's] death'').10 The defendant, here, therefore bore the burden of demonstrating how a further, specific investigation into the possible connection between the Lenox Avenue burglary and the victim's death reasonably may have led to additional evidence bearing on his guilt or innocence. He failed to meet his burden.
Toth testified during the defendant's offer of proof that, beyond speaking with the defendant, he did not attempt to establish a connection between the victim's death and the Lenox Avenue burglary. Toth also testified, however, that the defendant told him the people involved in the Lenox Avenue burglary were not the same individuals allegedly involved in the Wood Avenue home invasion.11 See footnote 6 of this opinion. Additionally, although the court precluded the defendant, during cross-examination, from asking Toth whether he asked Rivera-Torres if the Lenox Avenue burglars had Caribbean accents, Pattis represented that he was uncertain what Toth's answer would be. The defendant did not revisit this question during his subsequent offer of proof.
Cotto similarly testified that the defendant informed him that he did not believe the Lenox Avenue burglars were involved with the victim's death. In fact, defense counsel conceded that he could not connect the two incidents. And the only eyewitness to the September 9, 2010 Lenox Avenue burglary, Rivera-Torres, testified during the offer of proof that she noticed a single, familiar looking'' [six]-foot-[two], 220, 240 pound African-American male coming out of [the Lenox Avenue] apartment.'' The defendant, on the other hand, claimed that *619two unidentified Jamaican or Haitian males, standing at either five feet, eight inches or five feet, seven inches in height, shot the victim on May 23, 2011.
We realize that courts have noted that evidence of an inadequate police investigation need not meet the strict standard of establishing a direct connection to potential third-party culprits. See, e.g., Commonwealth v. Silva-Santiago , 453 Mass. 782, 800-803, 906 N.E.2d 299 (2009). A defendant who attempts to elicit evidence regarding the adequacy of a police investigation, however, ''must do more than simply *902seek to establish that the police could have done more.'' State v. Wright , supra, 322 Conn. at 284, 140 A.3d 939.
Here, the defendant's multiple offers of proof and the record fail to indicate how a further, specific investigation into the possible connection between the burglary and the victim's death may have led to additional evidence bearing on his guilt or innocence. See id., at 284, 288, 140 A.3d 939 ; see also Commonwealth v. Alcantara , supra, 471 Mass. at 562, 31 N.E.3d 561. The two incidents were separated by approximately eight months and allegedly involving individuals with distinct characteristics, and their only alleged connection was that they both took place at the shared residences of the defendant and the victim.12 Simply put, the defendant's proffer asked the court to engage in substantial speculation as to both the possible connection between the two incidents and how a further police investigation into that connection might have produced additional evidence bearing on the defendant's guilt or innocence.13 Under these circumstances, *620we agree that the defendant failed to establish the relevance of the proffered evidence. Furthermore, to the extent that this evidence had any probative value at all, we conclude that admitting it also could have diverted the jury's attention to a collateral matter, namely, speculation about a theorized connection between the unsolved Lenox Avenue burglary and the victim's death that allegedly involved unknown assailants. See State v. Wright , supra, at 284, 140 A.3d 939 (court must consider whether probative weight of evidence outweighed by unfair prejudice to state). Accordingly, the trial court properly excluded the evidence as irrelevant and, therefore, the defendant's claims fail.14
II
The defendant's final claim is that the trial court abused its discretion in admitting into evidence testimony regarding *903the victim's relationship with the defendant prior to her death. More specifically, he argues that the trial court improperly admitted hearsay testimony from Ariana Paneto and Kaila Oquendo under the state *621of mind exception to the hearsay rule. We decline to review this claim.
''It is well settled that, absent structural error, the mere fact that a trial court rendered an improper ruling does not entitle the party challenging that ruling to obtain a new trial. An improper ruling must also be harmful to justify such relief. ... The harmfulness of an improper ruling is material irrespective of whether the ruling is subject to review under an abuse of discretion standard or a plenary review standard. ... When the ruling at issue is not of constitutional dimensions, the party challenging the ruling bears the burden of proving harm.'' (Internal quotation marks omitted.) State v. Myers , 178 Conn. App. 102, 105-106, 174 A.3d 197 (2017).
''We do not reach the merits of [a] claim [where] the defendant has not briefed how he was harmed by the allegedly improper evidentiary ruling.'' (Internal quotation marks omitted.) State v. Toro , 172 Conn. App. 810, 817, 162 A.3d 63, cert. denied, 327 Conn. 905, 170 A.3d 2 (2017) ; see also State v. Myers , supra, 178 Conn. App. at 108, 174 A.3d 197 (''there must be some analysis of how the defendant was harmed from the claimed error given the other evidence before the jury''). ''It is also a well established principle that arguments cannot be raised for the first time in a reply brief.'' (Internal quotation marks omitted.) State v. Myers , supra, at 106, 174 A.3d 197.
The defendant, in his principal brief, does not address the harmfulness of the allegedly improper evidentiary rulings regarding the testimonies of Paneto and Oquendo. He argues that their testimonies did not relate to the victim's fear of the defendant and that ''there is no corroborating evidence of the victim's statements ... [and, therefore], the statements admitted in this case are far more prejudicial than probative, and should not have been before the jury.'' (Emphasis in original.) He maintains that he was ''irreparably prejudiced'' by *622the court's rulings and that the uncorroborated statements ''require[d] an impermissible inference concerning [his] motive.''
Even if we assume, without deciding, that the defendant is correct-the statements were improperly admitted and this also permitted the jury to make an impermissible inference concerning his motive-he does not analyze ''how [he] was harmed from the claimed error given the other evidence before the jury.'' State v. Myers , supra, 178 Conn. App. at 108, 174 A.3d 197. Nor does he address any of the relevant factors that courts consider when assessing harmlessness. See, e.g., State v. Toro , supra, 172 Conn. App. at 817, 162 A.3d 63 ; State v. Johnson , supra, 171 Conn. App. at 338, 157 A.3d 120. Simply put, he fails to argue how the testimonies of Paneto and Oquendo, along with any impermissible inference potentially drawn from them, substantially affected the verdict. See State v. Toro , supra, at 817, 162 A.3d 63 (''a nonconstitutional error is harmless when an appellate court has a fair assurance that the error did not substantially affect the verdict'' [internal quotation marks omitted] ). Accordingly, we decline to review his evidentiary claim.
The judgment is affirmed.
In this opinion the other judges concurred.